Bayesian Forecasts for Dynamic Demand of Finished

Revista Colombiana de Estadística Edición para autor(es)

Bayesian Forecasts for Dynamic Demand of Finished Products Pronósticos Bayesianos para Demanda Dinámica de Productos Terminados.

1, a

Marisol Valencia Cárdenas

1, b

, Juan Carlos Correa Morales

ABSTRACT: Bayesian processes are very usefull in industry applications, in special, in predictions related to the search of optimal plans for production and inventories. When there are few historical data to predict, the user is faced with possibilities which could lead to imprecise estimation, obtaining lack of stock or the opposite, an excess of inventories, both carrying bad consequences for the industry. We show a bayesian technique to predict demand of nished products, using few historical data, that can facilitate the temporal change that industry face, in order to be better prepared to drastic changes, showing comparisons with other known models, according to the indicators MAPE and RECM. KEYWORDS: Bayesian prediction, Demand forecasting, Prior distribution. RESUMEN: Los procesos bayesianos son de mucha utilidad para aplicaciones industriales, en especial, en predicciones relacionadas con la búsqueda de planes óptimos para producción e inventarios. cuando hay pocos datos históricos para realizar predicciones, el usuario se enfrenta con posiblidades que pueden llevar a estimaciones imprecisas, con lo cual se obtendrá una falta de inventario, o lo opuesto, un exceso, ambos llevando a malas consecuencias para la industria. Aquí mostramos una técnica bayesiana para elaborar predicción de demanda de producto terminado, en presencia de pocos datos, que puede facilitar el cambio temporal al que se enfrenta la industria, para prepararse mejor al cambio drástico, mostrando comparaciones, con otros modelos conocidos, acordes con los indicadores: MAPE y RECM. PALABRAS CLAVE: Predicción Bayesiana, Pronóstico de Demanda, Distribución a priori. 1. INTRODUCTION

Uncertainty in systems that represent the behavior of some industry processes are produced by changes in time, that generate variability in an unexpected form over some variables, for example, in inventory models (Gutiérrez & Vidal 2008, Simchi-Levi et al. 2008). Often it is necessary to quantify it in order to make decisions, using appropiate statistical tools (Rocquigny 2012). Some causes a Ms. en estadística. E-mail: [email protected] b Professor. E-mail: [email protected]

1

Marisol Valencia Cárdenas & Juan Carlos Correa Morales

2

of uncertainty could be: casual events, vulnerability of systems, control factors, and time, which is the principal reason to the dynamic variation, but also, characteristics on this could involve a time series on inputs, outputs and events of those systems (Rocquigny 2012). In these kind of processes, it is important to know that variables like demand have many forms to be predicted, and this can inuence the way to plan logistics (Gutiérrez & Vidal 2008, Simchi-Levi et al. 2008, Sarimveis et al. 2008). In general, an easy relation can be established as a balance at every period t − th, measured with the equation: It = It−1 + xt − Dt , where xt is the order quantity and Dt is the demand rate at t − th period. Manufacturing processes could be represented by models with xed factors, but if time is implicit, they are not very robust, because they do not incorporate uctuations on internal logistic, for example, variations in demand, causing probably the bullwhip eect, which can indicate strong variations due to drastic changes or the assumption of independence of demand in relation to other variables of these processes; or changes on prices, costs or other variables like the times until materials arrive (Bes & Sethi 1988, Gutiérrez & Vidal 2008, Sarimveis et al. 2008, Shoesmith & Pinder 2001). Classical models designed to characterize time series or to do forecasts like ARIMA, or exponential smoothing, require many data, or assume some hard theoretical structures which are not fullled, and make them non trust-able. These kind of reasons have obligated to many researchers to use other models, like those with bayesian techniques (Bolstad 1986, Zellner 1996, Kingdom 2004, Makridakis et al. 2011), leading to structures with equations like regression, or ARIMA, but using statistical distributions and the Bayes theorem, to estimate predictions. In this work, we show a characterization of conditions, that dene predictive bayesian distributions, in order to do forecasts associated to regression and Dynamic Bayesian Models, and these are applied to demand forecasting of four products of a confection industry. Then we compare the exposed models, estimating two error indicators: MAPE (Mean of the Absolute Percentage Error) and RMSE (Root of the Mean Square Error), in tted and forecasted values. We propose an innovation of a prior parameter for every time t to be forecasted, that, leads to renovate the posterior every time t, with expert knowledge.

