**1. Introduction**

184 Stochastic Modeling and Control

1982.

2010.

communication, 1991.

signals // Radio systems, № 81, 2004.

[3] Pervachev S.V., Perov A.I. Adaptive filtration of messages. - Moscow, Radio and

[4] Sejdzh E.P., White C.S. Optimum control of systems. М: Radio and communication,

[5] V.V. Khutortsev, I.V. Baranov. The optimization of observations control in problem of discrete searching for Poisson model of the observed objects flow // Radiotechnika, №3,

[6] Harisov V. N., Anikin A.L. Observations optimal control in problem of detection of

A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. From the very beginning of the application of optimization to these problems, it was recognized that analysts of natural and technological systems are almost always confronted with uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very large-scale optimization models. Decision-making under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multi-period or multi-stage setting.

Approaches to optimization under uncertainty have followed a variety of modeling philosophies, including expectation minimization, minimization of deviations from goals, minimization of maximum costs, and optimization over soft constraints. The main approaches to optimization under uncertainty are stochastic programming (recourse models, robust stochastic programming, and probabilistic models), fuzzy programming (flexible and possibilistic programming), and stochastic dynamic programming.

This paper is devoted to improvement of statistical decisions in revenue management systems. Revenue optimization – or revenue management as it is also called – is a relatively new field currently receiving much attention of researchers and practitioners. It focuses on how a firm should set and update pricing and product availability decisions across its various selling channels in order to maximize its profitability. The most familiar example probably comes from the airline industry, where tickets for the same flight may be sold at

© 2012 Nechval and Purgailis, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Nechval and Purgailis, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

many different fares throughout the booking horizon depending on product restrictions as well as the remaining time until departure and the number of unsold seats. Since the tickets for a flight have to be sold before the plane takes off, the product is perishable and cannot be stored for future use. The use of the above strategies has transformed the transportation and hospitality industries, and has become increasingly important in retail, telecommunications, entertainment, financial services, health care and manufacturing. In parallel, pricing and revenue optimization has become a rapidly expanding practice in consulting services, and a growing area of software and IT development, where the revenue optimization systems are tightly integrated in the existing Supply Chain Management solutions.

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 187

<sup>θ</sup> 1 kl <sup>θ</sup> 1 k <sup>θ</sup> l k f (x , ..., x , x ) f (x , ..., x )g (x |x ), (1)

θ i θ k

j 0 θ k θ k

j 0 θ k θ k (m k)! m l F (x ) F (x ) f (x ) ( 1) (l k 1)!(m l)! <sup>j</sup> 1 F (x ) 1 F (x ) (3)

m k

m! f (x )[1 F (x )] , (m k)! (2)

θ k θ k θ k

j θ l θ l

j θ l θ k θ l

θ i θ l θ k θ l θ l

f (x , ..., x , x ) f (x |x , ..., x ) <sup>g</sup> (x |x ), f (x , ..., x ) (5)

lk1 m l

lk1 m l θ l θ k θ l θ k θ l

situation, the problem is to predict future events in a sample or process based on early data from that sample or process. For the new-within-sample prediction situation, the problem is to predict future events in a sample or process based on early data from that sample or process as well as on a past data sample from the same process or population. Some mathematical

**2. Mathematical preliminaries for the within-sample prediction situation** 

*Theorem 1.* Let X1 ... Xk be the first k ordered observations (order statistics) in a sample of size m from a continuous distribution with some probability density function f (x) and distribution function F (x), where is a parameter (in general, vector). Then the joint probability density function of X1 ... Xk and the *l*th order statistics Xl (1 k < l m) is

> <sup>θ</sup> 1 k f (x ,..., x ) k

(m k)! F (x ) F (x ) F (x ) F (x ) f (x ) g (x |x ) <sup>1</sup>

<sup>θ</sup> 1 kl f (x , ..., x , x )

<sup>θ</sup> 1 k <sup>θ</sup> l k f (x , ..., x )g (x |x ). (4)

 <sup>θ</sup> 1 kl θ l1 k θ l k θ 1 k

m! f (x )[F (x ) F (x )] f (x )[1 F (x )] (l k 1)!(m l)!

i 1

k

*Proof.* The joint density of *X*<sup>1</sup> ... *Xk* and *X*<sup>l</sup> is given by

represents the conditional probability density function of Xl given Xk=xk.

i 1

(m k)! lk1 1 F (x ) f (x ) ( 1) (l k 1)!(m l)! <sup>j</sup> 1 F (x ) 1 F (x )

(l k 1)!(m l)! 1 F (x ) 1 F (x ) 1 F (x )

mlj lk1

lk1j m l

preliminaries for the within-sample prediction situation are given below.

given by

where

θ l k

It follows from (4) that

Most stochastic models, which are used in revenue optimization systems, are developed under the assumptions that the parameter values of the models are known with certainty. When these models are applied to solve real-world problems, the parameters are estimated and then treated as if they were the true values. The risk associated with using estimates rather than the true parameters is called estimation risk and is often ignored. When data are limited and/or unreliable, estimation risk may be significant, and failure to incorporate it into the model design may lead to serious errors. Its explicit consideration is important since decision rules which are optimal in the absence of uncertainty need not even be approximately optimal in the presence of such uncertainty.

In this paper, we consider the cases where it is known that the underlying probability distributions belong to a parameterized family of distributions. However, unlike in the Bayesian approach, we do not assume any prior knowledge on the parameter values. The primary purpose of the paper is to introduce the idea of embedding of sample statistics (say, sufficient statistics or maximum likelihood estimators) in a performance index of revenue optimization problem. In this case, we find the optimal stochastic control policy directly. We demonstrate the fact that the traditional approach, which separates the estimation and the optimization tasks in revenue optimization systems (i.e., when we use the estimates as if they were the true parameters) can often lead to pure results. It will be noted that the optimal statistical decision rules depend on data availability.

For constructing the improved statistical decisions, a new technique of invariant embedding of sample statistics in a performance index is proposed (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b). This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, an invariant embedding technique is independent of the choice of priors, i.e., subjectivity of investigator is eliminated from the problem. The technique allows one to eliminate unknown parameters from the problem and to find the improved invariant statistical decision rules, which has smaller risk than any of the well-known traditional statistical decision rules.

In order to obtain improved statistical decisions for revenue management under parametric uncertainty, it can be considered the three prediction situations: "new-sample" prediction, "within-sample" prediction, and "new-within-sample" prediction. For the new-sample prediction situation, the data from a past sample are used to make predictions on a future unit or sample of units from the same process or population. For the within-sample prediction situation, the problem is to predict future events in a sample or process based on early data from that sample or process. For the new-within-sample prediction situation, the problem is to predict future events in a sample or process based on early data from that sample or process as well as on a past data sample from the same process or population. Some mathematical preliminaries for the within-sample prediction situation are given below.

### **2. Mathematical preliminaries for the within-sample prediction situation**

*Theorem 1.* Let X1 ... Xk be the first k ordered observations (order statistics) in a sample of size m from a continuous distribution with some probability density function f (x) and distribution function F (x), where is a parameter (in general, vector). Then the joint probability density function of X1 ... Xk and the *l*th order statistics Xl (1 k < l m) is given by

$$\mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{1'},...,\mathbf{x}\_{\mathbf{k}'},\mathbf{x}\_{\mathbf{l}}) = \mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{1'},...,\mathbf{x}\_{\mathbf{k}}) \mathbf{g}\_{\boldsymbol{\theta}\boldsymbol{\theta}}(\mathbf{x}\_{\mathbf{l}} \mid \mathbf{x}\_{\mathbf{k}})\_{\boldsymbol{\prime}} \tag{1}$$

where

186 Stochastic Modeling and Control

many different fares throughout the booking horizon depending on product restrictions as well as the remaining time until departure and the number of unsold seats. Since the tickets for a flight have to be sold before the plane takes off, the product is perishable and cannot be stored for future use. The use of the above strategies has transformed the transportation and hospitality industries, and has become increasingly important in retail, telecommunications, entertainment, financial services, health care and manufacturing. In parallel, pricing and revenue optimization has become a rapidly expanding practice in consulting services, and a growing area of software and IT development, where the revenue optimization systems are

Most stochastic models, which are used in revenue optimization systems, are developed under the assumptions that the parameter values of the models are known with certainty. When these models are applied to solve real-world problems, the parameters are estimated and then treated as if they were the true values. The risk associated with using estimates rather than the true parameters is called estimation risk and is often ignored. When data are limited and/or unreliable, estimation risk may be significant, and failure to incorporate it into the model design may lead to serious errors. Its explicit consideration is important since decision rules which are optimal in the absence of uncertainty need not even be

In this paper, we consider the cases where it is known that the underlying probability distributions belong to a parameterized family of distributions. However, unlike in the Bayesian approach, we do not assume any prior knowledge on the parameter values. The primary purpose of the paper is to introduce the idea of embedding of sample statistics (say, sufficient statistics or maximum likelihood estimators) in a performance index of revenue optimization problem. In this case, we find the optimal stochastic control policy directly. We demonstrate the fact that the traditional approach, which separates the estimation and the optimization tasks in revenue optimization systems (i.e., when we use the estimates as if they were the true parameters) can often lead to pure results. It will be noted that the

For constructing the improved statistical decisions, a new technique of invariant embedding of sample statistics in a performance index is proposed (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b). This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, an invariant embedding technique is independent of the choice of priors, i.e., subjectivity of investigator is eliminated from the problem. The technique allows one to eliminate unknown parameters from the problem and to find the improved invariant statistical decision rules, which has

In order to obtain improved statistical decisions for revenue management under parametric uncertainty, it can be considered the three prediction situations: "new-sample" prediction, "within-sample" prediction, and "new-within-sample" prediction. For the new-sample prediction situation, the data from a past sample are used to make predictions on a future unit or sample of units from the same process or population. For the within-sample prediction

smaller risk than any of the well-known traditional statistical decision rules.

tightly integrated in the existing Supply Chain Management solutions.

approximately optimal in the presence of such uncertainty.

optimal statistical decision rules depend on data availability.

$$\mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{1},...,\mathbf{x}\_{k}) = \frac{\mathbf{m}!}{(\mathbf{m}-\mathbf{k})!} \prod\_{i=1}^{k} \mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{i}) [1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{k})]^{\mathbf{m}-\mathbf{k}}\tag{2}$$

 lk1 m l θ l θ k θ l θ k θ l θ l k θ k θ k θ k (m k)! F (x ) F (x ) F (x ) F (x ) f (x ) g (x |x ) <sup>1</sup> (l k 1)!(m l)! 1 F (x ) 1 F (x ) 1 F (x ) mlj lk1 j θ l θ l j 0 θ k θ k (m k)! lk1 1 F (x ) f (x ) ( 1) (l k 1)!(m l)! <sup>j</sup> 1 F (x ) 1 F (x ) lk1j m l j θ l θ k θ l j 0 θ k θ k (m k)! m l F (x ) F (x ) f (x ) ( 1) (l k 1)!(m l)! <sup>j</sup> 1 F (x ) 1 F (x ) (3)

represents the conditional probability density function of Xl given Xk=xk.

*Proof.* The joint density of *X*<sup>1</sup> ... *Xk* and *X*<sup>l</sup> is given by

$$\mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{1},...,\mathbf{x}\_{\mathbf{k}},\mathbf{x}\_{\mathbf{l}}) = \frac{\mathbf{m}!}{(\mathbf{l}-\mathbf{k}-\mathbf{l})!(\mathbf{m}-\mathbf{l})!} \prod\_{i=1}^{\mathbf{k}} \mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{i}) [\mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{1}) - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{k})]^{\mathbf{l}-\mathbf{k}-\mathbf{l}} \mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{1}) [1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{1})]^{\mathbf{m}-\mathbf{l}}$$

$$= \mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{1},...,\mathbf{x}\_{\mathbf{k}}) \mathbf{g}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\mathbf{l}} \mid \mathbf{x}\_{\mathbf{k}}).\tag{4}$$

It follows from (4) that

$$\mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\mathrm{l}} \mid \mathbf{x}\_{1}, \dots, \mathbf{x}\_{\mathrm{k}}) = \frac{\mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{1}, \dots, \mathbf{x}\_{\mathrm{k}}, \mathbf{x}\_{\mathrm{l}})}{\mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{1}, \dots, \mathbf{x}\_{\mathrm{k}})} = \mathbf{g}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\mathrm{l}} \mid \mathbf{x}\_{\mathrm{k}}) \,\tag{5}$$

i.e., the conditional distribution of Xl, given Xi = xi for all i = 1,…, k, is the same as the conditional distribution of Xl , given only Xk = xk. This ends the proof. □

*Corollary 1.1.* The conditional probability distribution function of *Xl* given *Xk*=*xk* is

$$\mathbf{P}\_{\theta} \left\{ \mathbf{X}\_{1} \le \mathbf{x}\_{1} \mid \mathbf{X}\_{\mathbf{k}} = \mathbf{x}\_{\mathbf{k}} \right\} = 1 - \frac{(\mathbf{m} - \mathbf{k})!}{(1 - \mathbf{k} - 1)!(\mathbf{m} - \mathbf{l})!}$$

$$\times \sum\_{j=0}^{\mathrm{l}-\mathrm{k}-1} \binom{\mathrm{l} - \mathrm{k}-\mathrm{l}}{\mathrm{j}} \frac{(-\mathrm{l})^{j}}{\mathrm{m} - \mathrm{l} + \mathrm{l} + \mathrm{j}} \left[ \frac{\mathrm{I} - \mathrm{F}\_{\theta}(\mathbf{x}\_{\mathrm{l}})}{\mathrm{I} - \mathrm{F}\_{\theta}(\mathbf{x}\_{\mathrm{k}})} \right]^{\mathrm{m} - \mathrm{l} + \mathrm{l} + \mathrm{j}}$$

$$= \frac{(\mathrm{m} - \mathrm{k})!}{(\mathrm{l} - \mathrm{k} - \mathrm{l})!(\mathrm{m} - \mathrm{l})!} \sum\_{j=0}^{\mathrm{m}-\mathrm{l}} \binom{\mathrm{m} - \mathrm{l}}{j} \frac{(-\mathrm{l})^{j}}{\mathrm{I} - \mathrm{k} + \mathrm{j}} \left[ \frac{\mathrm{F}\_{\theta}(\mathbf{x}\_{\mathrm{l}}) - \mathrm{F}\_{\theta}(\mathbf{x}\_{\mathrm{k}})}{\mathrm{I} - \mathrm{F}\_{\theta}(\mathbf{x}\_{\mathrm{k}})} \right]^{\mathrm{l} - \mathrm{k} + \mathrm{j}}.\tag{6}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 189

F (x ) F (x ) , 1 k m 1. 1 F (x ) (10)

1 x f (x) exp , x 0, θ 0, θ θ (11)

θ (12)

j l k

l k x x, (13)

l k

j l k l k

j 0 1 lk1

 lk1

Β(l k,(m l 1) j

l k <sup>j</sup> m l <sup>j</sup>

<sup>1</sup> m l ( 1) x x 1 exp . Β(l k,(m l 1) l k j <sup>j</sup> <sup>θ</sup> (14)

 m k

In order to use the results of Theorem 1, we consider, for illustration, the exponential

 <sup>θ</sup> <sup>x</sup> F (x) 1 exp , x 0, θ 0.

