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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
Jo
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Me
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a Z
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ico
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
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
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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
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
José Mezquita Zapico 3
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.
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
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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
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
José Mezquita Zapico 5
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.
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
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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.
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
José Mezquita Zapico 7
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)
La formulación del modelo es:
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.
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
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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)
La formulación del modelo es:
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)
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José Mezquita Zapico 9
La formulación del modelo es:
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
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
10 Escuela Técnica Superior de Ingenieros Industriales (UPM)
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
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
José Mezquita Zapico 11
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,
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
12 Escuela Técnica Superior de Ingenieros Industriales (UPM)
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.
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
José Mezquita Zapico 13
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.
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
14 Escuela Técnica Superior de Ingenieros Industriales (UPM)
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.
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
José Mezquita Zapico 15
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).
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
16 Escuela Técnica Superior de Ingenieros Industriales (UPM)
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
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
José Mezquita Zapico 17
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.
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
18 Escuela Técnica Superior de Ingenieros Industriales (UPM)
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).
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
José Mezquita Zapico 19
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.
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
20 Escuela Técnica Superior de Ingenieros Industriales (UPM)
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.
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
José Mezquita Zapico 21
Comparación de los modelos del problema de aprovisionamiento conjunto con demanda dinámica
22 Escuela Técnica Superior de Ingenieros Industriales (UPM)
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
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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:
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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:
Comparison of the joint replenishment problem approaches with dynamic
demand 6
∑ 𝑥𝑘𝑖𝑗
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:
∑ 𝑥𝑘𝑖𝑗
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.
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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)
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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).
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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)
Comparison of the joint replenishment problem approaches with dynamic
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:
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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)
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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).
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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
Products Periods TBO MSR
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:
Comparison of the joint replenishment problem approaches with dynamic
demand 26
Table 8: Exponential regression analysis for the BM computational time
Regression statistics
Multiple R 0.927
R Square 0.860
Adjusted R Square 0.859
Standard error 0.694
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
Comparison of the joint replenishment problem approaches with dynamic
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
Products Periods TBO MSR
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
Regression statistics
Multiple R 0.899
R Square 0.809
Adjusted R Square 0.809
Standard error 0.662
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
error t Stat p-value Lower 95%
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
Comparison of the joint replenishment problem approaches with dynamic
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
Products Periods TBO MSR
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
Comparison of the joint replenishment problem approaches with dynamic
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.
Regression statistics
Multiple R 0.958
R Square 0.917
Adjusted R Square 0.917
Standard error 0.486
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
error t Stat p-value
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
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
demand 37
constraints, the more calculations are in each node of the B&B tree what increases
the memory required.
Comparison of the joint replenishment problem approaches with dynamic
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).
Comparison of the joint replenishment problem approaches with dynamic
demand 39
Reference List
Arkin, E., Joneja, D., & Roundy, R. (1989). Computational complexity of
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.
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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
Comparison of the joint replenishment problem approaches with dynamic
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 BM-Modified R&G-Modified
ERF-Modified
Comparison of the joint replenishment problem approaches with dynamic
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 BM-Modified R&G-Modified
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 BM-Modified R&G-Modified
ERF-Modified
Comparison of the joint replenishment problem approaches with dynamic
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 BM-Modified R&G-Modified
ERF-Modified
Comparison of the joint replenishment problem approaches with dynamic
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
Regression statistics
Multiple R 0.929
R Square 0.863
Adjusted R
Square 0.863
Standard error 0.663
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
error t Stat p-value
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
Regression statistics
Multiple R 0.905
R Square 0.819
Adjusted R Square 0.818
Standard error 0.659
Observations 2561
ANOVA
Degrees
of
freedom
Sum of
squares
Mean of
the
squares F
Significance
F
Comparison of the joint replenishment problem approaches with dynamic
demand 46
Regression
statistics 6 5004.54 834.09 1922.35 0
Residual 2554 1108.15 0.43
Total 2560 6112.70
Coefficients
Standard
error t Stat p-value
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
Regression statistics
Multiple R 0.950
R Square 0.902
Adjusted R
Square 0.902
Standard error 0.527
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
error t Stat p-value
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
Comparison of the joint replenishment problem approaches with dynamic
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.
Comparison of the joint replenishment problem approaches with dynamic
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
Regression statistics
Multiple R 0.932
R Square 0.868
Adjusted R
Square 0.863
Standard error 0.652
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
error t Stat p-value Lower 95%
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
Regression statistics
Multiple R 0.920
R Square 0.847
Adjusted R Square 0.840
Standard error 0.723
Comparison of the joint replenishment problem approaches with dynamic
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
error t Stat p-value
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
Comparison of the joint replenishment problem approaches with dynamic
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
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München, den 20.07.2016
José Mezquita Zapico