2. Model proposed for prediction

The general regression model has many kind of variations, but it can be presented in a general form to understand the general mathematical process (1).

yt = β0 + β1 x1 + ... + βk xk + ϵt (1) Here, ϵt ∼ N (0, σ 2 ). In order to understand the theory of the bayesian process, we follow some steps, according to Zellner (1996): 1. Dene Likelihood function for vector Y , Revista Colombiana de Estadística, Edición para autor(es)

Bayesian Forecasts for Dynamic Demand of Finished Products

3

that contains all the information of the serie: Y = (y1 , y2 ....., yT ) and Y−1 = (y0 , y1 ....., yT −1 ), a vector of lag 1, and we use the matricial form of the general model: Y = Xβ . 2. Dene Prior distribution: informative or non-informative. 3. Derive the posterior distribution as the product of prior distribution and likelihood function. Using a non informative prior distribution: 1/σ shown in Zellner (1996), pg.73, nal predictive bayesian distribution is next one (Zellner 1996) 1 f (yt+1 |yt ) ∝ [ ](ν+q)/2 ˆ ′ H(Y+1 − X+1 β) ˆ ν + (Y+1 − X+1 β)

Where βˆ = (X ′ X)−1 X ′ Y , and H = s12 (I − X+1 M X+1 ) (Zellner, 1996). besides:ν = T − p, p:Number of parameters, q=Number of forecasting position, so for yT +1 , with this model, it will be: (ν + 1)/2 = (T − 2 + 1)/2 = (T − 1)/2, and for yT +2 , (ν + 2)/2 = (T − 2 + 2)/2 = T /2

• Likelihood function Assuming we know the value of y0 , and if we collect all the observations in a vector: Y = (y0 , y1 , y2 ....yT ), and with the expression Xβ = λY−1 + αXt : ′

L(y|y0 , β) ∝ σ −T exp[− 2σ2 (Y −Xβ) (Y −Xβ)] 1

• Bayesian process-with normal Prior Supposing a Normal distribution as the prior for β and σ1 parameters, N (β, σ1 ), with σ1 = σ0 σ [

1 ξ(β, σ) ∝ exp σ1 [

1 ξ(β, σ) ∝ exp σ0 σ

−

1 2 2σ1

(β−β0 )′ (β−β0 )

]

] − 2(σ 1σ)2 (β−β0 )′ (β−β0 ) 0

• Posterior distribution With the product between these two, the posterior distribution can be expressed as: And changing: τ =

1 σ2 ,

τ0 =

1 σ02

1/2

ξ(β, τ |τ0 , β0 , y0 , yt ) ∝ τ (T +1)/2 τ0

ξ(β, τ |τ0 , β0 , y0 , yt ) ∝ τ

T +1 2

1

τ

exp[− 2 (Y −Xβ)

τ

τ02 exp− 2 [(Y −Xβ)

′

′

(Y −Xβ)− 20 (β−β0 )′ (β−β0 )] ττ

(Y −Xβ)+τ0 (β−β0 )′ (β−β0 )]

After an algebra process of the exponent, the posterior distribution can be expressed as next equation. Revista Colombiana de Estadística, Edición para autor(es)


4

ξ(β, τ |τ0 , β0 , Y0 , Y ) ∝ τ

T +1 2

˜

′

−1

exp− 2 A[(β−β) A τ

˜ (X ′ X+τ0 )(β−β)+1 ]

˜ (X ′ X + τ0 )−1 )Ga( T +3 , Expo) Which is a Normal-Gamma: N (β, 2 Where β˜ = (X ′ X + τ0 )−1 (X ′ X βˆ + τ0 β0 ), if A = β0′ τ0 β0 + Y ′ Y [ ] ˜ ′ A−1 (X ′ X + τ0 )(β − β) ˜ + 1 /2 Expo: A (β − β)

• Predictive Distribution Where Y+ , is the h dimensional vector to be predicted, and X+ is the design matrix with hxp dimension. Here is also necessary the normal distribution of the original data to calculate predictive distribution: f (Y+ |β, τ ) = Nh (X+ β, τ I). If S = Y − Xβ and S+ = Y+ − X+ β .