*Theorem 2.* Let X1 ... Xk be the first k ordered observations (order statistics) in a sample of size m from the exponential distribution (11). Then the conditional probability density

1 1 lk1 (m l 1 j)(x x ) g (x |x ) ( 1) exp Β(l k,(m l 1) <sup>j</sup> θ θ

 lk1j m l

1 1 m l x x x x ( 1) 1 exp exp , Β(l k,(m l 1) <sup>j</sup> θθ θ

and the conditional probability distribution function of the lth order statistics Xl given Xk = xk is

 

 

l k ( 1) (m l 1 j)(x x ) exp ml1j <sup>θ</sup>

P X x |X x <sup>θ</sup> l lk k

1

j

*Proof.* It follows from (3) and (6), respectively. □

*Corollary 2.1.* If l = k + 1,

j 0

 

θ m θ k θ k

<sup>θ</sup>

function of the lth order statistics Xl (1 k < l m) given Xk = xk is

j 0

lk1

j 0

**2.1. Exponential distribution** 

distribution with the probability density function

and the probability distribution function

θ l k

*Corollary 1.2*. If l = k + 1,

$$\mathbf{g}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\boldsymbol{\kappa}+1}\mid\mathbf{x}\_{\boldsymbol{\kappa}}) = (\mathbf{m}-\mathbf{k}) \begin{bmatrix} 1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\boldsymbol{\kappa}+1}) \\ 1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\boldsymbol{\kappa}}) \end{bmatrix}^{\mathbf{m}-\mathbf{k}-1} \frac{\mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\boldsymbol{\kappa}+1})}{1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\boldsymbol{\kappa}})} $$
 
$$= (\mathbf{m}-\mathbf{k}) \sum\_{j=0}^{\mathbf{m}-\mathbf{k}-1} \binom{\mathbf{m}-\mathbf{k}-1}{j} (-\mathbf{1})^{j} \left[ \frac{\mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\boldsymbol{\kappa}+1}) - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\boldsymbol{\kappa}})}{1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\boldsymbol{\kappa}})} \right]^{\dagger} \frac{\mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\boldsymbol{\kappa}+1})}{1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\boldsymbol{\kappa}})}, \ 1 \le \mathbf{k} \le \mathbf{m}-1, \tag{7}$$

and

$$\mathbf{P}\_{\theta} \left\{ \mathbf{X}\_{k+1} \le \mathbf{x}\_{k+1} \mid \mathbf{X}\_{k} = \mathbf{x}\_{k} \right\} = 1 - \left[ \frac{\mathbf{1} - \mathbf{F}\_{\theta}(\mathbf{x}\_{k+1})}{\mathbf{1} - \mathbf{F}\_{\theta}(\mathbf{x}\_{k})} \right]^{\mathbf{m} - \mathbf{k}}$$

$$= (\mathbf{m} - \mathbf{k}) \sum\_{j=0}^{\mathbf{m} - \mathbf{k} - 1} \binom{\mathbf{m} - \mathbf{k} - 1}{j} \frac{(-1)^{j}}{1 + j} \left[ \frac{\mathbf{F}\_{\theta}(\mathbf{x}\_{k+1}) - \mathbf{F}\_{\theta}(\mathbf{x}\_{k})}{1 - \mathbf{F}\_{\theta}(\mathbf{x}\_{k})} \right]^{\mathbf{l} + j}, 1 \le \mathbf{k} \le \mathbf{m} - 1. \tag{8}$$

*Corollary 1.3*. If l = m,

$$\mathbf{g}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\rm m} \mid \mathbf{x}\_{\rm k}) = (\mathbf{m} - \mathbf{k}) \sum\_{j=0}^{\mathbf{m}-\mathbf{k}-1} \binom{\mathbf{m} - \mathbf{k} - 1}{j} (-1)^{j}$$

$$\times \left[ \frac{\mathbf{1} - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\rm m})}{1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\rm k})} \right] \frac{\mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\rm m})}{1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\rm k})} = (\mathbf{m} - \mathbf{k}) \left[ \frac{\mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\rm m}) - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\rm k})}{1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\rm k})} \right]^{\mathbf{m} - \mathbf{k} - 1} \frac{\mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\rm m})}{1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\rm k})}, \ 1 \le \mathbf{k} \le \mathbf{m} - 1,\tag{9}$$

and

$$\mathbb{P}\_{\theta} \left( \mathbf{X}\_{\mathrm{m}} \leq \mathbf{x}\_{\mathrm{m}} \mid \mathbf{X}\_{\mathrm{k}} = \mathbf{x}\_{\mathrm{k}} \right) = 1 - (\mathbf{m} - \mathbf{k}) \sum\_{j=0}^{\mathrm{m}-\mathrm{k}-1} \binom{\mathrm{m} - \mathrm{k}-1}{\mathrm{j}} \frac{(-1)^{j}}{1+\mathrm{j}} \left[ \frac{1 \cdot \mathrm{F}\_{\theta}(\mathbf{x}\_{\mathrm{m}})}{1 \cdot \mathrm{F}\_{\theta}(\mathbf{x}\_{\mathrm{k}})} \right]^{1+\mathrm{j}}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 189

$$\mathbf{x} = \left[ \frac{\mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\mathrm{m}}) - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\mathrm{k}})}{1 - \mathbf{F}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\mathrm{k}})} \right]^{\mathrm{m} - \mathrm{k}} \quad 1 \le \mathrm{k} \le \mathrm{m} - 1. \tag{10}$$

#### **2.1. Exponential distribution**

188 Stochastic Modeling and Control

*Corollary 1.2*. If l = k + 1,

*Corollary 1.3*. If l = m,

 

j θ m θ m θ k θ k

θ m mk k

1 F (x ) f (x ) 1 F (x ) 1 F (x )

and

and

i.e., the conditional distribution of Xl, given Xi = xi for all i = 1,…, k, is the same as the

 <sup>θ</sup> l lk k (m k)! P X x |X x 1 (l k 1)!(m l)!

ml1 <sup>j</sup> lk1 <sup>j</sup>

lk1 ( 1) 1 F (x ) j m l 1 j 1 F (x )

θ l

 

θ k θ k

mk1 θ k 1 θ k 1

j θ k 1 θ k θ k 1

mk1 F (x ) F (x ) f (x ) (m k) ( 1) , 1 k m 1, <sup>j</sup> 1 F (x ) 1 F (x ) (7)

θ k 1 θ k

mk1 ( 1) F (x ) F (x ) (m k) , 1 k m 1. <sup>j</sup> 1 j 1 F (x ) (8)

mk1

 mk1 θ m θ k θ m

θ k θ k F (x ) F (x ) f (x ) (m k) , 1 k m 1, 1 F (x ) 1 F (x ) (9)

m k

j

1+j <sup>j</sup> θ m θ k

(-1) 1- F (x ) 1 + j 1- F (x )

θ k 1

θ k

θ l θ k θ k

l k j

F (x ) F (x ) . 1 F (x ) (6)

j 0 θ k

m l j

mk1

 

<sup>g</sup> (x |x ) (m k) ( 1) <sup>j</sup>

j 0

j 0

 mk1

j 0 θ k θ k

1 F (x ) P X x |X x 1 1 F (x )

j 0 θ k

 1 j mk1 <sup>j</sup>

<sup>j</sup> mk1

θ k1 k1 k k

θ m k

mk1 P X x |X x 1 (m k) <sup>j</sup>

1 F (x ) f (x ) g (x |x ) (m k) 1 F (x ) 1 F (x )

 

j 0 (m k)! m l ( 1) (l k 1)!(m l)! l k j j

conditional distribution of Xl , given only Xk = xk. This ends the proof. □

θ k1 k

*Corollary 1.1.* The conditional probability distribution function of *Xl* given *Xk*=*xk* is

In order to use the results of Theorem 1, we consider, for illustration, the exponential distribution with the probability density function

$$\text{tf}\_{\theta}(\mathbf{x}) = \frac{1}{\theta} \exp\left(-\frac{\mathbf{x}}{\theta}\right), \quad \mathbf{x} \ge \mathbf{0}, \ \theta > \mathbf{0},\tag{11}$$

and the probability distribution function

$$F\_{\theta}(\mathbf{x}) = 1 - \exp\left(-\frac{\mathbf{x}}{\theta}\right), \quad \mathbf{x} \ge 0, \ \theta > 0. \tag{12}$$

*Theorem 2.* Let X1 ... Xk be the first k ordered observations (order statistics) in a sample of size m from the exponential distribution (11). Then the conditional probability density function of the lth order statistics Xl (1 k < l m) given Xk = xk is

$$\log\_{\theta}(\mathbf{x}\_{\mathrm{l}} \mid \mathbf{x}\_{\mathrm{k}}) = \frac{1}{\mathrm{B}(\mathrm{l} - \mathbf{k}\_{\mathrm{r}}(\mathrm{m} - \mathrm{l} + 1))} \sum\_{j=0}^{\mathrm{l} - \mathrm{k} - 1} \binom{\mathrm{l} - \mathrm{k} - 1}{\mathrm{j}} (-1)^{j} \frac{1}{\theta} \exp\left(-\frac{(\mathrm{m} - \mathrm{l} + 1 + \mathrm{j})(\mathbf{x}\_{\mathrm{l}} - \mathbf{x}\_{\mathrm{k}})}{\theta}\right)$$

$$\mathbf{x} = \frac{1}{\mathrm{B}(\mathrm{l} - \mathbf{k}\_{\mathrm{r}}(\mathrm{m} - \mathrm{l} + 1))} \sum\_{j=0}^{\mathrm{m} - 1} \binom{\mathrm{m} - 1}{\mathrm{j}} (-1)^{j} \frac{1}{\theta} \left[1 - \exp\left(-\frac{\mathbf{x}\_{\mathrm{l}} - \mathbf{x}\_{\mathrm{k}}}{\theta}\right)\right]^{\mathrm{l} - \mathrm{k} - 1 + \mathrm{j}} \exp\left(\frac{\mathbf{x}\_{\mathrm{l}} - \mathbf{x}\_{\mathrm{k}}}{\theta}\right),$$

$$\mathbf{x}\_{\mathrm{l}} \ge \mathbf{x}\_{\mathrm{k} \prime} \tag{13}$$

and the conditional probability distribution function of the lth order statistics Xl given Xk = xk is

$$\begin{aligned} \mathbf{P}\_{\boldsymbol{\Theta}} \left\{ \mathbf{X}\_{\text{l}} \le \mathbf{x}\_{\text{l}} \, \middle| \, \mathbf{X}\_{\text{k}} = \mathbf{x}\_{\text{k}} \right\} &= 1 - \frac{1}{\mathbf{B} (\mathbf{l} - \mathbf{k}\_{\star} (\mathbf{m} - \mathbf{l} + 1))} \sum\_{j=0}^{\mathbf{l} - \mathbf{k} - 1} \binom{\mathbf{l} - \mathbf{k} - \mathbf{1}}{\mathbf{j}} \\\\ & \times \frac{(-1)^{\mathbf{j}}}{\mathbf{m} - \mathbf{l} + 1 + \mathbf{j}} \exp \left( - \frac{(\mathbf{m} - \mathbf{l} + 1 + \mathbf{j})(\mathbf{x}\_{\text{l}} - \mathbf{x}\_{\text{k}})}{\boldsymbol{\Theta}} \right) \\\\ &= \frac{1}{\mathbf{B} (\mathbf{l} - \mathbf{k}\_{\star} (\mathbf{m} - \mathbf{l} + 1))} \sum\_{j=0}^{\mathbf{m} - \mathbf{l}} \binom{\mathbf{m} - \mathbf{l}}{\mathbf{j}} \frac{(-1)^{j}}{\mathbf{l} - \mathbf{k} + \mathbf{j}} \left[ 1 - \exp \left( - \frac{\mathbf{x}\_{\text{l}} - \mathbf{x}\_{\text{k}}}{\boldsymbol{\Theta}} \right) \right]^{\mathbf{l} - \mathbf{k} + \mathbf{j}}. \end{aligned}$$

*Proof.* It follows from (3) and (6), respectively. □ *Corollary 2.1.* If l = k + 1,

$$\mathbf{g}\_{\theta}(\mathbf{x}\_{k+1} \mid \mathbf{x}\_{k}) = (\mathbf{m} - \mathbf{k}) \frac{1}{\theta} \exp\left(-\frac{(\mathbf{m} - \mathbf{k})(\mathbf{x}\_{k+1} - \mathbf{x}\_{k})}{\theta}\right)$$

$$= (\mathbf{m} - \mathbf{k}) \sum\_{j=0}^{\mathbf{m}-\mathbf{k}-1} \binom{\mathbf{m} - \mathbf{k}-1}{\mathbf{j}} (-\mathbf{1})^{j} \frac{\mathbf{1}}{\theta} \left[1 - \exp\left(-\frac{\mathbf{x}\_{k+1} - \mathbf{x}\_{k}}{\theta}\right)\right]^{\mathbf{j}} \exp\left(\frac{\mathbf{x}\_{k+1} - \mathbf{x}\_{k}}{\theta}\right),$$

$$\mathbf{x}\_{k+1} \ge \mathbf{x}\_{k'} \quad \mathbf{1} \le \mathbf{k} \le \mathbf{m} - \mathbf{1},\tag{15}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 191

l k k

s

 lk1

j 0 1 lk1

j 0

k

S X (m k)X (20)

lk1 j

j

s s ks lk <sup>g</sup> (x |x ,v), (22)

/ V Sk θ (23)

.

<sup>1</sup> k 1 f(v) v exp( v), v 0. Γ(k) (24)

k k

k1 k x x , 1 k m 1, (26)

x x 1

(k 1)

(21)

 k

 

vexp v

is the pivotal quantity, the probability density function of which is given by

2010b; 2010c; 2010d; 2010e; 2011a; 2011b), we reduce (13) to

θ l k

s k1 k

i 1

ki k

is the sufficient statistic for , and the predictive probability distribution function of the *l*th

<sup>k</sup>

<sup>1</sup> lk1 ( 1) P X x |X x 1 Β(l k,(m l 1) <sup>j</sup> ml1j

 

*Proof.* Using the technique of invariant embedding (Nechval et al., 1999; 2004; 2008; 2010a;

<sup>g</sup> (x |x ) ( 1) Β(l k,(m l 1) <sup>j</sup>

l k k k

 kk k <sup>k</sup> s lk s lk s lk s lk 0

<sup>g</sup> (x |x ) k(m k) 1 (m k) , s s

<sup>k</sup>

g (x |x ) E{g (x |x ,v)} g (x |x ,v)f(v)dv g (x |x ) (25)

k1 k

(m l 1 j)(x x ) 1

x x 1 (m l 1 j) .

where

where

Then

This ends the proof. □

*Corollary 3.1*. If l = k + 1,

order statistics Xl is given by

s l lk k

and

$$\mathbf{P}\_{\theta} \left( \mathbf{X}\_{\mathbf{k}+1} \le \mathbf{x}\_{\mathbf{k}+1} \mid \mathbf{X}\_{\mathbf{k}} = \mathbf{x}\_{\mathbf{k}} \right) = 1 - \exp \left( -\frac{(\mathbf{m} - \mathbf{k})(\mathbf{x}\_{\mathbf{k}+1} - \mathbf{x}\_{\mathbf{k}})}{\theta} \right)$$

$$= (\mathbf{m} - \mathbf{k}) \sum\_{j=0}^{\mathbf{m}-\mathbf{k}-1} \binom{\mathbf{m} - \mathbf{k}-1}{j} \frac{(-1)^{j}}{1+j} \left[ 1 - \exp \left( -\frac{\mathbf{x}\_{\mathbf{k}+1} - \mathbf{x}\_{\mathbf{k}}}{\theta} \right) \right]^{1+j}, \ 1 \le \mathbf{k} \le \mathbf{m} - 1. \tag{16}$$

*Corollary 2.2*. If l = m,

$$\mathbf{g}\_{\boldsymbol{\theta}}(\mathbf{x}\_{\mathrm{m}} \mid \mathbf{x}\_{\mathrm{k}}) = (\mathbf{m} - \mathbf{k})$$

$$\times \sum\_{\mathbf{j}=0}^{\mathbf{m}-\mathbf{k}-1} \binom{\mathbf{m} - \mathbf{k} - 1}{\mathbf{j}} (-1)^{\mathbf{j}} \frac{1}{\boldsymbol{\theta}} \exp\left(-\frac{(\mathbf{1} + \mathbf{j})(\mathbf{x}\_{\mathrm{m}} - \mathbf{x}\_{\mathrm{k}})}{\boldsymbol{\theta}}\right)$$

$$= (\mathbf{m} - \mathbf{k}) \frac{1}{\boldsymbol{\theta}} \Big[ 1 - \exp\left(-\frac{\mathbf{x}\_{\mathrm{m}} - \mathbf{x}\_{\mathrm{k}}}{\boldsymbol{\theta}}\right) \right]^{\mathbf{m} - \mathbf{k} - 1} \exp\left(-\frac{\mathbf{x}\_{\mathrm{m}} - \mathbf{x}\_{\mathrm{k}}}{\boldsymbol{\theta}}\right), \ \mathbf{x}\_{\mathrm{m}} \ge \mathbf{x}\_{\mathrm{k}'} \quad 1 \le \mathbf{k} \le \mathbf{m} - 1,\tag{17}$$

and

$$\mathbb{P}\_{\theta}\left\{\mathbf{X}\_{\mathbf{m}} \le \mathbf{x}\_{\mathbf{m}} \, \middle| \, \mathbf{X}\_{\mathbf{k}} = \mathbf{x}\_{\mathbf{k}} \right\} = 1 - (\mathbf{m} - \mathbf{k}) \sum\_{j=0}^{\mathbf{m}-\mathbf{k}-1} \binom{\mathbf{m} - \mathbf{k}-1}{j} \frac{(-1)^{j}}{1+j} \exp\left(-\frac{(1+j)(\mathbf{x}\_{\mathbf{m}} - \mathbf{x}\_{\mathbf{k}})}{\theta}\right)$$

$$= \left[1 - \exp\left(-\frac{\mathbf{x}\_{\mathbf{m}} - \mathbf{x}\_{\mathbf{k}}}{\theta}\right)\right]^{\mathbf{m}-\mathbf{k}}, \quad 1 \le \mathbf{k} \le \mathbf{m} - 1. \tag{18}$$

*Theorem 3.* Let X1 ... Xk be the first k ordered observations (order statistics) in a sample of size m from the exponential distribution (11), where the parameter is unknown. Then the predictive probability density function of the lth order statistics Xl (1 k < l m) is given by

$$\log\_{\mathbf{s}\_{\mathbf{k}}} \left( \mathbf{x}\_{\mathbf{l}} \, \middle| \, \mathbf{x}\_{\mathbf{k}} \right) = \frac{\mathbf{k}}{\mathbf{B} (\mathbf{l} - \mathbf{k}\_{\mathbf{s}}) \mathbf{m} - \mathbf{l} + 1} \sum\_{j=0}^{\mathbf{l} - \mathbf{k} - 1} \binom{\mathbf{l} - \mathbf{k} - \mathbf{l}}{\mathbf{j}} (-1)^{j}$$

$$\times \left[ \mathbf{1} + (\mathbf{m} - \mathbf{l} + \mathbf{1} + \mathbf{j}) \frac{\mathbf{x}\_{\mathbf{l}} - \mathbf{x}\_{\mathbf{k}}}{\mathbf{s}\_{\mathbf{k}}} \right]^{-(\mathbf{k} + 1)} \frac{1}{\mathbf{s}\_{\mathbf{k}}} \, \middle| \, \mathbf{x}\_{\mathbf{l}} \ge \mathbf{x}\_{\mathbf{k}} \, \right. \tag{19}$$

where

190 Stochastic Modeling and Control

*Corollary 2.2*. If l = m,

and

and

j 0

j 0

θ k1 k

mk1

j 0

s lk

θ m mk k

θ k1 k1 k k

k1 k

k1 k x x , 1 k m 1, (15)

k1 k

j m k

mk1 ( 1) x x (m k) 1 exp , 1 k m 1. <sup>j</sup> 1 j <sup>θ</sup> (16)

k1 k

m k

j

l k

m k

(19)

m k x x 1 exp , 1 k m 1. θ (18)

j k1 k k1 k

<sup>1</sup> (m k)(x x ) g (x |x ) (m k) exp θ θ

mk1 <sup>1</sup> xx xx (m k) ( 1) 1 exp exp , j θθ θ

 <sup>j</sup> mk1

 1 j mk1 <sup>j</sup>

<sup>θ</sup> m k g (x |x ) (m k)

j 0 mk1 ( 1) (1 j)(x x ) P X x |X x 1 (m k) exp <sup>j</sup> 1 j <sup>θ</sup>

*Theorem 3.* Let X1 ... Xk be the first k ordered observations (order statistics) in a sample of size m from the exponential distribution (11), where the parameter is unknown. Then the predictive probability density function of the lth order statistics Xl (1 k < l m) is given by

<sup>k</sup>

<sup>g</sup> (x |x ) ( 1) Β(l k,(m l 1) <sup>j</sup>

l k

x x <sup>1</sup> 1 (m l 1 j) , x x , s s

mk1 m k m k

mk1 <sup>1</sup> (1 j)(x x ) ( 1) exp <sup>j</sup> θ θ

<sup>1</sup> x x x x (m k) 1 exp exp , x x , 1 k m 1, θθ θ (17)

m k

mk1 j

lk1

(k 1)

k k

j 0 k lk1

(m k)(x x ) P X x |X x 1 exp <sup>θ</sup>

$$\mathbf{S}\_{\mathbf{k}} = \sum\_{i=1}^{\mathbf{k}} \mathbf{X}\_{\mathbf{i}} + (\mathbf{m} - \mathbf{k}) \mathbf{X}\_{\mathbf{k}} \tag{20}$$

is the sufficient statistic for , and the predictive probability distribution function of the *l*th order statistics Xl is given by

$$\Pr\_{\mathbf{s}\_{\mathbf{k}}}\left\{\mathbf{X}\_{\mathbf{l}} \le \mathbf{x}\_{\mathbf{l}} \, | \, \mathbf{X}\_{\mathbf{k}} = \mathbf{x}\_{\mathbf{k}}\right\} = 1 - \frac{1}{\mathrm{B}(\mathbf{l} - \mathbf{k}, \mathrm{(m - 1 + 1)})} \sum\_{\mathbf{j}=0}^{\mathrm{l}-\mathrm{k}-1} \binom{\mathrm{l} - \mathrm{k} - 1}{\mathrm{j}} \frac{(-1)^{\mathrm{j}}}{\mathrm{m} - \mathrm{l} + 1 + \mathrm{j}}$$

$$\times \left[ \mathbf{1} + (\mathrm{m} - \mathrm{l} + 1 + \mathrm{j}) \frac{\mathbf{x}\_{\mathbf{l}} - \mathbf{x}\_{\mathbf{k}}}{\mathbf{s}\_{\mathbf{k}}} \right]^{-\mathrm{k}}.\tag{21}$$

*Proof.* Using the technique of invariant embedding (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b), we reduce (13) to

$$\mathbf{g}\_{\boldsymbol{\Theta}}(\mathbf{x}\_{\boldsymbol{1}} \mid \mathbf{x}\_{\boldsymbol{k}}) = \frac{1}{\mathbf{B}(1 - \mathbf{k}, \mathbf{(m - 1 + 1)})} \sum\_{j=0}^{\mathrm{l} - \mathrm{k} - 1} \binom{\mathrm{l} - \mathrm{k} - 1}{\mathrm{j}} (-1)^{\mathrm{j}}$$

$$\mathbf{x} \times \mathrm{vexp} \left( -\frac{(\mathbf{m - 1 + 1 + \mathrm{j})(\mathbf{x}\_{\boldsymbol{1}} - \mathbf{x}\_{\boldsymbol{k}})}{\mathbf{s}\_{\boldsymbol{k}}} \mathbf{v} \right) \frac{1}{\mathbf{s}\_{\boldsymbol{k}}} = \mathbf{g}\_{\mathrm{s}\_{\boldsymbol{k}}}(\mathbf{x}\_{\boldsymbol{1}} \mid \mathbf{x}\_{\boldsymbol{k}}, \mathbf{v}), \tag{22}$$

where

$$\mathbf{V} = \mathbf{S}\_{\mathbf{k}} \; / \; \Theta \tag{23}$$

is the pivotal quantity, the probability density function of which is given by

$$\mathbf{f}(\mathbf{v}) = \frac{1}{\Gamma(\mathbf{k})} \mathbf{v}^{\mathbf{k}-1} \exp(-\mathbf{v}), \quad \mathbf{v} \ge \mathbf{0}. \tag{24}$$

Then

$$\mathbf{g}\_{\mathbf{s}\_{\mathbf{k}}} \left( \mathbf{x}\_{\mathbf{l}} \mid \mathbf{x}\_{\mathbf{k}} \right) = \mathbb{E} \{ \mathbf{g}\_{\mathbf{s}\_{\mathbf{k}}} \left( \mathbf{x}\_{\mathbf{l}} \mid \mathbf{x}\_{\mathbf{k}'} \mathbf{v} \right) \} = \int\_{0}^{\eta} \mathbf{g}\_{\mathbf{s}\_{\mathbf{k}}} \left( \mathbf{x}\_{\mathbf{l}} \mid \mathbf{x}\_{\mathbf{k}'} \mathbf{v} \right) \mathbf{f}(\mathbf{v}) \mathrm{d}\mathbf{v} = \mathbf{g}\_{\mathbf{s}\_{\mathbf{k}}} \left( \mathbf{x}\_{\mathbf{l}} \mid \mathbf{x}\_{\mathbf{k}} \right). \tag{25}$$

This ends the proof. □

*Corollary 3.1*. If l = k + 1,

$$\mathbf{g}\_{\mathbf{s}\_{\mathbf{k}}} \left( \mathbf{x}\_{\mathbf{k}+1} \, \vert \, \mathbf{x}\_{\mathbf{k}} \right) = \mathbf{k} (\mathbf{m} - \mathbf{k}) \left[ 1 + (\mathbf{m} - \mathbf{k}) \frac{\mathbf{x}\_{\mathbf{k}+1} - \mathbf{x}\_{\mathbf{k}}}{\mathbf{s}\_{\mathbf{k}}} \right]^{-(\mathbf{k}+1)} \frac{1}{\mathbf{s}\_{\mathbf{k}}} \, \mathbf{l} \tag{26}$$
 
$$\mathbf{x}\_{\mathbf{k}+1} \ge \mathbf{x}\_{\mathbf{k}} \, \; \mathbf{l} \le \mathbf{k} \le \mathbf{m} - 1 \, \tag{26}$$

#### 192 Stochastic Modeling and Control

and

$$P\_{\mathbf{s}\_{\mathbf{k}}}\left\{\mathbf{X}\_{\mathbf{k}+1} \le \mathbf{x}\_{\mathbf{k}+1} \mid \mathbf{X}\_{\mathbf{k}} = \mathbf{x}\_{\mathbf{k}}\right\} = 1 - \left[1 + (\mathbf{m} - \mathbf{k}) \frac{\mathbf{x}\_{\mathbf{k}+1} - \mathbf{x}\_{\mathbf{k}}}{\mathbf{s}\_{\mathbf{k}}}\right]^{-\mathbf{k}}, \text{ } 1 \le \mathbf{k} \le \mathbf{m} - 1. \tag{27}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 193

k 1

<sup>i</sup> ik1Y is given by

<sup>i</sup> ik1Y is given by

(35)

(38)

(36)

(m k)y <sup>G</sup> <sup>y</sup> |k 1 exp . θ (32)

1 z z

<sup>g</sup> (z |k) (m k) 1 exp exp , z 0, 1 k m 1; θθ θ (33)

 m k

<sup>z</sup> G (z |k) 1 exp , 1 k m 1. θ (34)

k k y 1

k 1

k

s

 

j m

mz 0, 1 k m 1; (37)

k

s s

(k 1)

k 1

j 0 k k

(k 1) mk1

m

 

g (y |k) k(m k) 1 (m k) , y 0;

<sup>k</sup>

mk1 <sup>z</sup> <sup>1</sup> g (z |k) k(m k) ( 1) 1 (1 j) , <sup>j</sup> s s

j 0 k m k <sup>z</sup> G (z |k) ( 1) 1 j , 1 k m 1. <sup>j</sup> <sup>s</sup>

 <sup>k</sup> <sup>k</sup> m k

Most of the inventory management literature assumes that demand distributions are specified explicitly. However, in many practical situations, the true demand distributions are not known, and the only information available may be a time-series of historic demand data. When the demand distribution is unknown, one may either use a parametric approach

j m

<sup>k</sup>

<sup>y</sup> G (y |k) 1 1 (m k) .

s k1 k 1

mk1 m m

θ m m

Predictive probability density function of Yk+1, k{1, …, m 1}, is given by

<sup>k</sup>

Predictive probability distribution function of Yk+1, k{1, …, m 1}, is given by

s k1

Predictive probability density function of Zm is given by

Predictive probability distribution function of Zm is given by

s m

**3. Stochastic inventory control problem** 

s m

θ k 1

Conditional probability density function of Zm <sup>m</sup>

Conditional probability distribution function of Zm <sup>m</sup>

θ m

*Corollary 3.2*. If l = m,

$$\mathbf{g\_{s\_k}}(\mathbf{x\_m}|\mathbf{x\_k}) = \mathbf{k(m-k)} \sum\_{j=0}^{m-k-1} \binom{m-k-1}{j} (-1)^j \left[ 1 + (1+j) \frac{\mathbf{x\_m} - \mathbf{x\_k}}{\mathbf{s\_k}} \right]^{-(k+1)} \frac{1}{\mathbf{s\_k}},$$

$$\mathbf{x\_m} \ge \mathbf{x\_{k'}} \quad 1 \le \mathbf{k} \le m-1,\tag{28}$$

and

$$\mathbf{P}\_{\mathbf{s}\_{k}}\left\{\mathbf{X}\_{\mathbf{m}}\leq\mathbf{x}\_{\mathbf{m}}\,\middle|\,\mathbf{X}\_{\mathbf{k}}=\mathbf{x}\_{\mathbf{k}}\right\}=\mathbf{1}-\left\{\mathbf{m}-\mathbf{k}\right\}\sum\_{j=0}^{\mathbf{m}-\mathbf{k}-1}\binom{\mathbf{m}-\mathbf{k}-1}{j}\frac{(-1)^{j}}{1+j}\binom{\mathbf{x}\_{\mathbf{m}}-\mathbf{x}\_{\mathbf{k}}}{\mathbf{s}\_{\mathbf{k}}}^{-\mathbf{x}}$$

$$=\sum\_{j=0}^{\mathbf{m}-\mathbf{k}}\binom{\mathbf{m}-\mathbf{k}}{j}(-1)^{j}\left[1+j\frac{\mathbf{x}\_{\mathbf{m}}-\mathbf{x}\_{\mathbf{k}}}{\mathbf{s}\_{\mathbf{k}}}\right]^{-\mathbf{k}},\ 1\leq\mathbf{k}\leq\mathbf{m}-1.\tag{29}$$

#### **2.2. Cumulative customer demand**

The primary purpose of this paper is to introduce the idea of cumulative customer demand in inventory control problems to deal with the order statistics from the underlying distribution. It allows one to use the above results to improve statistical decisions for inventory control problems under parametric uncertainty.

*Assumptions.* The customer demand at the *i*th period represents a random variable Yi, i{1, …, m}. It is assumed (for the cumulative customer demand) that the random variables

$$\mathbf{X}\_1 = \mathbf{Y}\_1, \dots, \mathbf{X}\_k = \sum\_{\mathbf{i}=}^k \mathbf{Y}\_{\mathbf{i}'}, \dots, \mathbf{X}\_1 = \sum\_{\mathbf{i}=1}^l \mathbf{Y}\_{\mathbf{i}'}, \dots, \mathbf{X}\_m = \sum\_{\mathbf{i}=1}^m \mathbf{Y}\_{\mathbf{i}} \tag{30}$$

represent the order statistics (X1 … Xm) from the exponential distribution (11).

*Inferences.* For the above case, we have the following inferences.

Conditional probability density function of Yk+1, k{1, …, m 1}, is given by

$$\log\_{\theta} \left( \mathbf{y}\_{k+1} \mid \mathbf{k} \right) = \frac{\mathbf{m} - \mathbf{k}}{\theta} \exp \left( -\frac{(\mathbf{m} - \mathbf{k}) \mathbf{y}\_{k+1}}{\theta} \right) \quad \mathbf{y}\_{k+1} \ge \mathbf{0};\tag{31}$$

Conditional probability distribution function of Yk+1, k{1, …, m 1}, is given by

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 193

$$\mathbf{G}\_{\theta} \left\{ \mathbf{y}\_{k+1} \,|\, \mathbf{k} \right\} = 1 - \exp \left( -\frac{(\mathbf{m} - \mathbf{k}) \mathbf{y}\_{k+1}}{\theta} \right). \tag{32}$$

Conditional probability density function of Zm <sup>m</sup> <sup>i</sup> ik1Y is given by

192 Stochastic Modeling and Control

*Corollary 3.2*. If l = m,

s mk

**2.2. Cumulative customer demand** 

inventory control problems under parametric uncertainty.

*Inferences.* For the above case, we have the following inferences.

s m mk k

s k1 k1 k k

j m k

m k x x , 1 k m 1, (28)

<sup>k</sup> mk1 <sup>j</sup>

j 0 k

m k

(29)

k

(27)

j 0 k k

(k 1) mk1

k1 k

k

s

x x P X x |X x 1 1 (m k) , 1 k m 1.

<sup>k</sup>

mk1 x x <sup>1</sup> g (x |x ) k(m k) ( 1) 1 (1 j) , <sup>j</sup> s s

j m k

The primary purpose of this paper is to introduce the idea of cumulative customer demand in inventory control problems to deal with the order statistics from the underlying distribution. It allows one to use the above results to improve statistical decisions for

*Assumptions.* The customer demand at the *i*th period represents a random variable Yi, i{1, …, m}. It is assumed (for the cumulative customer demand) that the random

 

θ k 1 k 1

represent the order statistics (X1 … Xm) from the exponential distribution (11).

Conditional probability density function of Yk+1, k{1, …, m 1}, is given by

Conditional probability distribution function of Yk+1, k{1, …, m 1}, is given by

11 k i l i m i

i i 1 i 1

X Y , ..., X Y , ..., X Y , ..., X Y (30)

m k (m k)y <sup>g</sup> (y |k) exp , y 0; θ θ (31)

k 1

kl m

m k x x ( 1) 1 j , 1 k m 1. <sup>j</sup> <sup>s</sup>

mk1 ( 1) x x P X x |X x 1 (m k) 1 (1 j) <sup>j</sup> 1j s

 <sup>k</sup> m k

j 0 k

<sup>k</sup>

<sup>k</sup>

and

and

variables

$$\log\_{\boldsymbol{\theta}}(\mathbf{z}\_{\mathrm{m}} \mid \mathbf{k}) = (\mathbf{m} - \mathbf{k}) \frac{1}{\boldsymbol{\theta}} \left[ 1 - \exp\left( -\frac{\mathbf{z}\_{\mathrm{m}}}{\boldsymbol{\theta}} \right) \right]^{\mathbf{m} - \mathbf{k} - 1} \exp\left( -\frac{\mathbf{z}\_{\mathrm{m}}}{\boldsymbol{\theta}} \right) \; \; \mathbf{z}\_{\mathrm{m}} \ge \mathbf{0}, \; \; 1 \le \mathbf{k} \le \mathbf{m} - 1; \tag{33}$$

Conditional probability distribution function of Zm <sup>m</sup> <sup>i</sup> ik1Y is given by

$$\mathbf{G}\_{\theta}(\mathbf{z}\_{\rm m} \mid \mathbf{k}) = \left[ 1 - \exp\left( -\frac{\mathbf{z}\_{\rm m}}{\Theta} \right) \right]^{\rm m - k}, \quad 1 \le \mathbf{k} \le \mathbf{m} - 1. \tag{34}$$

Predictive probability density function of Yk+1, k{1, …, m 1}, is given by

$$\log\_{\mathbf{s}\_{\mathbf{k}}} \left( \mathbf{y}\_{\mathbf{k}+1} \, \big|\, \mathbf{k} \right) = \mathbf{k} (\mathbf{m} - \mathbf{k}) \Big[ \mathbf{1} + (\mathbf{m} - \mathbf{k}) \frac{\mathbf{y}\_{\mathbf{k}+1}}{\mathbf{s}\_{\mathbf{k}}} \Big]^{-(\mathbf{k}+1)} \frac{1}{\mathbf{s}\_{\mathbf{k}}} \, \mathbf{y}\_{\mathbf{k}+1} \ge \mathbf{0};\tag{35}$$

Predictive probability distribution function of Yk+1, k{1, …, m 1}, is given by

$$\mathbf{G}\_{\mathbf{s}\_{\mathbf{k}}} \left( \mathbf{y}\_{\mathbf{k}+1} \mid \mathbf{k} \right) = 1 - \left[ 1 + (\mathbf{m} - \mathbf{k}) \frac{\mathbf{y}\_{\mathbf{k}+1}}{\mathbf{s}\_{\mathbf{k}}} \right]^{-\mathbf{k}} \,. \tag{36}$$

Predictive probability density function of Zm is given by

$$\mathrm{g\_{s\_k}}(\mathrm{z\_m} \mid \mathbf{k}) = \mathrm{k}(\mathbf{m} - \mathbf{k}) \sum\_{j=0}^{\mathbf{m}-\mathbf{k}-1} \binom{\mathbf{m}-\mathbf{k}-1}{j} (-\mathbf{1})^j \left[ 1 + (\mathbf{1} + \mathbf{j}) \frac{\mathbf{z\_m}}{\mathbf{s\_k}} \right]^{-(\mathbf{k}+1)} \frac{\mathbf{1}}{\mathbf{s\_k}},$$
 
$$\mathrm{z\_m} \ge 0, \quad \mathbf{1} \le \mathbf{k} \le \mathbf{m} - \mathbf{1}; \tag{37}$$

Predictive probability distribution function of Zm is given by

$$\mathbf{G}\_{\mathbf{s}\_{\mathbf{k}}} \left( \mathbf{z}\_{\mathbf{m}} \mid \mathbf{k} \right) = \sum\_{j=0}^{\mathbf{m}-\mathbf{k}} \binom{\mathbf{m}-\mathbf{k}}{j} (-1)^{j} \left[ 1 + j \frac{\mathbf{z}\_{\mathbf{m}}}{\mathbf{s}\_{\mathbf{k}}} \right]^{-\mathbf{k}}, \ 1 \le \mathbf{k} \le \mathbf{m} - 1. \tag{38}$$

#### **3. Stochastic inventory control problem**

Most of the inventory management literature assumes that demand distributions are specified explicitly. However, in many practical situations, the true demand distributions are not known, and the only information available may be a time-series of historic demand data. When the demand distribution is unknown, one may either use a parametric approach

#### 194 Stochastic Modeling and Control

(where it is assumed that the demand distribution belongs to a parametric family of distributions) or a non-parametric approach (where no assumption regarding the parametric form of the unknown demand distribution is made).