∫ ∫ f (Y+ |y0 , Y ) =

1

∫ ∫

f (Y+ |y0 , Y ) ∝ τ02

τ

1 2

f (Y+ |Y0 , Y ) ∝ τ0

1

f (Y+ |Y0 , Y ) ∝ τ02

∫

ℜ

f (Y+ |β, τ )ξ(β, τ |Y0 , Y )dτ dβ

0

T +2 2

∫ ∫

∞

exp− 2 [(S τ

Γ

[ T +4 ]

D

2 T +4 2

′

′ S)+(S+ S+ )+τ0 (β−β0 )′ (β−β0 )]

dτ dβ

T +4

∗

T +2 τ D 2 [ ] τ 2 exp− 2 [D] dτ dβ Γ T +4 2

[ ′ ]− T +4 ′ 2 (S S) + (S+ S+ ) + τ0 (β − β0 )′ (β − β0 ) dβ

Expanding the expression of the parenthesis: ′ ′ D = Y ′ Y −2βX ′ Y +β ′ X ′ Xβ+Y+′ Y+ −2βX+ Y+ +β ′ X+ X+ β+β ′ τ0 β−2β ′ β0 τ0 +β0′ τ0 β0

′ ′ D = β ′ (X+ X+ +X ′ X+τ0 )β−2β ′ (X ′ Y +X+ Y+ +τ0 β0 )+Y ′ Y +Y+′ Y+ +β0′ τ0 β0

[ ] ′ D = β ′ β − 2β ′ M −1 (X ′ Y + X+ Y+ + τ0 β0 ) + M −1 (Y ′ Y + Y+′ Y+ + β0′ τ0 β0 ) M ′ ′ Where M −1 = (X+ X+ + X ′ X + τ0 )−1 . Let βn = M −1 (X ′ Y + X+ Y+ + β0 τ0 )

D = (β − βn )′ M (β − βn ) − βn′ M βn + Y ′ Y + Y+′ Y+ + β0′ τ0 β0 Revista Colombiana de Estadística, Edición para autor(es)

Bayesian Forecasts for Dynamic Demand of Finished Products 1 2

f (Y+ |Y0 , Y ) ∝ τ0

∫

[

(β − βn )′ M (β − βn ) + Y ′ Y + Y+′ Y+ + β0′ τ0 β0 − βn′ M βn

5

]− T +4 2

dβ

Let A = Y ′ Y + Y+′ Y+ + β0′ τ0 β0 − βn′ M βn After solving: 1

∫

f (Y+ |Y0 , Y ) ∝ τ02

[

]− T +4 2 dβ (β − βn )′ M (β − βn ) + A

Integral which is a Student T. 1

f (Y+ |y0 , Y ) ∝ τ02

∫

[ ( )]− T +4 2 A (β − βn )′ A−1 M (β − βn ) + 1 dβ

1 [ ]− T +4 2 f (Y+ |y0 , Y ) ∝ τ02 Y ′ Y + Y+′ Y+ + β0′ τ0 β0 − βn′ M βn

Final expression is also a Student T, but we can change the internal terms, to get a function of Y+ . Expanding expression in the parenthesis: I = Y+′ Y+ − βn′ M βn + Y ′ Y + β0′ τ0 β0 ′ ′ M −1 = (X+ X+ + X ′ X + τ0 )−1 and βn = M −1 (X ′ Y + X+ Y+ + β0 τ0 )

′ ′ I = Y+′ Y+ −M −1 (X ′ Y +X+ Y+ +β0 τ0 )M M −1 (X ′ Y +X+ Y+ +β0 τ0 )+Y ′ Y +β0′ τ0 β0

′ ′ I = Y+′ Y+ + Y ′ Y + β0′ τ0 β0 − M −1 ((X ′ Y )′ (X ′ Y ) + (X+ Y+ )′ (X+ Y+ ) + (β0 τ0 )′ β0 τ0

′ ′ +2(X ′ Y )′ (X+ Y+ ) + 2(X ′ Y )′ (β0 τ0 ) + 2(X+ Y+ )′ (β0 τ0 ))

I = Y+′ (I−X+ M −1 X+ )Y+ −2Y+ X+ M −1 (X ′ Y +β0 τ0 )+β0′ τ0 β0 −M −1 (Y X ′ X ′ Y +τ0 β0′ β0 τ0 +2X ′ Y β0 τ0 )+Y ′ Y