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 195

(39)

 

θ (41)

<sup>1</sup> <sup>θ</sup> k1 2 <sup>θ</sup> k 1 c P {Y u} c (1 P {Y u}) 0 (42)

(43)

mk c (44)

mk c (45)

 k 1 1 k 1

u Y c if Y u, <sup>θ</sup> C(u) Y u c if Y u.

2 k 1

 

<sup>1</sup> E {C(u)} c (u <sup>y</sup> )g (y |k)dy c (y u)g (y |k)dy . θ (40)

θ

θ 1 k1 θ k1 k1 2 k1 θ k1 k1

The function <sup>θ</sup> E {C(u)} can be shown to be convex in u, thus having a unique minimum.

1 θ k1 k1 2 θ k1 k1

<sup>c</sup> P {Y u} .

θ 2 θ k1 1 2 k1 θ k1 k1

 1 2

*Parametric uncertainty.* Consider the case when the parameter is unknown. To find the best invariant decision rule BI u , we use the invariant embedding technique (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b) to transform (39) to the form, which is depended only on the pivotal quantities V, V1, and the ancillary factor . In statistics, a pivotal quantity or pivot is a function of observations and unobservable

c c ln 1 .

<sup>1</sup> E {C(u )} c E {Y } (c c ) y g (y |k)dy <sup>θ</sup>

<sup>θ</sup> <sup>c</sup> u ln 1

θ k 1

 2

c c

1 2

2 1

0

1

 u

 

c g (y |k)dy c g (y |k)dy 0

Taking the first derivative of <sup>θ</sup> E {C(u)} with respect to u and equating it to zero, we get

0 u

0 u

k 1

The expected cost for the (k+1)th period, E{C(u)}, is expressed as

u

u

1

It follows from (31), (32), (40), and (43) that

or

or

and

Under the parametric approach, one may choose to estimate the unknown parameters or choose a prior distribution for the unknown parameters and apply the Bayesian approach to incorporating the demand data available. Scarf (1959) and Karlin (1960) consider a Bayesian framework for the unknown demand distribution. Specifically, assuming that the demand distribution belongs to the family of exponential distributions, the demand process is characterized by the prior distribution on the unknown parameter. Further extension of this approach is presented in (Azoury, 1985). Application of the Bayesian approach to the censored demand case is given in (Ding et al., 2002; Lariviere & Porteus, 1999). Parameter estimation is first considered in (Conrad, 1976) and recent developments are reported in (Agrawal & Smith, 1996; Nahmias, 1994). Liyanage & Shanthikumar (2005) propose the concept of operational statistics and apply it to a single period newsvendor inventory control problem.

This section deals with inventory items that are in stock during a single time period. At the end of the period, leftover units, if any, are disposed of, as in fashion items. Two models are considered. The difference between the two models is whether or not a setup cost is incurred for placing an order. The symbols used in the development of the models include:

c = setup cost per order,

c1= holding cost per held unit during the period,

c2= penalty cost per shortage unit during the period,

g (yk+1|k) = conditional probability density function of customer demand, Yk+1, during the (k+1)th period,

= parameter (in general, vector),

u = order quantity,

q = inventory on hand before an order is placed.

## **3.1. No-setup model (Newsvendor model)**

This model is known in the literature as the *newsvendor* model (the original classical name is the *newsboy* model). It deals with stocking and selling newspapers and periodicals. The assumptions of the model are:


The model determines the optimal value of u that minimizes the sum of the expected holding and shortage costs. Given optimal u (= u\* ), the inventory policy calls for ordering u\* q if q < u\* ; otherwise, no order is placed.

If Yk+1 u, the quantity u Yk+1 is held during the (k+1)th period. Otherwise, a shortage amount Yk+1 u will result if Yk+1> u. Thus, the cost per the (k+1)th period is

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 195

$$\mathbf{C}(\mathbf{u}) = \begin{cases} \mathbf{c}\_1 \frac{\mathbf{u} - \mathbf{Y}\_{\mathbf{k}+1}}{\theta} & \text{if} \quad \mathbf{Y}\_{\mathbf{k}+1} \le \mathbf{u}\_{\mathbf{\prime}} \\ \mathbf{c}\_2 \frac{\mathbf{Y}\_{\mathbf{k}+1} - \mathbf{u}}{\theta} & \text{if} \quad \mathbf{Y}\_{\mathbf{k}+1} > \mathbf{u}. \end{cases} \tag{39}$$

The expected cost for the (k+1)th period, E{C(u)}, is expressed as

$$\mathbb{E}\_{\theta} \left\{ \mathbb{C} \{ \mathbf{u} \} \right\} = \frac{1}{6} \Big| \mathop{\mathbf{c}}\_{\text{1}} \Big[ \mathop{\mathbf{u}} - \mathop{\mathbf{y}}\_{\text{k}+1} \big] \mathop{\mathbf{g}}\_{\theta} \big( \mathbf{y}\_{\text{k}+1} \big| \mathop{\mathbf{k}} \big) \mathrm{dy}\_{\text{k}+1} + \mathop{\mathbf{c}}\_{2} \Big[ \mathop{\mathbf{y}}\_{\text{k}+1} \big( \mathbf{y}\_{\text{k}+1} - \mathbf{u} \big) \big\mathfrak{g}\_{\theta} \big( \mathbf{y}\_{\text{k}+1} \big| \big( \mathbf{k} \big) \mathrm{dy}\_{\text{k}+1} \bigg]. \tag{40}$$

The function <sup>θ</sup> E {C(u)} can be shown to be convex in u, thus having a unique minimum. Taking the first derivative of <sup>θ</sup> E {C(u)} with respect to u and equating it to zero, we get

$$\frac{1}{\Theta} \left( \mathop{\mathbf{c}}\_{\mathbf{0}} \limits\_{\mathbf{0}}^{\mathbf{u}} \mathbf{g}\_{\theta} (\mathbf{y}\_{\mathbf{k}+1} \mid \mathbf{k}) \mathbf{d} \mathbf{y}\_{\mathbf{k}+1} - \mathop{\mathbf{c}}\_{\mathbf{0}} \limits\_{\mathbf{u}}^{\mathbf{v}} \mathbf{g}\_{\theta} (\mathbf{y}\_{\mathbf{k}+1} \mid \mathbf{k}) \mathbf{d} \mathbf{y}\_{\mathbf{k}+1} \right) = \mathbf{0} \tag{41}$$

or

194 Stochastic Modeling and Control

c = setup cost per order,

= parameter (in general, vector),

assumptions of the model are:

2. No setup cost is incurred.

(k+1)th period,

u = order quantity,

received.

q if q < u\*

u\*

c1= holding cost per held unit during the period, c2= penalty cost per shortage unit during the period,

q = inventory on hand before an order is placed.

**3.1. No-setup model (Newsvendor model)** 

holding and shortage costs. Given optimal u (= u\*

; otherwise, no order is placed.

form of the unknown demand distribution is made).

(where it is assumed that the demand distribution belongs to a parametric family of distributions) or a non-parametric approach (where no assumption regarding the parametric

Under the parametric approach, one may choose to estimate the unknown parameters or choose a prior distribution for the unknown parameters and apply the Bayesian approach to incorporating the demand data available. Scarf (1959) and Karlin (1960) consider a Bayesian framework for the unknown demand distribution. Specifically, assuming that the demand distribution belongs to the family of exponential distributions, the demand process is characterized by the prior distribution on the unknown parameter. Further extension of this approach is presented in (Azoury, 1985). Application of the Bayesian approach to the censored demand case is given in (Ding et al., 2002; Lariviere & Porteus, 1999). Parameter estimation is first considered in (Conrad, 1976) and recent developments are reported in (Agrawal & Smith, 1996; Nahmias, 1994). Liyanage & Shanthikumar (2005) propose the concept of operational

This section deals with inventory items that are in stock during a single time period. At the end of the period, leftover units, if any, are disposed of, as in fashion items. Two models are considered. The difference between the two models is whether or not a setup cost is incurred for placing an order. The symbols used in the development of the models include:

g (yk+1|k) = conditional probability density function of customer demand, Yk+1, during the

This model is known in the literature as the *newsvendor* model (the original classical name is the *newsboy* model). It deals with stocking and selling newspapers and periodicals. The

1. Demand occurs instantaneously at the start of the period immediately after the order is

The model determines the optimal value of u that minimizes the sum of the expected

If Yk+1 u, the quantity u Yk+1 is held during the (k+1)th period. Otherwise, a shortage

amount Yk+1 u will result if Yk+1> u. Thus, the cost per the (k+1)th period is

), the inventory policy calls for ordering

statistics and apply it to a single period newsvendor inventory control problem.

$$\mathbf{c}\_1 \mathbf{P}\_\theta \{ \mathbf{Y}\_{k+1} \le \mathbf{u} \} - \mathbf{c}\_2 (1 - \mathbf{P}\_\theta \{ \mathbf{Y}\_{k+1} \le \mathbf{u} \}) = \mathbf{0} \tag{42}$$

or

$$P\_{\theta} \{ \mathbf{Y}\_{k+1} \le \mathbf{u} \} = \frac{\mathbf{c}\_2}{\mathbf{c}\_1 + \mathbf{c}\_2}. \tag{43}$$

It follows from (31), (32), (40), and (43) that

$$\mathbf{u}^\* = \frac{\theta}{\mathbf{m} - \mathbf{k}} \ln \left( \mathbf{1} + \frac{\mathbf{c}\_2}{\mathbf{c}\_1} \right) \tag{44}$$

and

$$\mathrm{E}\_{\theta}\{\mathrm{C}(\mathbf{u}^{\*})\} = \frac{1}{\theta} \Big( \mathrm{c}\_{2}\mathrm{E}\_{\theta}\{\mathrm{Y}\_{\mathrm{k}+1}\} - (\mathrm{c}\_{1} + \mathrm{c}\_{2}) \Big\|\_{\mathrm{y}}^{\mathrm{u}} \mathrm{y}\_{\mathrm{k}+1} \mathrm{g}\_{\theta}(\mathrm{y}\_{\mathrm{k}+1} \,|\,\mathrm{k}) \mathrm{dy}\_{\mathrm{k}+1} \Big)$$

$$= \frac{\mathrm{c}\_{1}}{\mathrm{m} - \mathrm{k}} \ln \Big( 1 + \frac{\mathrm{c}\_{2}}{\mathrm{c}\_{1}} \Big). \tag{45}$$

*Parametric uncertainty.* Consider the case when the parameter is unknown. To find the best invariant decision rule BI u , we use the invariant embedding technique (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b) to transform (39) to the form, which is depended only on the pivotal quantities V, V1, and the ancillary factor . In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters whose probability distribution does not depend on unknown parameters. Note that a pivotal quantity need not be a statistic—the function and its value can depend on parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic.

Transformation of C(u) based on the pivotal quantities V, V1 is given by

$$\mathbf{C}^{(1)}(\eta \text{ \textquotedblleft}) = \begin{cases} \mathbf{c}\_1(\eta \text{V} - \mathbf{V}\_1) & \text{if} \quad \mathbf{V}\_1 \le \eta \text{V}, \\ \mathbf{c}\_2(\mathbf{V}\_1 - \eta \mathbf{V}) & \text{if} \quad \mathbf{V}\_1 > \eta \text{V}, \end{cases} \tag{46}$$

where

$$\mathbf{u}\eta = \frac{\mathbf{u}}{\mathbf{S}\_{\mathbf{k}}}\,'\tag{47}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 197

c (k 1) c <sup>1</sup> 1 E{C (η )}. mk c (54)

(1)

E{E {(C(u)}} E {C(u)}f(v)dv E {C (u)}, (56)

 

k 1

k k y 1

s s

(k 2)

(55)

(58)

(59)

(57)

and the expected cost, if we use uBI, is given by

equation (40) as follows:

Then it follows from (55) that

(1)

where

BI

u

u

u

s

cost per the (k+1)th period is reduced to

<sup>θ</sup> E {C(u )}

 

It will be noted that, on the other hand, the invariant embedding technique (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b) allows one to transform

θ 1 k1 θ k1 k1 2 k1 θ k1 k1

0 u <sup>1</sup> E {C(u)} c (u y )g (y |k)dy c (y u)g (y |k)dy <sup>θ</sup>

> 

k 0 u

Minimizing the expected prediction cost for the (k+1)th period,

<sup>k</sup>

(2) k

θ θ s 0

k 0 k k 1 1 v(m k)y c (u y )v (m k)exp dy s s s

 <sup>2</sup> k 1 2 k1 k 1 u k k v(m k)y <sup>1</sup> c (y u)v (m k)exp dy . s s

1/(k 1) 1 2 (1) 1

 

 

> 2 k 1 1 k1 k 1

<sup>k</sup>

kk k

s 1 k1 s k1 k1 2 k1 s k1 k1

represents the expected prediction cost for the (k+1)th period. It follows from (57) that the

k 1 1 k 1

2 k 1

u Y c if Y u, s /k C (u) Y u c if Y u, s /k

k 1

k

and the predictive probability density function of Yk+1 (compatible with (40)) is given by

 

g (y |k) (k 1)(m k) 1 (m k) , y 0.

s k1 k 1

<sup>k</sup> E {C (u)} c (u y )g (y |k)dy c (y u)g (y |k)dy

$$\mathbf{V}\_{1} = \frac{\mathbf{Y}\_{\mathbf{k}+1}}{\Theta} \sim \mathbf{g}(\mathbf{v}\_{1} \mid \mathbf{k}) = (\mathbf{m} - \mathbf{k}) \exp[- (\mathbf{m} - \mathbf{k})\mathbf{v}\_{1}] \quad \mathbf{v}\_{1} \ge \mathbf{0}. \tag{48}$$

Then E{C(1)()} is expressed as

$$\mathbb{E}[\mathbf{C}^{(1)}(\mathbf{r})] = \bigcap\_{\mathbf{0} \neq \mathbf{0}}^{\oplus} \left( \mathbf{c}\_{1} \int\_{0}^{\mathbf{r} \mathbf{v}} (\mathbf{\eta} \ \mathbf{v} - \mathbf{v}\_{1}) \mathbf{g}(\mathbf{v}\_{1} \ | \mathbf{k}) \mathbf{dv}\_{1} + \mathbf{c}\_{2} \int\_{\mathbf{r} \mathbf{v}}^{\mathbf{v}} (\mathbf{v}\_{1} - \mathbf{v}\_{2}) \mathbf{g}(\mathbf{v}\_{1} \ | \mathbf{k}) \mathbf{dv}\_{1} \right) \mathbf{f}(\mathbf{v}) \mathbf{dv}. \tag{49}$$

The function E{C(1)()} can be shown to be convex in , thus having a unique minimum. Taking the first derivative of E{C(1)()} with respect to and equating it to zero, we get

$$\int\_{0}^{\circ} \mathbf{v} \left( \mathbf{c}\_{1} \int\_{0}^{\eta \mathbf{v}} \mathbf{g}(\mathbf{v}\_{1} \mid \mathbf{k}) \mathbf{d} \mathbf{v}\_{1} - \mathbf{c}\_{2} \int\_{\eta \mathbf{v}}^{\circ} \mathbf{g}(\mathbf{v}\_{1} \mid \mathbf{k}) \mathbf{d} \mathbf{v}\_{1} \right) \mathbf{f}(\mathbf{v}) \mathbf{d} \mathbf{v} = \mathbf{0} \tag{50}$$

or

$$\frac{\int\_{0}^{\omega} \text{vP}(\mathbf{V}\_{1} \le \eta \mathbf{v}) \text{f}(\mathbf{v}) d\mathbf{v}}{\int\_{0}^{\omega} \text{vf}(\mathbf{v}) d\mathbf{v}} = \frac{\mathbf{c}\_{2}}{\mathbf{c}\_{1} + \mathbf{c}\_{2}}.\tag{51}$$

It follows from (47), (49), and (51) that the optimum value of is given by

$$\ln \mathbf{q}^\* = \frac{1}{\mathbf{m} - \mathbf{k}} \left[ \left( 1 + \frac{\mathbf{c}\_2}{\mathbf{c}\_1} \right)^{1/(\mathbf{k}+1)} - 1 \right] \tag{52}$$

the best invariant decision rule is

$$\mathbf{u}^{\rm BI} = \mathbf{r}\_{\parallel} \mathbf{S}\_{\mathbf{k}} = \frac{\mathbf{S}\_{\mathbf{k}}}{\mathbf{m} - \mathbf{k}} \left[ \left( 1 + \frac{\mathbf{c}\_2}{\mathbf{c}\_1} \right)^{1/(\mathbf{k}+1)} - 1 \right] \tag{53}$$

and the expected cost, if we use uBI, is given by

$$\mathbb{E}\_{\theta} \{ \mathbb{C}(\mathbf{u}^{\text{BI}}) \} \begin{aligned} &= \frac{\mathbf{c}\_{1}(\mathbf{k}+1)}{\mathbf{m}-\mathbf{k}} \Bigg[ \Big( 1 + \frac{\mathbf{c}\_{2}}{\mathbf{c}\_{1}} \Bigg)^{1/(\mathbf{k}+1)} - 1 \Bigg] = \mathbb{E} \{ \mathbb{C}^{(1)}(\mathbf{r}^{\*}) \}. \end{aligned} \tag{54}$$

It will be noted that, on the other hand, the invariant embedding technique (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b) allows one to transform equation (40) as follows:

$$\mathrm{E}\_{0}\{\mathbf{C}(\mathbf{u})\} = \frac{1}{\mathsf{0}} \Big( \mathrm{c}\_{1} \Big[ \mathrm{u} - \mathrm{y}\_{\mathrm{k}+1} \big] \mathrm{g}\_{0} (\mathrm{y}\_{\mathrm{k}+1} \mid \mathbf{k}) \mathrm{dy}\_{\mathrm{k}+1} + \mathrm{c}\_{2} \Big[ \mathrm{y}\_{\mathrm{k}+1} - \mathrm{u} \big] \mathrm{g}\_{0} (\mathrm{y}\_{\mathrm{k}+1} \mid \mathbf{k}) \mathrm{dy}\_{\mathrm{k}+1} \Big) \,\mathrm{

$$= \frac{1}{\mathsf{s}\_{\mathrm{k}}} \Big( \mathrm{c}\_{1} \Big[ \mathrm{u} - \mathrm{y}\_{\mathrm{k}+1} \big] \mathrm{v}^{2} (\mathrm{m} - \mathrm{k}) \exp \Big( - \frac{\mathrm{v} (\mathrm{m} - \mathrm{k}) \mathrm{y}\_{\mathrm{k}+1}}{\mathrm{s}\_{\mathrm{k}}} \Big) \frac{1}{\mathrm{s}\_{\mathrm{k}}} \mathrm{d} \mathbf{y}\_{\mathrm{k}+1} \Big] \,\mathrm{d} \mathbf{y}\_{\mathrm{k}+1} \,\tag{55}$$

$$+ \mathrm{c}\_{2} \Big[ \mathrm{y}\_{\mathrm{k}+1} - \mathrm{u} \big) \mathrm{v}^{2} (\mathrm{m} - \mathrm{k}) \exp \Big( - \frac{\mathrm{v} (\mathrm{m} - \mathrm{k}) \mathrm{y}\_{\mathrm{k}+1}}{\mathrm{s}\_{\mathrm{k}}} \Big) \frac{1}{\mathrm{s}\_{\mathrm{k}}} \mathrm{d} \mathbf{y}\_{\mathrm{k}+1} \Big]. \tag{56}$$
$$

Then it follows from (55) that

$$\mathrm{E}\{\mathrm{E}\_{\theta}\{\mathrm{\{C(u)\}}\} = \oint\_{0}^{\infty} \mathrm{E}\_{\theta}\{\mathrm{C(u)}\} \mathrm{f}\{\mathrm{v}\} \mathrm{dv} = \mathrm{E}\_{s\_{k}} \{\mathrm{C}^{\{1\}}(\mathrm{u})\} . \tag{56}$$

where

196 Stochastic Modeling and Control

an ancillary statistic.

Then E{C(1)()} is expressed as

(1)

the best invariant decision rule is

()

ηv

ηv

where

or

parameters whose probability distribution does not depend on unknown parameters. Note that a pivotal quantity need not be a statistic—the function and its value can depend on parameters of the model, but its distribution must not. If it is a statistic, then it is known as

(1) 11 1

 k u

 k 1 1 1 1 1

 

The function E{C(1)()} can be shown to be convex in , thus having a unique minimum. Taking the first derivative of E{C(1)()} with respect to and equating it to zero, we get

1 1 1 12 1 1 1

E{C } c ( η η v v )g(v |k)dv c (v ηv)g(v |k)dv f(v)dv. (49)

<sup>Y</sup> V ~ g(v |k) (m k)exp[ (m k)v ], v 0. θ (48)

v c g(v |k)dv c g(v |k)dv f(v)dv 0 (50)

1/(k 1)

S c

2 1 1 2

<sup>η</sup> <sup>1</sup> 1 , mk c (52)

<sup>u</sup> <sup>η</sup> S 1 1 , mk c (53)

1/(k 1)

1

<sup>c</sup> . c c

(51)

2 1 1 c (η V ) if V ηV, <sup>C</sup> <sup>η</sup> c (V ηV) if V ηV, ) 

<sup>V</sup> ( (46)

<sup>η</sup> , S (47)

Transformation of C(u) based on the pivotal quantities V, V1 is given by

0 0 ηv

0 0 ηv

1

0

It follows from (47), (49), and (51) that the optimum value of is given by

vP(V ηv)f(v)dv

vf(v)dv

1 c

BI k 2 k

1 1 12 1 1

0 2

$$\mathbf{E}\_{\mathbf{s}\_{\mathbf{k}}}\left\{\mathbf{C}^{(\mathrm{I})}(\mathbf{u})\right\} = \frac{\mathbf{k}}{\mathbf{s}\_{\mathbf{k}}} \Bigg\{\mathbf{c}\_{\mathbf{l}} \Bigg\} \Big(\mathbf{u} - \mathbf{y}\_{\mathbf{k}+1}\big) \mathbf{g}\_{\mathbf{s}\_{\mathbf{k}}}^{\bullet}\Big(\mathbf{y}\_{\mathbf{k}+1} \big|\mathbf{k}\big) \mathbf{dy}\_{\mathbf{k}+1} + \mathbf{c}\_{\mathbf{2}} \big[\mathbf{y}\_{\mathbf{k}+1} - \mathbf{u}\big] \mathbf{g}\_{\mathbf{s}\_{\mathbf{k}}}^{\bullet}\Big(\mathbf{y}\_{\mathbf{k}+1} \big|\mathbf{k}\big) \mathbf{dy}\_{\mathbf{k}+1} \Bigg\} \tag{57}$$

represents the expected prediction cost for the (k+1)th period. It follows from (57) that the cost per the (k+1)th period is reduced to

$$\mathbf{C}^{(2)}(\mathbf{u}) = \begin{cases} \mathbf{c}\_1 \frac{\mathbf{u} - \mathbf{Y}\_{\mathbf{k}+1}}{\mathbf{s}\_\mathbf{k} / \mathbf{k}} & \text{if} \quad \mathbf{Y}\_{\mathbf{k}+1} \le \mathbf{u}\_\prime \\\\ \mathbf{c}\_2 \frac{\mathbf{Y}\_{\mathbf{k}+1} - \mathbf{u}}{\mathbf{s}\_\mathbf{k} / \mathbf{k}} & \text{if} \quad \mathbf{Y}\_{\mathbf{k}+1} > \mathbf{u}\_\prime \end{cases} \tag{58}$$

and the predictive probability density function of Yk+1 (compatible with (40)) is given by

$$\mathbf{g\_{s\_k}^{\bullet}}(\mathbf{y\_{k+1}}\mid\mathbf{k}) = (\mathbf{k}+1)(\mathbf{m}-\mathbf{k}) \left[1+(\mathbf{m}-\mathbf{k})\frac{\mathbf{y\_{k+1}}}{\mathbf{s\_k}}\right]^{-(\mathbf{k}+2)} \frac{1}{\mathbf{s\_k}},\ \mathbf{y\_{k+1}}\geq 0. \tag{59}$$

Minimizing the expected prediction cost for the (k+1)th period,

#### 198 Stochastic Modeling and Control

$$\mathbb{E}\_{\mathbf{s}\_{\mathbf{k}}}\{\mathbf{C}^{(2)}(\mathbf{u})\} = \frac{\mathbf{k}}{\mathbf{s}\_{\mathbf{k}}} \Big| \Big\{ \mathbf{c}\_{1} \Big[ (\mathbf{u} - \mathbf{y}\_{\mathbf{k}+1}) \mathbf{g}\_{\mathbf{s}\_{\mathbf{k}}}^{\bullet}(\mathbf{y}\_{\mathbf{k}+1} \mid \mathbf{k}) \mathbf{d} \mathbf{y}\_{\mathbf{k}+1} + \mathbf{c}\_{2} \Big[ (\mathbf{y}\_{\mathbf{k}+1} - \mathbf{u}) \mathbf{g}\_{\mathbf{s}\_{\mathbf{k}}}^{\bullet}(\mathbf{y}\_{\mathbf{k}+1} \mid \mathbf{k}) \mathbf{d} \mathbf{y}\_{\mathbf{k}+1} \Big] \Big\}, \tag{60}$$

with respect to u, we obtain uBI immediately, and

$$\mathbb{E}\_{\mathbf{s}\_{\mathbf{k}}}\{\mathbf{C}^{\langle 2\rangle}\langle \mathbf{u}^{\text{BI}}\rangle\} = \frac{\mathbf{c}\_{1}(\mathbf{k}+1)}{\mathbf{m}-\mathbf{k}} \left[ \left(1 + \frac{\mathbf{c}\_{2}}{\mathbf{c}\_{1}}\right)^{1/(\mathbf{k}+1)} - 1 \right].\tag{61}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 199

θ θ = E {C(u )} E {C(u )} 0.838. (68)

must satisfy (43). Because c is constant, the

and may be

it follows from the above that ML u is inadmissible in relation to BI u . If, say, k=1 and

Thus, in this case, the use of BI u leads to a reduction in the expected cost of about 16.2 % as

The present model differs from the one in Section 3.1 in that a setup cost c is incurred. Using

θ θ E {C(u)} c E {C(u)}

 

<sup>1</sup> c c (u <sup>y</sup> )g (y |k)dy c (y u)g (y |k)dy . θ (69)

. In Fig.1,

θθ θ E {C(s)} E {C(S)} c E {C(S)}, s S. (70)

 

1 k1 θ k1 k1 2 k1 θ k1 k1

ML BI Rel.eff.{u , u ,q} BI ML

compared with ML u . The absolute expected cost will be proportional to

the same notation, the total expected cost per the (k+1)th period is

0 u

*S*) optimal ordering policy in a single-period model with setup cost

Assume that q is the amount on hand before an order is placed. How much should be ordered? This question is answered under three conditions: 1) q < s; 2) s q S; 3) q > S.

100 2 1 c /c , we have that

**3.2. Setup model (s-S policy)** 

u

As shown in Section 3.1, the optimum value u\*

minimum value of <sup>θ</sup> E {C(u)} must also occur at u\*

, and the value of s (< S) is determined from the equation

The equation yields another value s1 (> S), which is discarded.

considerable.

S = u\*

**Figure 1.** (*s*

It should be remarked that the cost per the (k+1)th period, (2) C (u), can also be transformed to

$$\mathbf{C}^{(3)}(\mathfrak{n}) = \begin{cases} \mathbf{c}\_1 \mathbf{k} \left( \frac{\mathbf{u}}{\mathbf{s}\_k} - \frac{\mathbf{Y}\_{k+1}}{\mathbf{s}\_k} \right) & \text{if } \quad \frac{\mathbf{Y}\_{k+1}}{\mathbf{s}\_k} \le \frac{\mathbf{u}}{\mathbf{s}\_k} \\\ \mathbf{c}\_2 \mathbf{k} \left( \frac{\mathbf{Y}\_{k+1}}{\mathbf{s}\_k} - \frac{\mathbf{u}}{\mathbf{s}\_k} \right) & \text{if } \quad \frac{\mathbf{Y}\_{k+1}}{\mathbf{s}\_k} > \frac{\mathbf{u}}{\mathbf{s}\_k} \\\ \mathbf{c}\_2 \mathbf{k} \left( \frac{\mathbf{Y}\_{k+1}}{\mathbf{s}\_k} - \frac{\mathbf{u}}{\mathbf{s}\_k} \right) & \text{if } \quad \frac{\mathbf{Y}\_{k+1}}{\mathbf{s}\_k} > \frac{\mathbf{u}}{\mathbf{s}\_k} \end{cases} = \begin{cases} \mathbf{c}\_1 \mathbf{k}(\mathfrak{n} - \mathbf{W}) & \text{if } \quad \mathbf{W} \le \mathfrak{n} \\\ \mathbf{c}\_2 \mathbf{k}(\mathbf{W} - \mathfrak{n}) & \text{if } \quad \mathbf{W} > \mathfrak{n} \end{cases} \tag{62}$$

where the probability density function of the ancillary statistic W (compatible with (40)) is given by

$$\log\_{s\_k}^{\circ}(\mathbf{w} \mid \mathbf{k}) = (\mathbf{k} + 1)(\mathbf{m} - \mathbf{k}) \left[ 1 + (\mathbf{m} - \mathbf{k})\mathbf{w} \right]^{-(\mathbf{k} + 2)}, \text{ w} \ge 0. \tag{63}$$

Then the best invariant decision rule BI <sup>k</sup> <sup>u</sup> <sup>η</sup> S , where <sup>η</sup> minimizes

$$\mathbb{E}\{\mathbf{C}^{(\mathfrak{J})}\mathbf{(\eta)}\} = \mathbb{k}\left(\underset{\mathfrak{o}}{\mathrm{c}}\limits\_{\mathfrak{l}}^{\mathfrak{h}}(\mathfrak{q}-\mathbf{w})\mathbf{g}^{\circ}(\mathbf{w}\mid\mathbf{k})\mathrm{d}\mathbf{w} + \underset{\mathfrak{o}}{\mathrm{c}}\limits\_{\mathfrak{l}}^{\overline{\mathfrak{o}}}(\mathbf{w}-\mathfrak{q})\mathbf{g}^{\circ}(\mathbf{w}\mid\mathbf{k})\mathrm{d}\mathbf{w}\right). \tag{64}$$

*Comparison of statistical decision rules.* For comparison, consider the maximum likelihood decision rule that may be obtained from (44),

$$\mathbf{u}^{\text{ML}} = \frac{\hat{\boldsymbol{\theta}}}{\mathbf{m} - \mathbf{k}} \ln \left( \mathbf{1} + \frac{\mathbf{c}\_2}{\mathbf{c}\_1} \right) = \mathbf{\eta}\_{\text{j}}^{\text{ML}} \mathbf{S}\_{\text{k} \text{ } \boldsymbol{\prime}} \tag{65}$$

where <sup>k</sup> θ S /k is the maximum likelihood estimator of ,

$$\left| \mathbf{n} \right|^{\text{ML}} = \frac{1}{\mathbf{m} - \mathbf{k}} \ln \left( 1 + \frac{\mathbf{c}\_2}{\mathbf{c}\_1} \right)^{1/\mathbf{k}}.\tag{66}$$

Since BI u and ML u belong to the same class,

$$\mathbf{C} = \{ \mathbf{u} : \mathbf{u} = \mathbf{r} \mathbf{S}\_k \} \tag{67}$$

it follows from the above that ML u is inadmissible in relation to BI u . If, say, k=1 and 100 2 1 c /c , we have that

$$\text{Rel.eff.}\{\mathbf{u}^{\text{ML}}, \mathbf{u}^{\text{BI}}, \mathbf{q}\} = \text{E}\_{\theta}\{\mathbf{C}(\mathbf{u}^{\text{BI}})\} \Big/ \text{E}\_{\theta}\{\mathbf{C}(\mathbf{u}^{\text{ML}})\} = 0.838.\tag{68}$$

Thus, in this case, the use of BI u leads to a reduction in the expected cost of about 16.2 % as compared with ML u . The absolute expected cost will be proportional to and may be considerable.

#### **3.2. Setup model (s-S policy)**

,

(62)

(60)

 

 

1 c (k 1) c <sup>1</sup> 1 .

1/(k 1)

(k 2)

<sup>s</sup> g (w|k) (k 1)(m k) 1 (m k)w , w 0. (63)

<sup>k</sup> <sup>u</sup> <sup>η</sup> S , where <sup>η</sup> minimizes

 

E{C (η)} k c (η w)g (w|k)dw c (w η)g (w|k)dw . (64)

j k

1/k

1

mk c , (65)

mk c (66)

C {u : u <sup>k</sup> ηS }, (67)

mk c (61)

kk k

1 2

It should be remarked that the cost per the (k+1)th period, (2) C (u), can also be transformed

k 1 k 1 2

ss ss c k(η W) if W <sup>η</sup> C (η) Y Y u u c k(W η) if W η, c k if

where the probability density function of the ancillary statistic W (compatible with (40)) is

kk kk

ss ss

1 2 0 η

*Comparison of statistical decision rules.* For comparison, consider the maximum likelihood

ML 2 ML

1 c η ln 1 .

ML 2

<sup>θ</sup> <sup>c</sup> u ln 1 <sup>η</sup> <sup>S</sup>

1

k1 k1

(3) kk kk 1

u u Y Y c k if

η

<sup>k</sup> θ S /k is the maximum likelihood estimator of ,

s 1 k1 s k1 k1 2 k1 s k1 k1

<sup>k</sup> E {C (u)} c (u y )g (y |k)dy c (y u)g (y |k)dy

k 0 u

198 Stochastic Modeling and Control

(2)

to

given by

where 

u

with respect to u, we obtain uBI immediately, and

k

1

2

k

Then the best invariant decision rule BI

decision rule that may be obtained from (44),

Since BI u and ML u belong to the same class,

(3)

(2) BI <sup>s</sup> E {C (u )}

s

The present model differs from the one in Section 3.1 in that a setup cost c is incurred. Using the same notation, the total expected cost per the (k+1)th period is

$$\mathcal{E}\_{\theta} \{ \mathbf{C}(\mathbf{u}) \} = \mathbf{c} + \mathcal{E}\_{\theta} \{ \mathbf{C}(\mathbf{u}) \}$$

$$\mathbf{y} = \mathbf{c} + \frac{1}{\theta} \Big| \mathbf{c}\_1 \Big[ (\mathbf{u} - \mathbf{y}\_{k+1}) \mathbf{g}\_{\theta} (\mathbf{y}\_{k+1} \mid \mathbf{k}) \mathbf{d} \mathbf{y}\_{k+1} + \mathbf{c}\_2 \Big[ (\mathbf{y}\_{k+1} - \mathbf{u}) \mathbf{g}\_{\theta} (\mathbf{y}\_{k+1} \mid \mathbf{k}) \mathbf{d} \mathbf{y}\_{k+1} \Big]. \tag{69}$$

As shown in Section 3.1, the optimum value u\* must satisfy (43). Because c is constant, the minimum value of <sup>θ</sup> E {C(u)} must also occur at u\* . In Fig.1,

S = u\* , and the value of s (< S) is determined from the equation

**Figure 1.** (*sS*) optimal ordering policy in a single-period model with setup cost

$$\mathbb{E}\_{\theta} \{ \mathbf{C}(\mathbf{s}) \} = \mathbb{E}\_{\theta} \{ \overline{\mathbf{C}}(\mathbf{S}) \} = \mathbf{c} + \mathbb{E}\_{\theta} \{ \mathbf{C}(\mathbf{S}) \}, \quad \mathbf{s} < \mathbf{S}. \tag{70}$$

The equation yields another value s1 (> S), which is discarded.