[ ] ′ ′ −1 ′ ′ I = Y+′ Y+ − 2Y+ X+ (I − M −1 X+ X+ ) M −1 (X ′ Y + β0 τ0 ) (1 − M −1 X+ X+ )+ +(β0′ τ0 β0 − M −1 τ0 β0′ β0 τ0 ) + (Y ′ (I − M −1 X ′ X)Y − 2M −1 X ′ Y β0 τ0 ) [ ] ′ ′ I = (Y+ − Yn )′ (Y+ − Yn ) − Yn′ Yn (1 − M −1 X+ X+ ) +(β0′ τ0 β0 −M −1 τ0 β0′ β0 τ0 )+(Y ′ Y −2M −1 (I−M −1 X ′ X)−1 X ′ Y β0 τ0 )(I−M −1 X ′ X) [ ] ′ ′ I = (Y+ − Yn )′ (Y+ − Yn ) − Yn′ Yn (1 − M −1 X+ X+ ) Revista Colombiana de Estadística, Edición para autor(es)

6

Marisol Valencia Cárdenas & Juan Carlos Correa Morales [ ] ′ +(β0′ τ0 β0 − M −1 τ0 β0′ β0 τ0 ) + (Y − Ym )′ (Y − Ym ) − Ym Ym (I − M −1 X ′ X)

Where: Yn = (I − X+ M −1 X+ )−1 X+ M −1 (X ′ Y + β0 τ0 ), using a denition of inverted dierence in matrices (Zellner 1996).

Yn = (I + X+ (X ′ X)−1 X+ )M −1 X+ (X ′ Y + β0 τ0 ) ′ ′ So, Yn = (I + (X ′ X)−1 X+ X+ )(X ′ X + τ + X+ X+ )−1 X+ (X ′ Y + β0 τ0 ) = (I + ′ −1 ′ ′ −1 ′ −1 (X X) X+ X+ )(I + (X X + τ0 ) X+ X+ ) X+ (X ′ X + τ0 )−1 (X ′ Y + β0 τ0 )

τ0 is a very small quantity, so, Yn = (X ′ X + τ0 )−1 X+ (X ′ Y + β0 τ0 ) = X+ β˜, and Ym = M −1 (I − M −1 X ′ X)−1 X ′ β0 τ0 . Some terms disappear due to the proportionality, and set A = (Y −Ym )′ (Y − Ym ), simplifying the nal predictive distribution. [ ] [ ] ′ ′ ′ )+ (Y − Ym )′ (Y − Ym ) − Ym Ym (I−M −1 X ′ X) X+ I = (Y+ − Yn )′ (Y+ − Yn ) − Yn′ Yn (1−M −1 X+ [ ] I = (Y+ − Yn )′ (Y+ − Yn ) + (Y − Ym )′ (Y − Ym ) f (Y+ |Y0 , Y ) ∝

[[

]]− T +4 2 (Y+ − Yn )′ (Y+ − Yn ) + A

[ ]− T +4 2 f (Y+ |Y0 , Y ) ∝ ((Y+ − Yn )′ A−1 (Y+ − Yn ) + 1)A

Which is a Student T, with mean: Yn = X+ β˜, with ν degrees of freedom, ν ν A= ν−2 (Y − Ym )′ (Y − Ym ), a kind similar to the result of and variance, ν−2 Zellner (1996), with non informative prior. The equation of the regression model is determinant for the design matrix, but the past mean of the predictive distribution Yn will determine the random value of the T distribution, every time to be predicted. 2.1. Comparison among models

A, B, C and D represent products in confection sector, and we compared tted and forecasted values of sales for the models expressed in tables 1 and 2, between next models:

• Bayesian predictive distribution with Normal prior distribution • Dynamic Linear Model with Kalman Filter • Classical models: ARIMA and exponential smoothing Revista Colombiana de Estadística, Edición para autor(es)


7

2.2. Dynamic Linear Systems

As the same word means, dynamic involves movement, which reects changes for some variable with the pass of time, as the unique motivator for those changes (Sarimveis et al. 2008). This implies that systems have implicitly some changes that produce uncertainty, caused by the time. A Dynamic Linear System is a general mechanism for univariate and multivariate systems. Specically, in statistical areas, the standard form of such dynamics involves one step and linear dependence, and involves a parameter, like θ. Some authors have worked on these processes (Meinhold & Singpurwalla 1983, ?) in order to generate estimations related to the probability distributions they involve. According to (Meinhold & Singpurwalla 1983) next form is the general representation of a Dynamic Linear Model (DLM).