Assume that q is the amount on hand before an order is placed. How much should be ordered? This question is answered under three conditions: 1) q < s; 2) s q S; 3) q > S.

*Case 1* (q < s). Because q is already on hand, its equivalent cost is given by <sup>θ</sup> E {C(q)}. If any additional amount u q (u > q) is ordered, the corresponding cost given u is <sup>θ</sup> E {C(u)} , which includes the setup cost c. From Fig. 1, we have

$$\min\_{\mathbf{u}\sim\mathbf{q}} \mathcal{E}\_{\theta} \{ \overline{\mathcal{C}}(\mathbf{u}) \} = \mathcal{E}\_{\theta} \{ \overline{\mathcal{C}}(\mathbf{S}) \} < \mathcal{E}\_{\theta} \{ \mathcal{C}(\mathbf{q}) \}. \tag{71}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 201

An airline, typically, offers tickets for many origin–destination itineraries in various fare classes. These fare classes not only include business and economy class, which are settled in separate parts of the plane, but also include fare classes for which the difference in fares is explained by different conditions regarding for example cancellation options or overnight stay arrangements. Therefore the seats on a flight are products, which can be offered to different customer segments for different prices. Since the tickets for a flight have to be sold before the plane takes off, the product is perishable and cannot be stored for future use. The

At the heart of airline revenue management lies the airline seat inventory control problem. It is common practice for airlines to sell a pool of identical seats at different prices according to different booking classes to improve revenues in a very competitive market. In other words, airlines sell the same seat at different prices according to different types of travelers (first class, business and economy) and other conditions. The question then arises whether to offer seats at a relatively low price at a given time with a given number of seats remaining or to wait for the possible arrival of a higher paying customer. Assigning seats in the same compartment to different fare classes of passengers in order to improve revenues is a major problem of airline seat inventory control. This problem has been considered in numerous papers. For details, the reader is referred to a review of yield management, as well as perishable asset revenue management, by Weatherford et al. (1993), and a review of relevant

The problem of finding an optimal airline seat inventory control policy for multi-leg flight with multiple fare classes, which allows one to maximize the expected profit of this flight, is one of the most difficult problems of air transport logistics. On the one hand, one must have reasonable assurance that the requirements of customers for reservations will be met under most circumstances. On the other hand, one is confronted with the limitation of the capacity of the cabin, as well as with a host of other less important constraints. The problem is normally solved by the application of judgment based on past experience. The question arises whether or not it is possible to construct a simple mathematical theory of the above problem, which will allow one better to use the available data based upon airline statistics. Two models (dynamic model and static one) of airline data may be considered. In the dynamic model, the problem is formulated as a sequential decision process. In this case, an optimal dynamic reservation policy is used at each stage prior to departure time for multileg flights with several classes of passenger service. The essence of determining the optimal dynamic reservation policy is maximization of the expected gain of the flight, which is carried out at each stage prior to departure time using the available data. The term (dynamic reservation policy) is used in this paper to mean a decision rule, based on the available data, for determining whether to accept a given reservation request made at a particular time for some future date. An optimal static reservation policy is based on the static model. The models proposed here contain a simple and natural treatment of the airline reservation

same is true for most other service industries, such as hotels, hospitals and schools.

**4.1. Airline seat inventory control** 

mathematical models by Belobaba (1987).

process and may be appropriate in practice.

Thus, the optimal inventory policy in this case is to order S q units.

*Case 2* (s q S). From Fig. 1, we have

$$\mathbb{E}\_{\theta} \{ \mathbf{C}(\mathbf{q}) \} \le \min\_{\mathbf{u} > \mathbf{q}} \mathbb{E}\_{\theta} \{ \overline{\mathbf{C}}(\mathbf{u}) \} = \mathbb{E}\_{\theta} \{ \overline{\mathbf{C}}(\mathbf{S}) \}. \tag{72}$$

Thus, it is not advantageous to order in this case and u\* = q.

*Case 3* (q > S). From Fig. 1, we have for u > q,

$$\mathbb{E}\_{\theta} \{ \mathbf{C}(\mathbf{q}) \} < \mathbb{E}\_{\theta} \{ \overline{\mathbf{C}}(\mathbf{u}) \}. \tag{73}$$

This condition indicates that, as in case (2), is not advantageous to place an order that is, u\* = q.

The optimal inventory policy, frequently referred to as the *s S policy*, is summarized as

$$\begin{aligned} \text{If } \mathbf{x} < \mathbf{S}, \text{order} & \mathbf{S} - \mathbf{x}, \\ \text{If } \mathbf{x} \ge \mathbf{s}, \text{do not order.} \end{aligned} \tag{74}$$

The optimality of the s S policy is guaranteed because the associated cost function is convex.

*Parametric uncertainty.* In the case when the parameter is unknown, the total expected prediction cost for the (*k*+1)th period,

$$\mathbb{E}\_{\mathbf{s}\_k} \{ \overline{\mathbf{C}}^{\{\mathbf{l}\}} (\mathbf{u}) \} = \mathbf{c} + \mathbb{E}\_{\mathbf{s}\_k} \{ \mathbf{C}^{\{\mathbf{l}\}} (\mathbf{u}) \}$$

$$= \mathbf{c} + \frac{\mathbf{k}}{\mathbf{s}\_k} \left( \mathbf{c}\_1 \Big[ \mathbf{u} - \mathbf{y}\_{k+1} \big] \mathbf{g}\_{\mathbf{s}\_k}^\bullet \{ \mathbf{y}\_{k+1} \big| \, \mathbf{k} \big] \mathbf{dy}\_{k+1} + \mathbf{c}\_2 \Big[ \big( \mathbf{y}\_{k+1} - \mathbf{u} \big) \mathbf{g}\_{\mathbf{s}\_k}^\bullet \{ \mathbf{y}\_{k+1} \big| \, \mathbf{k} \big] \mathbf{dy}\_{k+1} \right) \,\tag{75}$$

is considered in the same manner as above.

#### **4. Airline revenue management problem**

The process of revenue management has become extremely important within the airline industry. It consists of setting fares, setting overbooking limits, and controlling seat inventory to increase revenues. It has allowed the airlines to survive deregulation by allowing them to respond to competitors' deep discount fares on a rational basis.

An airline, typically, offers tickets for many origin–destination itineraries in various fare classes. These fare classes not only include business and economy class, which are settled in separate parts of the plane, but also include fare classes for which the difference in fares is explained by different conditions regarding for example cancellation options or overnight stay arrangements. Therefore the seats on a flight are products, which can be offered to different customer segments for different prices. Since the tickets for a flight have to be sold before the plane takes off, the product is perishable and cannot be stored for future use. The same is true for most other service industries, such as hotels, hospitals and schools.

#### **4.1. Airline seat inventory control**

200 Stochastic Modeling and Control

u\* = q.

convex.

which includes the setup cost c. From Fig. 1, we have

*Case 2* (s q S). From Fig. 1, we have

*Case 3* (q > S). From Fig. 1, we have for u > q,

Thus, the optimal inventory policy in this case is to order S q units.

u q

Thus, it is not advantageous to order in this case and u\* = q.

*Parametric uncertainty.* In the case when the parameter

prediction cost for the (*k*+1)th period,

u

is considered in the same manner as above.

**4. Airline revenue management problem** 

*Case 1* (q < s). Because q is already on hand, its equivalent cost is given by <sup>θ</sup> E {C(q)}. If any additional amount u q (u > q) is ordered, the corresponding cost given u is <sup>θ</sup> E {C(u)} ,

θ θθ

This condition indicates that, as in case (2), is not advantageous to place an order that is,

The optimality of the s S policy is guaranteed because the associated cost function is

If x S, order S x,

The optimal inventory policy, frequently referred to as the *s S policy*, is summarized as

 k k (1) (1) s s E {C (u)} c E {C (u)}

k 0 u

 θ θθ u q

min E {C(u)} E {C(S)} E {C(q)}. (71)

E {C(q)} min E {C(u)} E {C(S)}. (72)

θ θ E {C(q)} E {C(u)}. (73)

If x s, do not order. (74)

is unknown, the total expected

(75)

 

 k k

<sup>k</sup> c c (u <sup>y</sup> )g (y |k)dy c (y u)g (y |k)dy , <sup>s</sup>

The process of revenue management has become extremely important within the airline industry. It consists of setting fares, setting overbooking limits, and controlling seat inventory to increase revenues. It has allowed the airlines to survive deregulation by

allowing them to respond to competitors' deep discount fares on a rational basis.

1 k1 s k1 k1 2 k1 s k1 k1

At the heart of airline revenue management lies the airline seat inventory control problem. It is common practice for airlines to sell a pool of identical seats at different prices according to different booking classes to improve revenues in a very competitive market. In other words, airlines sell the same seat at different prices according to different types of travelers (first class, business and economy) and other conditions. The question then arises whether to offer seats at a relatively low price at a given time with a given number of seats remaining or to wait for the possible arrival of a higher paying customer. Assigning seats in the same compartment to different fare classes of passengers in order to improve revenues is a major problem of airline seat inventory control. This problem has been considered in numerous papers. For details, the reader is referred to a review of yield management, as well as perishable asset revenue management, by Weatherford et al. (1993), and a review of relevant mathematical models by Belobaba (1987).

The problem of finding an optimal airline seat inventory control policy for multi-leg flight with multiple fare classes, which allows one to maximize the expected profit of this flight, is one of the most difficult problems of air transport logistics. On the one hand, one must have reasonable assurance that the requirements of customers for reservations will be met under most circumstances. On the other hand, one is confronted with the limitation of the capacity of the cabin, as well as with a host of other less important constraints. The problem is normally solved by the application of judgment based on past experience. The question arises whether or not it is possible to construct a simple mathematical theory of the above problem, which will allow one better to use the available data based upon airline statistics. Two models (dynamic model and static one) of airline data may be considered. In the dynamic model, the problem is formulated as a sequential decision process. In this case, an optimal dynamic reservation policy is used at each stage prior to departure time for multileg flights with several classes of passenger service. The essence of determining the optimal dynamic reservation policy is maximization of the expected gain of the flight, which is carried out at each stage prior to departure time using the available data. The term (dynamic reservation policy) is used in this paper to mean a decision rule, based on the available data, for determining whether to accept a given reservation request made at a particular time for some future date. An optimal static reservation policy is based on the static model. The models proposed here contain a simple and natural treatment of the airline reservation process and may be appropriate in practice.

#### **4.2. Expected marginal seat revenue model (EMSR)**

Different approaches were developed for solving the airline seat inventory control problem. The most important and widely used model – EMSR – was originally proposed by Littlewood (1972). To explain the basic ideas of the Littlewood model, we will consider the seat allocation problem on a nonstop flight with two airfares. We denote by U the number of seats in the aircraft, by uL the number of seats reserved for passengers with a lower fare and by p the probability that a passenger who would pay the higher fare cannot find a seat because it was sold to a passenger paying a lower fare. A rise in variable uL means a rise in probability p. A rise in the number of seats for passengers with the lower fare, i.e. a rise in variable p, decreases the number of seats for passengers with the higher fare, i.e., variable U – uL decreases. A reduction in variable U – uL increases the probability that a passenger paying the higher fare cannot find a vacant seat on their desired flight because it has been sold to a passenger with a lower fare. Probability p is given by

$$\mathbf{p} = \int\_{\mathbf{U} - \mathbf{u}\_{\perp}}^{\mathbf{o}} \mathbf{f}\_{\boldsymbol{\theta}}(\mathbf{x}) d\mathbf{x},\tag{76}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 203

<sup>δ</sup> x x f(x|β,δ) exp (x 0), ββ β (82)

<sup>h</sup> h z β, (83)

12 r z (z , z , ..., z ) , (85)

<sup>X</sup> Z , i 1, ..., r, β (86)

δ are the maximum likelihood estimators of and based on the first r

β x (n r)x r , (87)

γ, (84)

Richter (1982) gave a marginal analysis, which proved that (77) gives an optimal allocation (assuming certain continuity conditions). Optimal policies for more than two classes have been presented independently by Curry (1990), Wollmer (1992), Brumelle & McGill (1993),

*Parametric uncertainty.* In order to solve (79) under parametric uncertainty, it can be used, for

*Theorem 4*. Let X1 ... Xr be the first r ordered past observations from a previous sample of

 δ 1 δ

where both distribution parameters (β – scale, - shape) are positive. Then a lower onesided conditional prediction limit h on the lth order statistic <sup>f</sup> Xl in a set of m future

> 1/δ

 <sup>f</sup> (r) f 1/δ (r) <sup>l</sup> l h Pr{X h|z } Pr{X z β|z }

<sup>r</sup> <sup>r</sup> <sup>r</sup> r 2 vv v 2i i r 2

are ancillary statistics, any r2 of which form a functionally independent set (for notational convenience we include all of z1, …, zr in (85); zr-1 and zr can be expressed as function of z1,

ordered past observations (X1 … Xr) from a sample of size n from the two-parameter

 

 1/δ <sup>r</sup> δ δ i r

 δ i

v z z (n r)z dv

<sup>r</sup> <sup>r</sup> l1 k <sup>r</sup> r 2 <sup>v</sup> <sup>j</sup> <sup>v</sup> v v 2 i hi r 2

m k v z ( 1) (m k j)z z (n r)z dv k j

2 2 2 2

 

22 2

0 i 1 k0 j0 i 1

0 i 1 i 1

i

i 1

(r)

Weibull distribution, which can be found from solution of

and Nechval et al. (2006).

example, the following results.

where zh satisfies the equation

…, zr-2 only),

β and

size n from the two-parameter Weibull distribution

ordered observations also from the distribution (82) is given by

where the ( ) *f x* is the underlying probability density function for the total number of requests *X* of passengers who would pay the higher fare, is the parameter (in general, vector). Littlewood (1972) proposed the following way to determine the reservation level *uL* to which reservations are accepted for passengers with the lower fare. We denote by *c*1 the revenue from passengers with the lower fare and by *c*2 the revenue from passengers with the higher fare. Let *uL* seats be reserved for low fare passengers. The revenue per lower fare seat is *c*1. Expected revenue from a potential high fare passenger is c2p. Passengers with lower fares should be accepted until

$$\mathbf{c}\_1 \ge \mathbf{c}\_2 \mathbf{p}\_{\prime} \tag{77}$$

i.e.,

$$\mathbf{p} \le \frac{\mathbf{c}\_1}{\mathbf{c}\_2}.\tag{78}$$

Reservation level *uL* is determined by solving the following equality:

$$\Pr\left\{\left.\chi>\text{h}\right\}=\chi\right.\tag{79}$$

where

$$\mathbf{h} = \mathbf{U} - \mathbf{u}\_{\mathrm{L}'} \tag{80}$$

$$\chi = \frac{\mathbf{c}\_1}{\mathbf{c}\_2}.\tag{81}$$

Richter (1982) gave a marginal analysis, which proved that (77) gives an optimal allocation (assuming certain continuity conditions). Optimal policies for more than two classes have been presented independently by Curry (1990), Wollmer (1992), Brumelle & McGill (1993), and Nechval et al. (2006).

*Parametric uncertainty.* In order to solve (79) under parametric uncertainty, it can be used, for example, the following results.

*Theorem 4*. Let X1 ... Xr be the first r ordered past observations from a previous sample of size n from the two-parameter Weibull distribution

$$\text{if } \mathbf{f}(\mathbf{x} \mid \boldsymbol{\beta}, \boldsymbol{\delta}) = \frac{\boldsymbol{\delta}}{\boldsymbol{\beta}} \left( \frac{\mathbf{x}}{\boldsymbol{\beta}} \right)^{\boldsymbol{\delta} - 1} \exp \left[ - \left( \frac{\mathbf{x}}{\boldsymbol{\beta}} \right)^{\boldsymbol{\delta}} \right] \text{ ( $\mathbf{x} > \mathbf{0}$ ),}\tag{82}$$

where both distribution parameters (β – scale, - shape) are positive. Then a lower onesided conditional prediction limit h on the lth order statistic <sup>f</sup> Xl in a set of m future ordered observations also from the distribution (82) is given by

$$\mathbf{h} = \mathbf{z}\_{\mathbf{h}}^{1/\bar{\delta}} \hat{\boldsymbol{\beta}} \tag{83}$$

where zh satisfies the equation

202 Stochastic Modeling and Control

where the ( ) *f x*

i.e.,

where

fares should be accepted until

**4.2. Expected marginal seat revenue model (EMSR)** 

sold to a passenger with a lower fare. Probability p is given by

requests *X* of passengers who would pay the higher fare,

Reservation level *uL* is determined by solving the following equality:

Different approaches were developed for solving the airline seat inventory control problem. The most important and widely used model – EMSR – was originally proposed by Littlewood (1972). To explain the basic ideas of the Littlewood model, we will consider the seat allocation problem on a nonstop flight with two airfares. We denote by U the number of seats in the aircraft, by uL the number of seats reserved for passengers with a lower fare and by p the probability that a passenger who would pay the higher fare cannot find a seat because it was sold to a passenger paying a lower fare. A rise in variable uL means a rise in probability p. A rise in the number of seats for passengers with the lower fare, i.e. a rise in variable p, decreases the number of seats for passengers with the higher fare, i.e., variable U – uL decreases. A reduction in variable U – uL increases the probability that a passenger paying the higher fare cannot find a vacant seat on their desired flight because it has been

p f (x)dx, (76)

1 2 c c p, (77)

Pr X h γ, (79)

, <sup>L</sup> hUu (80)

(81)

(78)

is the parameter (in general,

 L θ U u

vector). Littlewood (1972) proposed the following way to determine the reservation level *uL* to which reservations are accepted for passengers with the lower fare. We denote by *c*1 the revenue from passengers with the lower fare and by *c*2 the revenue from passengers with the higher fare. Let *uL* seats be reserved for low fare passengers. The revenue per lower fare seat is *c*1. Expected revenue from a potential high fare passenger is c2p. Passengers with lower

> <sup>1</sup> 2 c p . c

> <sup>1</sup> 2 c <sup>γ</sup> . <sup>c</sup>

is the underlying probability density function for the total number of

$$\Pr\{\mathbf{Y}\_1^\ell > \mathbf{h} \mid \mathbf{z}^{(t)}\} = \Pr\{\mathbf{X}\_1^\ell > \mathbf{z}\_1^{1/\delta} \hat{\boldsymbol{\beta}} \mid \mathbf{z}^{(t)}\}$$

$$= \frac{\int\_0^\infty \mathbf{v}\_1^{r-2} \prod\_{i=1}^\tau \mathbf{z}\_i^{\mathbf{v}\_2} \sum\_{k=0}^{1-1} \binom{\mathbf{m}}{k} \sum\_{j=0}^k \binom{\mathbf{k}}{j} (-\mathbf{l})^j \left( (\mathbf{m} - \mathbf{k} + \mathbf{j}) \mathbf{z}\_1^{\mathbf{v}\_2} + \sum\_{i=1}^\tau \mathbf{z}\_i^{\mathbf{v}\_2} + (\mathbf{n} - \mathbf{r}) \mathbf{z}\_r^{\mathbf{v}\_2} \right)^{-\tau} \mathrm{d}\mathbf{v}\_2}{\int\_0^\infty \mathbf{v}\_2^{r-2} \prod\_{i=1}^\tau \mathbf{z}\_i^{\mathbf{v}\_2} \left( \sum\_{i=1}^\tau \mathbf{z}\_i^{\mathbf{v}\_2} + (\mathbf{n} - \mathbf{r}) \mathbf{z}\_r^{\mathbf{v}\_2} \right)^{-\tau} \mathrm{d}\mathbf{v}\_2} = \mathbf{y}\_\prime \tag{84}$$

$$\mathbf{z}^{(t)} = \left( \mathbf{z}\_1, \mathbf{z}\_2, \dots, \mathbf{z}\_{\mathbf{r}} \right), \tag{85}$$

$$\mathbf{z}^{(t)} = \mathbf{z}^{(t)}$$

$$Z\_{\mathbf{i}} = \left(\frac{\chi\_{\mathbf{i}}}{\hat{\beta}}\right)^{\delta}, \quad \mathbf{i} = \mathbf{1}, \dots, \mathbf{r},\tag{86}$$

are ancillary statistics, any r2 of which form a functionally independent set (for notational convenience we include all of z1, …, zr in (85); zr-1 and zr can be expressed as function of z1, …, zr-2 only), β and δ are the maximum likelihood estimators of and based on the first r ordered past observations (X1 … Xr) from a sample of size n from the two-parameter Weibull distribution, which can be found from solution of

$$\hat{\boldsymbol{\beta}} = \left( \left[ \sum\_{i=1}^{\mathbf{r}} \mathbf{x}\_i^{\bar{\boldsymbol{\beta}}} + (\mathbf{n} - \mathbf{r}) \mathbf{x}\_{\mathbf{r}}^{\bar{\boldsymbol{\beta}}} \right] \bigg/ \mathbf{r} \right)^{1/\bar{\boldsymbol{\beta}}} \tag{87}$$

#### 204 Stochastic Modeling and Control

and

$$\hat{\boldsymbol{\delta}} = \left[ \left( \sum\_{i=1}^{\mathbf{r}} \mathbf{x}\_i^{\bar{\boldsymbol{\delta}}} \text{l} \mathbf{n} \mathbf{x}\_i + (\mathbf{n} - \mathbf{r}) \mathbf{x}\_r^{\bar{\boldsymbol{\delta}}} \text{l} \mathbf{n} \mathbf{x}\_r \right) \left( \sum\_{i=1}^{\mathbf{r}} \mathbf{x}\_i^{\bar{\boldsymbol{\delta}}} + (\mathbf{n} - \mathbf{r}) \mathbf{x}\_r^{\bar{\boldsymbol{\delta}}} \right)^{-1} - \frac{1}{\mathbf{r}} \sum\_{i=1}^{\mathbf{r}} \text{l} \mathbf{n} \mathbf{x}\_i \right]^{-1},\tag{88}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 205

j v 1 h

Pr{X h|z } Pr{X h|v ,v }f(v ,v |z )dv dv . (95)

2

 <sup>2</sup>

<sup>f</sup>

h

 f f (r) (r) l l 12 12 1 2

m k <sup>h</sup> v z ( 1) (m k j) z (n r)z dv k j <sup>β</sup>

<sup>r</sup> <sup>r</sup> <sup>r</sup> r 2 vv v 2i i r 2

v z z (n r)z dv

*Remark 1.* If l=m=1, then the result of Theorem 4 can be used to construct the static policy of

*Theorem 5*. Let X1 ... Xk be the first k ordered early observations from a sample of size m from the two-parameter Weibull distribution (82). Then a lower one-sided conditional

prediction limit h on the lth order statistic Xl (l > k) in the same sample is given by

 1/δ

<sup>l</sup> lk k <sup>h</sup> Pr{X h|z } Pr X / X h / X z Pr{W w |z }

δ δ (k) (k) (k)

<sup>r</sup> δv <sup>r</sup> l1 k <sup>r</sup> r 2 <sup>v</sup> <sup>j</sup> v v 2 i i r2

0 0

0 i 1 k0 j0 i 1

0 i 1 i 1

Now v1 can be integrated out of (95) in a straightforward way to give

<sup>m</sup> h h 1 exp exp <sup>k</sup> β β

<sup>k</sup> m k δ δ l 1

m k ( 1) exp[ v (m k j)z ] k j

 

h

 δ

<sup>f</sup> (r) <sup>l</sup> Pr{X h|z }

2 2 2

 

22 2

k0 j0

l1 k

k 0

where

we have from (91) and (93) that

This completes the proof. □

airline seat inventory control.

where wh satisfies the equation

l 12 Pr{X h|v ,v }, (93)

z , β (94)

h k h w X, (97)

.

(96)

(Observe that an upper one-sided conditional prediction limit h on the lth order statistic <sup>f</sup> Xl from a set of m future ordered observations f f X X 1 m may be obtained from a lower one-sided conditional prediction limit by replacing by 1.)

*Proof*. The joint density of X1 ... Xr is given by

$$\text{f(x}\_{1},...,\text{x}\_{\text{r}} \mid \beta,\delta) = \frac{\text{n!}}{(\text{n}-\text{r})!} \prod\_{i=1}^{\text{r}} \frac{\partial}{\partial \left(\frac{\text{x}\_{i}}{\beta}\right)^{\delta-1}} \exp\left(-\left(\frac{\text{x}\_{i}}{\beta}\right)^{\delta}\right) \exp\left(-\text{(n-r)}\left(\frac{\text{x}\_{\text{r}}}{\beta}\right)^{\delta}\right). \tag{89}$$

Let <sup>β</sup> , δ be the maximum likelihood estimates of , , respectively, based on X1 … Xr from a complete sample of size n, and let

$$\mathbf{V}\_1 = \left(\frac{\hat{\boldsymbol{\beta}}}{\hat{\boldsymbol{\beta}}}\right)^{\boldsymbol{\delta}}, \ \mathbf{V}\_2 = \frac{\boldsymbol{\delta}}{\hat{\boldsymbol{\delta}}}, \ \text{and} \ \mathbf{Z}\_i = \left(\frac{\boldsymbol{\chi}\_i}{\hat{\boldsymbol{\beta}}}\right)^{\bar{\boldsymbol{\delta}}}, \ \mathbf{i} = \mathbf{1}, \dots, \mathbf{r}, \tag{90}$$

Parameters and in (90) are scale and shape parameters, respectively, and it is well known that if β and δ are estimates of and , possessing certain invariance properties, then V1 and V2 are the pivotal quantities whose distributions depend only on n. Most, if not all, proposed estimates of and possess the necessary properties; these include the maximum likelihood estimates and various linear estimates.

Using (90) and the invariant embedding technique (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b), we then find in a straightforward manner, that the joint density of V1, V2, conditional on fixed **z** 12 r (z , z , ..., z ), is

$$\mathbf{f}(\mathbf{v}\_1, \mathbf{v}\_2 \mid \mathbf{z}^{(t)})$$

$$\mathbf{f} = \mathbf{g}^\bullet(\mathbf{z}^{(t)}) \mathbf{v}\_2^{\prime - 2} \prod\_{i=1}^r \mathbf{z}\_i^{\mathbf{v}\_2} \mathbf{v}\_1^{\prime - 1} \exp\left(-\mathbf{v}\_1 \left[\sum\_{i=1}^r \mathbf{z}\_i^{\mathbf{v}\_2} + (\mathbf{n} - \mathbf{r}) \mathbf{z}\_\mathbf{r}^{\mathbf{v}\_2}\right]\right), \mathbf{v}\_1 \in \{0, \infty\}, \mathbf{v}\_2 \in \{0, \infty\},\tag{91}$$

where

$$\mathcal{G}^{\bullet}(\mathbf{z}^{\text{(r)}}) = \left[ \int\_{0}^{\mathbf{z}} \Gamma(\mathbf{r}) \mathbf{v}\_{2}^{\text{r}-2} \prod\_{i=1}^{\mathbf{r}} \mathbf{z}\_{i}^{\text{v}\_{2}} \left( \sum\_{i=1}^{\mathbf{r}} \mathbf{z}\_{i}^{\text{v}\_{2}} + (\mathbf{n} - \mathbf{r}) \mathbf{z}\_{\mathbf{r}}^{\text{v}\_{2}} \right)^{-\text{r}} \text{d}\mathbf{v}\_{2} \right]^{-1} \tag{92}$$

is the normalizing constant. Writing

$$\Pr\{\mathbf{X}\_{\parallel}^{\mathbf{f}} > \mathbf{h} \mid \boldsymbol{\beta}, \boldsymbol{\delta}\} = \sum\_{\mathbf{k}=0}^{\mathrm{l}-1} \binom{\mathrm{m}}{\mathrm{k}} [\mathrm{F}(\mathbf{h} \mid \boldsymbol{\beta}, \boldsymbol{\delta})]^{\mathrm{k}} [1 - \mathrm{F}(\mathbf{h} \mid \boldsymbol{\beta}, \boldsymbol{\delta})]^{\mathrm{m}-\mathrm{k}}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 205

$$\mathbf{I} = \sum\_{\mathbf{k}=0}^{\mathrm{l}-1} \binom{\mathrm{m}}{\mathrm{k}} \left[ 1 - \exp\left(-\left(\frac{\mathrm{h}}{\beta}\right)^{\delta}\right) \right]^{\mathrm{k}} \left[ \exp\left(-\left(\frac{\mathrm{h}}{\beta}\right)^{\delta}\right) \right]^{\mathrm{m}-\mathrm{k}}$$

$$= \sum\_{\mathbf{k}=0}^{\mathrm{l}-1} \binom{\mathrm{m}}{\mathrm{k}} \sum\_{\mathbf{j}=0}^{\mathrm{k}} \binom{\mathrm{k}}{\mathrm{j}} (-1)^{\mathrm{j}} \exp[-\mathrm{v}\_{1}(\mathrm{m}-\mathrm{k}+\mathrm{j})\mathrm{z}\_{\mathrm{h}}^{\mathrm{v}\_{2}}]$$

$$= \mathrm{Pr}[\mathbf{X}\_{1}^{\mathrm{f}} > \mathrm{h} \mid \mathbf{v}\_{1}, \mathbf{v}\_{2}] \tag{93}$$

where

<sup>X</sup> Z , i 1, ..., r, β (90)

1 2

(88)

r

one-sided conditional prediction limit by replacing by 1.)

 

likelihood estimates and various linear estimates.

δ 1 2 β δ V , V , <sup>β</sup> <sup>δ</sup>

joint density of V1, V2, conditional on fixed **z** 12 r (z , z , ..., z ), is

l

i 1 i 1

<sup>r</sup> <sup>r</sup> (r) r2 r1 <sup>v</sup> v v 2 i1 1 i r

i 1

*Proof*. The joint density of X1 ... Xr is given by

1 r f(x , ..., x |β,δ)

from a complete sample of size n, and let

 

(Observe that an upper one-sided conditional prediction limit h on the lth order statistic <sup>f</sup> Xl from a set of m future ordered observations f f X X 1 m may be obtained from a lower

δ be the maximum likelihood estimates of , , respectively, based on X1 … Xr

and

i

Parameters and in (90) are scale and shape parameters, respectively, and it is well known

and V2 are the pivotal quantities whose distributions depend only on n. Most, if not all, proposed estimates of and possess the necessary properties; these include the maximum

Using (90) and the invariant embedding technique (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b), we then find in a straightforward manner, that the

, v (0, ), v (0, ),

<sup>2</sup>

 l 1

<sup>m</sup> Pr{X h|β,δ} [F(h|β,δ)] [1 F(h|β,δ)] <sup>k</sup>

0 i 1 i 1

k 0

(r) 1 2 f(v ,v |z )

2 2

2i i r 2

(z ) Γ(r)v z z (n r)z dv , (92)

 22 2 <sup>1</sup> <sup>r</sup> <sup>r</sup> <sup>r</sup> (r) r 2 vv v

f k m k

(z )v z v exp v z (n r)z (91)

<sup>δ</sup> <sup>1</sup> δ δ <sup>r</sup> ii r

 

 δ i

δ are estimates of and , possessing certain invariance properties, then V1

n! <sup>δ</sup> xx x exp exp (n r) . (n r)! ββ β <sup>β</sup> (89)

<sup>1</sup> <sup>1</sup> r rr δ δ δδ ii rr i r i i 1 i 1 i 1 <sup>1</sup> <sup>δ</sup> x lnx (n r)x lnx x (n r)x lnx ,

204 Stochastic Modeling and Control

and

Let <sup>β</sup> ,

that if

where

β and

is the normalizing constant. Writing

$$\mathbf{z}\_{\mathbf{h}} = \left(\frac{\mathbf{h}}{\overline{\beta}}\right)^{\overline{\beta}},\tag{94}$$

we have from (91) and (93) that

$$\Pr\{\mathbf{X}\_{\mathrm{l}}^{\mathrm{f}} > \mathbf{h} \mid \mathbf{z}^{\mathrm{(r)}}\} = \bigcap\_{\mathbf{0} \neq \mathbf{0}}^{\mathrm{a} \mid \mathbf{v}} \Pr\{\mathbf{X}\_{\mathrm{l}}^{\mathrm{f}} > \mathbf{h} \mid \mathbf{v}\_{1}, \mathbf{v}\_{2}\} \mathbf{f} (\mathbf{v}\_{1}, \mathbf{v}\_{2} \mid \mathbf{z}^{\mathrm{(r)}}) \mathrm{d}\mathbf{v}\_{1} \mathrm{d}\mathbf{v}\_{2}. \tag{95}$$

Now v1 can be integrated out of (95) in a straightforward way to give

$$\Pr\{\mathcal{X}\_1^{\mathbf{f}} > \mathbf{h} \, | \, \mathbf{z}^{\mathbf{f}\mathbf{\hat{z}}}\}$$

$$=\frac{\begin{bmatrix} \int\_{0}^{\alpha} \mathbf{v}\_{2}^{\mathsf{r}-2} \prod\_{i=1}^{\mathsf{r}} \mathbf{z}\_{i}^{\mathsf{v}\_{2}} \sum\_{\mathbf{k}=0}^{\mathsf{l}-1} \binom{\mathsf{m}}{\mathsf{k}} \sum\_{j=0}^{\mathsf{k}} \binom{\mathsf{k}}{j} (-1)^{j} \left( (\mathsf{m}-\mathsf{k}+\mathsf{j}) \Big| \begin{matrix} \mathbf{h} \\ \widehat{\boldsymbol{\beta}} \end{matrix} \right)^{\mathsf{\delta}\mathbf{v}\_{2}} + \sum\_{i=1}^{\mathsf{r}} \mathbf{z}\_{i}^{\mathsf{v}\_{2}} + (\mathsf{m}-\mathsf{r}) \mathbf{z}\_{\mathsf{r}}^{\mathsf{v}\_{2}} \Big| \begin{matrix} \mathbf{w}\_{2} \\ \mathbf{w}\_{2} \end{matrix} \right)}{\int \mathbf{v}\_{2}^{\mathsf{r}-2} \prod\_{i=1}^{\mathsf{r}} \mathbf{z}\_{i}^{\mathsf{v}\_{2}} \left( \sum\_{i=1}^{\mathsf{r}} \mathbf{z}\_{i}^{\mathsf{v}\_{2}} + (\mathsf{m}-\mathsf{r}) \mathbf{z}\_{\mathsf{r}}^{\mathsf{v}\_{2}} \right)^{-\mathsf{r}} \mathrm{d} \mathbf{v}\_{2}}. \tag{96}$$

This completes the proof. □

*Remark 1.* If l=m=1, then the result of Theorem 4 can be used to construct the static policy of airline seat inventory control.