• Observation equation: Yt = Ft′ θt−1 + νt . Where νt ∼ N (0, Vt ) • System equation θt = Gt θt−1 + wt . Where wt ∼ N (0, Wt ) Here, Yt is a time series, function of random variable θt , which is the state of nature of t − th time. Variable νt is not directly involved with the observation, but wt with the system, so variances are not directly used for prior estimations. Ft y Gt are known matrices, and Gt is evolution matrix. In t − 1: (θt−1 |Yt−1 ) ∼ N (θˆt−1 , Σt ) Where θˆt−1 and Σt are initial mean and variance. In this case, Ft′ matrix do not control over the serie, and Gt describes a relation among time t − 1 and t for parameter θt . Other models can also incorporate more terms, like:

Yt = Ft′ θt−1 + Bt βt + νt The DLM proposed here was presented at (Valencia & Correa 2013), and will be compared to the results the other models. 3. Methodology

We propose to compare bayesian results, according to these variations:

• Two prior distributions: non informative, and informative: Normal Distribution. • Normal distribution will have variation in the parameter β0 , that will permit dierent prior information for every period t. • Models with dierent number , ( ) of covariables: yt = λ0 + λyt−1 (+2πtαtime ) yt = λ0 + λyt−1 + αSin 2πt , y = λ + λy + α time + α Sin t 0 t−1 1 2 12 12 Revista Colombiana de Estadística, Edición para autor(es)


8

And these results with the estimations of classical models: ARIMA, Exponential Smoothing; and a Dynamic Linear Model with Kalman Filter (Meinhold & Singpurwalla 1983, Valencia & Correa 2013). We also propose to do a variation, changing the value of the parameter β0 for every time t, this implication will lead to do variations in prior distribution for every period, and also, dierent forecast. An expert can give a percentage of change in order to produce this dierence every period. The process to nd the percentage for β0 at every time t, will be argued by a questionnaire. It consist on getting information about the changes in percentages, that will change values of such parameter. The questions to the expert were:

• In the month XX, do you think that sales of product YY will be: Reduced ( ) Augmented ( ) Equal ( ) • For the month XX, for that product, what is the value of the percentage of: decreasing ( ) increasing ( ) 4. Results

• Graphical results shown here, are associated to the use of the Normal prior distribution for the parameter β . • First pair of graphs will show tted values for sales of the product A, with prior values for the parameters: β0 and τ0 . • Second pair, shows forecasts estimated for this tting. • Then, a table is shown comparing error indicators of models with: diuse and Normal prior, dierent co-variables. 4.1. Forecasts

Graphics 1 and 2 present tted and forecasted values for sales of product A, for years 2010 and 2011. For the bayesian model, we used here an initial vector of parameters β0 and τ0 . Values of τ0 and β0 aect the forecasts, so, there must be adequate prior information to be more precise. ( • Model 1: y = β0 + β1 yt−1 + β2 sin 2πt 2 ) . MAPE of tted values: 10.8, and forecasted: 14.17. RECM of tted values: 17.1, and forecasted: 18.637. ( • Model 2: y = β0 + β1 yt−1 + β2 t + β3 sin 2πt 2 ) . MAPE of tted values:9.3 and forecasted:69. RECM of tted values: 14.6, and forecasted: 81.29. The second model does not t well in forecast values, we can see very similar results for all the products. Revista Colombiana de Estadística, Edición para autor(es)


9

Revista Colombiana de Estadística, Edición para autor(es)


10

4.2. Models with informative distribution

In table 1, rst two lines correspond to model with prior normal distribution for the parameter vector β ; using a vector of initial values for β0i =(11.6,50.3,50.95,51.2).

A B C D

A B C D

A B C D

ARIMA MAPE RECM Fitted Forecast Fitted Forecast 11,6% 21,4% 18,7 43,6 26,3% 29,1% 19,73 41,9 17,5% 54,5% 13,5 46,8 29,9% 17,2% 19,9 18,6 Exponential Smoothing MAPE RECM Fitted Forecast Fitted Forecast 12,1% 22,0% 18,6 44,8 25,9% 28,1% 19,3 40,9 15,1% 17,4% 12,3 18,3 23,3% 25,0% 19,5 19,2 DLM-KF MAPE RECM Fitted Forecasted Fitted Forecast 11,6% 17,6% 14,8 36,1 23,6% 33,3% 28,3 42,1 36,3% 48,3% 30,5 45,2 33,4% 25,0% 22,0 20,9

Table 2 shows results of the proposed model with the predictive bayesian model and the variations of the parameters. There we can see that model with t variable does not t well.



A B C D

11

β0 + β1 Yt−1 + β1 t + β3 Sin MAPE RECM Fitted Forecast Fitted Forecast 9,61% 37,0% 13,09 37,86 18,16% 135% 13,60 94,60 11,6% 54,2% 9,18 33,85 16,7% 101,3% 12,80 55,64

According to the variations of the parameter τ0 , the results are better in some instances, and the behavior is dierent according to the products, for example, for product A, the result is better if precision parameter is 0, 0001, for product C, if τ0 = 1, and for D, if τ0 = 2.

A B C D A B C D A B C D

Change for β0 in t No change for β0 MAPE RECM MAPE RECM Fitted Forecast Fitted Forecast Fitted Forecast Fitted Forecast β0 + β1 Yt−1 + β3 Sin τ0 = 0, 0001 10,8% 14,17% 17,08 18,64 10,8% 14,17% 17,08 18,64 27,4% 26,3% 19,49 18,51 26,6% 26,3% 19,28 18,52 17,6% 30,6% 14,28 18,97 12,6% 30,6% 9,62 18,97 33,8% 35,5% 23,28 20,46 21,9% 35,5% 16,67 20,47 τ0 = 1 11,8% 17,4% 19,69 25,09 11,5% 14,73% 17,72 18,94 26,9% 21,0% 19,78 20,4 26,6% 19,81% 20,28 19,41 22,9% 16,1% 16,69 13,08 17,3% 16,96% 12,17 12,94 37,0% 24,4% 24,22 14,33 25,5% 22,49% 18,02 13,80 τ0 = 2 11,1% 16,1% 19,17 25,49 16,2% 17,9% 23,46 22,57 27,6% 24,7% 19,69 17,57 30,3% 19,9% 21,69 19,95 28,6% 22,9% 19,89 17,36 16,5% 17,0% 13,98 12,88 38,6% 19,6% 24,8 12,83 38,4% 26,2% 26,61 14,83

5. Discussion

The DLM only showed acceptable results for the product A, but compared with the rest of models, there were bad results using variable t, but better for models using a lag and a sinusoidal variable of the regression bayesian model; also, with the innovation in the parameters of the vector β0i , but there is not a unique result for τ0 parameter which be adequate.


12


References

Bes, C. & Sethi, S. (1988), `Concepts of forecast and decision horizons: Applications to dynamic stochastic optimization problems', Mathematics of Operations Research 13(2), 295310. Bolstad, W. M. (1986), `Harrison-Stevens Forecasting and the Multiprocess Dynamic Linear Model', The American Statistician 40(2), 129135. Gutiérrez, V. & Vidal, C. (2008), `Modelos de Gestión de Inventarios en Cadenas de Abastecimiento: Revisión de la Literatura', Revista Facultad de Ingeniería de la Universidad .. . pp. 134149. Kingdom, U. (2004), `Forecast Calibration and Combination : A Simple Bayesian Approach for ENSO', (1977), 15041516. Makridakis, S., Hibon, M., Moser, C., Journal, S., Statistical, R. & Series, S. (2011), Àccuracy of Forecasting : An Empirical Investigation of Forecasting : An Empirical Accuracy Investigation', Journal of the Royal Statistical Society 142(2), 97145. Meinhold, R. J. & Singpurwalla, N. D. (1983), Ùnderstanding the Kalman Filter', The American Statistician 37(2), 123127. Rocquigny, E. (2012), Modelling Probability and Statistics.

Under Risk and Uncertainty,

Wiley Series in

Sarimveis, H., Patrinos, P., Tarantilis, C. D. & Kiranoudis, C. T. (2008), `Dynamic modeling and control of supply chain systems: A review', Computers & Operations Research 35(11), 35303561. *http://linkinghub.elsevier.com/retrieve/pii/S0305054807000366 Shoesmith, G. & Pinder, J. (2001), `Potential Inventory Cost Reductions Using Advanced Time Series Forecasting Techniques', Journal of the Operational Research Society 52(11), 12671275. Simchi-Levi, D., Kaminski, P. & Simchi-Levi, E. (2008), the Supply Chair, 3 edn, McGraw-Hill.

Designing and Managing

Valencia, M. & Correa, J. C. (2013), Ùn modelo dinámico bayesiano para pronóstico de energía diaria', Revista Ingeniería Industrial 12(2), 717. Zellner, A. (1996), An introduction edn, Wiley Classics Library.

to bayesian inference in econometrics, second