*Theorem 5*. Let X1 ... Xk be the first k ordered early observations from a sample of size m from the two-parameter Weibull distribution (82). Then a lower one-sided conditional prediction limit h on the lth order statistic Xl (l > k) in the same sample is given by

$$\mathbf{h} = \mathbf{w}\_{\mathbf{h}}^{1/\delta} \mathbf{\mathcal{X}}\_{\mathbf{k}'} \tag{97}$$

where wh satisfies the equation

$$\Pr\{\mathbf{X}\_{\mathrm{l}} > \mathbf{h} \, | \, \mathbf{z}^{\mathrm{(k)}}\} = \Pr\left\{ \left( \mathbf{X}\_{\mathrm{l}} / \mathbf{X}\_{\mathrm{k}} \right)^{\mathrm{\delta}} > \left( \mathbf{h} \, / \mathbf{X}\_{\mathrm{k}} \right)^{\mathrm{\delta}} \middle| \, \mathbf{z}^{\mathrm{(k)}} \right\} = \Pr\{\mathbf{W} > \mathbf{w}\_{\mathrm{h}} \, | \, \mathbf{z}^{\mathrm{(k)}}\}$$

206 Stochastic Modeling and Control

$$\mathbf{J} = \left[ \int\_{0}^{\infty} \mathbf{v}\_{2}^{k-2} \prod\_{i=1}^{\mathbf{k}} \mathbf{z}\_{i}^{\mathbf{v}\_{2}} \sum\_{j=0}^{\mathbf{l}-\mathbf{k}-1} \binom{\mathbf{l}-\mathbf{k}-1}{\mathbf{j}} \frac{(-\mathbf{l})^{\mathbf{l}-\mathbf{k}-\mathbf{l}-\mathbf{j}}}{\mathbf{m}-\mathbf{k}-\mathbf{j}} \binom{\mathbf{m}-\mathbf{k}-\mathbf{j}}{\mathbf{m}-\mathbf{k}-\mathbf{j}} (\mathbf{m}-\mathbf{k}-\mathbf{j}) (\mathbf{w}\_{\mathbf{l}} \mathbf{z}\_{\mathbf{k}})^{\mathbf{v}\_{2}} + \mathbf{j} \mathbf{z}\_{\mathbf{k}}^{\mathbf{v}\_{2}} + \sum\_{i=1}^{\mathbf{k}} \mathbf{z}\_{i}^{\mathbf{v}\_{2}} \right]^{-\mathbf{k}} \mathrm{d}\mathbf{v}\_{2} \right]$$

$$\times \left[ \int\_{0}^{\infty} \mathbf{v}\_{2}^{k-2} \prod\_{i=1}^{\mathbf{k}} \mathbf{z}\_{i}^{\mathbf{v}\_{2}} \sum\_{j=0}^{\mathbf{l}-\mathbf{k}-1} \binom{\mathbf{l}-\mathbf{k}-1}{\mathbf{j}} \frac{(-\mathbf{l})^{\mathbf{l}-\mathbf{k}-\mathbf{l}-\mathbf{j}}}{\mathbf{m}-\mathbf{k}-\mathbf{j}} \binom{\mathbf{m}-\mathbf{k}}{\mathbf{m}-\mathbf{k}-\mathbf{j}} \binom{\mathbf{m}}{\mathbf{m}-\mathbf{k}}^{\mathbf{v}\_{2}} + \sum\_{i=1}^{\mathbf{k}} \mathbf{z}\_{i}^{\mathbf{v}\_{2}} \right)^{-\mathbf{k}} \mathrm{d}\mathbf{v}\_{2} \right]^{-1} = \mathbf{y}\_{\prime} \tag{98}$$

$$\mathbf{z}^{(k)} = (z\_{1'} \; \dots \; z\_k)\_{\prime} \tag{99}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 207

l

k β δ <sup>Y</sup> V ,V ,W <sup>β</sup> <sup>δ</sup> <sup>Y</sup> (105)

1 2 v (0, ), v (0, ), w (1, ),

2 2

<sup>l</sup> <sup>x</sup> exp (m l) <sup>β</sup> . (104)

δ

δ be the maximum likelihood estimates of , , respectively, based on X1 ... Xk

, δ δ

<sup>δ</sup> Z (Y / i i β) , i=1(1)k. Using the invariant embedding technique (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b), we then find in a straightforward manner, that the joint density of V1, V2, W, conditional on fixed

> 

lk1 f(v ,v ,w|z ) (z )v z (wz ) v ( 1) <sup>j</sup>

2 2 2 <sup>k</sup> v v <sup>v</sup>

i 1

2 i ki 2

 <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>k</sup> <sup>k</sup> lk1 lk1j <sup>k</sup>

lk1 ( 1) Γ(k) (z ) v z (m k)z z dv <sup>j</sup> mkj (107)

(k) (k) h 1 2 1 2

 2 2 <sup>2</sup> <sup>k</sup> <sup>k</sup> lk1 lk1j <sup>k</sup> k 2 v v v v 2 i hk k i 2

lk1 ( 1) v z (m k j)(w z ) jz z dv

 <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>k</sup> <sup>k</sup> lk1 lk1j <sup>k</sup>

lk1 ( 1) v z (m k)z z dv , <sup>j</sup> mkj (108)

<sup>Y</sup> <sup>h</sup> Pr{Y h|z } Pr z

h

 

0 i 1 j 0 i 1

 

0 i 1 j 0 i 1

k 2 v v v 2 i ki 2

j mkj

0w 0 Pr{W w |z } f(v ,v ,w|z )dv dwdv

(k) l (k)

k k

Y Y

δ δ

 

(k) k 2 v v v

0 i 1 j 0 i 1

i 1 j 0

exp v (m k j)(wz ) jz z , (106)

k lk1 (k) (k) k 1 v v k lk1 j

1 2

is the normalizing constant. Using (106), we have that

l

1 kki

1 2 2 i k1

Let <sup>β</sup> ,

(k)

where

<sup>2</sup>

<sup>2</sup>

<sup>2</sup>

and

1 k z (z , ..., z ) , is

from a complete sample of size m, and let

$$Z\_{\mathbf{i}} = \left(\frac{\mathbf{Y}\_{\mathbf{i}}}{\hat{\boldsymbol{\beta}}}\right)^{\bar{\boldsymbol{\beta}}}, \quad \mathbf{i} = \mathbf{1}, \dots, \mathbf{k}, \tag{100}$$

$$\mathbf{w}\_{\mathbf{h}} = \left(\frac{\mathbf{h}}{\mathbf{Y}\_{\mathbf{k}}}\right)^{\bar{\delta}},\tag{101}$$

where β and δ are the maximum likelihood estimates of and based on the first k ordered past observations X1 ... Xk from a sample of size m from the two-parameter Weibull distribution (82), which can be found from solution of

$$\hat{\boldsymbol{\beta}} = \left( \left[ \sum\_{i=1}^{k} \mathbf{x}\_i^{\boldsymbol{\delta}} + (\mathbf{m} - \mathbf{k}) \mathbf{x}\_k^{\boldsymbol{\delta}} \right] \bigg/ \mathbf{k} \right)^{1/\bar{\delta}},\tag{102}$$

and

$$\hat{\boldsymbol{\Theta}} = \left[ \left( \sum\_{i=1}^{\mathbf{k}} \mathbf{x}\_i^{\bar{\boldsymbol{\Theta}}} \mathbf{l} \mathbf{m}\_i + (\mathbf{m} - \mathbf{k}) \mathbf{x}\_{\bar{\mathbf{k}}}^{\bar{\boldsymbol{\Theta}}} \mathbf{l} \mathbf{m}\_{\bar{\mathbf{k}}} \right) \left( \sum\_{i=1}^{\mathbf{k}} \mathbf{x}\_i^{\bar{\boldsymbol{\Theta}}} + (\mathbf{m} - \mathbf{k}) \mathbf{x}\_{\bar{\mathbf{k}}}^{\bar{\boldsymbol{\Theta}}} \right)^{-1} - \frac{1}{\mathbf{k}} \sum\_{i=1}^{\mathbf{k}} \mathbf{l} \mathbf{m}\_{\bar{\mathbf{i}}} \right]^{-1},\tag{103}$$

(Observe that an upper one-sided conditional prediction limit h on the lth order statistic Xl based on the first k ordered early-failure observations X1 ... Xk, where l > k, from the same sample may be obtained from a lower one-sided conditional prediction limit by replacing by 1)

*Proof.* The joint density of X1 ... Xk and Xl is given by

$$\begin{split} \text{f}(\mathbf{x}\_{1},...,\mathbf{x}\_{k},\mathbf{x}\_{1} \mid \boldsymbol{\beta},\boldsymbol{\delta}) &= \frac{\text{m}!}{(1-\text{k}-1)!(\text{m}-\text{l})!} \prod\_{i=1}^{k} \frac{\text{\(}\mathbf{x}\_{i}\text{}\right^{\delta}\text{\(}\frac{\mathbf{x}\_{i}}{\boldsymbol{\beta}}\text{\)}^{\delta-1} \exp\left(-\left(\frac{\mathbf{x}\_{i}}{\boldsymbol{\beta}}\right)^{\delta}\right) \\ & \times \left[\exp\left(-\left(\frac{\mathbf{x}\_{k}}{\boldsymbol{\beta}}\right)^{\delta}\right) - \exp\left(-\left(\frac{\mathbf{x}\_{1}}{\boldsymbol{\beta}}\right)^{\delta}\right)\right]^{\mathrm{l}-\text{k}-1} \frac{\text{\(}\mathbf{x}\_{1}\text{\)}^{\delta-1}}{\text{\(}\mathbf{\beta}\text{\)}^{\delta}} \exp\left(-\left(\frac{\mathbf{x}\_{1}}{\boldsymbol{\beta}}\right)^{\delta}\right) \end{split}$$

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 207

$$\times \exp\left(-\text{(m-l)}\left(\frac{\mathbf{x}\_{l}}{\beta}\right)^{\delta}\right). \tag{104}$$

Let <sup>β</sup> , δ be the maximum likelihood estimates of , , respectively, based on X1 ... Xk from a complete sample of size m, and let

$$\mathbf{V}\_1 = \left(\frac{\hat{\boldsymbol{\beta}}}{\boldsymbol{\beta}}\right)^{\boldsymbol{\delta}}, \ \mathbf{V}\_2 = \frac{\boldsymbol{\delta}}{\hat{\boldsymbol{\delta}}}, \ \mathbf{W} = \left(\frac{\mathbf{Y}\_1}{\mathbf{Y}\_k}\right)^{\boldsymbol{\delta}},\tag{105}$$

and <sup>δ</sup> Z (Y / i i β) , i=1(1)k. Using the invariant embedding technique (Nechval et al., 1999; 2004; 2008; 2010a; 2010b; 2010c; 2010d; 2010e; 2011a; 2011b), we then find in a straightforward manner, that the joint density of V1, V2, W, conditional on fixed (k) 1 k z (z , ..., z ) , is

$$\mathbf{f}(\mathbf{v}\_{1},\mathbf{v}\_{2},\mathbf{w}\mid\mathbf{z}^{(\mathbf{k})}) = \mathcal{G}(\mathbf{z}^{(\mathbf{k})})\mathbf{v}\_{2}^{\mathbf{k}-1}\prod\_{i=1}^{\mathbf{k}}\mathbf{z}\_{i}^{\mathbf{v}\_{2}}\langle\mathbf{w}\mathbf{z}\_{k}\rangle^{\mathbf{v}\_{2}}\mathbf{v}\_{1}^{\mathbf{k}}\sum\_{j=0}^{\mathbf{l}-\mathbf{k}-1}{\mathbf{j}}\binom{\mathbf{l}-\mathbf{k}-1}{\mathbf{j}}(-1)^{\mathbf{l}-\mathbf{k}-1-j}$$

$$\times \exp\left[-\mathbf{v}\_{1}\left(\langle\mathbf{m}-\mathbf{k}-\mathbf{j}\rangle\langle\mathbf{w}\mathbf{z}\_{k}\rangle^{\mathbf{v}\_{2}}+\langle\mathbf{z}\_{k}^{\mathbf{v}\_{2}}+\sum\_{i=1}^{\mathbf{k}}\mathbf{z}\_{i}^{\mathbf{v}\_{2}}\right)\right],\ \mathbf{v}\_{1}\in\langle0,\infty\rangle,\ \mathbf{v}\_{2}\in\langle0,\infty\rangle,\ \mathbf{w}\in\langle1,\infty\rangle,\tag{106}$$

where

206 Stochastic Modeling and Control

where

and

replacing by 1)

β and

<sup>2</sup>

<sup>2</sup>

 

 <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>k</sup> <sup>k</sup> lk1 lk1j <sup>k</sup>

<sup>1</sup> ( , ..., ), *<sup>k</sup>*

 

k h

ordered past observations X1 ... Xk from a sample of size m from the two-parameter

 

 1/δ <sup>k</sup> δ δ i k

(Observe that an upper one-sided conditional prediction limit h on the lth order statistic Xl based on the first k ordered early-failure observations X1 ... Xk, where l > k, from the same sample may be obtained from a lower one-sided conditional prediction limit by

δ δ δδ

 

<sup>1</sup> <sup>1</sup> k kk

ii kk i k i i 1 i 1 i 1

<sup>1</sup> <sup>δ</sup> x lnx (m k)x lnx x (m k)x lnx , k (103)

 

i 1

m! δ x x exp (l k 1)!(m l)! ββ β

 δ

δ are the maximum likelihood estimates of and based on the first k

β x (m k)x k , (102)

<sup>δ</sup> <sup>1</sup> <sup>δ</sup> <sup>k</sup> i i

> 

δ 1 δ l l δ x x exp ββ β

 

 δ i

h

0 i 1 j 0 i 1

 

( )

0 i 1 j 0 i 1

i

i 1

Weibull distribution (82), which can be found from solution of

*Proof.* The joint density of X1 ... Xk and Xl is given by

<sup>k</sup> <sup>l</sup> x x exp exp β β

lk1 δ δ

1 kl f(x , ..., x ,x |β,δ)

k 2 v v v 2 i ki 2

j mkj

 2 2 <sup>2</sup> <sup>k</sup> <sup>k</sup> lk1 lk1j <sup>k</sup> k 2 v v v v 2 i hk k i 2

lk1 ( 1) v z (m k)z z dv γ, <sup>j</sup> mkj (98)

*<sup>k</sup>* **z** *z z* (99)

<sup>Y</sup> Z , i 1, ..., k, β (100)

w , Y (101)

lk1 ( 1) v z (m k j)(w z ) jz z dv

$$\mathcal{G}(\mathbf{z}^{(k)}) = \left[ \int\_0^\mathbf{v} \mathbf{v}\_2^{k-2} \prod\_{i=1}^\mathbf{k} \mathbf{z}\_i^{\mathbf{v}\_2} \sum\_{j=0}^{\mathbf{l}-\mathbf{k}-1} \binom{\mathbf{l}-\mathbf{k}-\mathbf{l}}{\mathbf{j}} \frac{(-1)^{\mathbf{l}-\mathbf{k}-\mathbf{l}-\mathbf{j}} \Gamma(\mathbf{k})}{\mathbf{m}-\mathbf{k}-\mathbf{j}} \left( (\mathbf{m}-\mathbf{k}) \mathbf{z}\_{\mathbf{k}}^{\mathbf{v}\_2} + \sum\_{i=1}^\mathbf{k} \mathbf{z}\_i^{\mathbf{v}\_2} \right)^{-\mathbf{k}} \mathrm{d}\mathbf{v}\_2 \right]^{-1} \tag{107}$$

is the normalizing constant. Using (106), we have that

 δ δ (k) l (k) l k k <sup>Y</sup> <sup>h</sup> Pr{Y h|z } Pr z Y Y h (k) (k) h 1 2 1 2 0w 0 Pr{W w |z } f(v ,v ,w|z )dv dwdv <sup>2</sup> 2 2 <sup>2</sup> <sup>k</sup> <sup>k</sup> lk1 lk1j <sup>k</sup> k 2 v v v v 2 i hk k i 2 0 i 1 j 0 i 1 lk1 ( 1) v z (m k j)(w z ) jz z dv j mkj <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>k</sup> <sup>k</sup> lk1 lk1j <sup>k</sup> k 2 v v v 2 i ki 2 0 i 1 j 0 i 1 lk1 ( 1) v z (m k)z z dv , <sup>j</sup> mkj (108)

and the proof is complete. □

*Remark 2.* If l=m, then the result of Theorem 5 can be used to construct the dynamic policy of airline seat inventory control.

Stochastic Control and Improved Statistical Decisions in Revenue Optimization Systems 209

Agrawal, N. & Smith, S.A. (1996). Estimating Negative Binomial Demand for Retail Inventory Management with Unobservable Lost Sales. *Naval Res. Logist*., Vol. 43, pp. 839

Azoury, K.S. (1985). Bayes Solution to Dynamic Inventory Models under Unknown Demand

Belobaba, P.P. (1987). Survey Paper: Airline Yield Management – an Overview of Seat

Brumelle, S.L. & J.I. McGill (1993). Airline Seat Allocation with Multiple Nested Fare

Conrad, S.A. (1976). Sales Data and the Estimation of Demand. *Oper. Res. Quart*., Vol. 27, pp.

Curry, R.E. (1990). Optimal Airline Seat Allocation with Fare Classes Nested by Origins and

Ding, X.; Puterman, M.L. & Bisi, A. (2002). The Censored Newsvendor and the Optimal

Karlin, S. (1960). Dynamic Inventory Policy with Varying Stochastic Demands. *Management* 

Lariviere, M.A. & Porteus, E.L. (1999). Stalking Information: Bayesian Inventory Management with Unobserved Lost Sales. *Management Sci*., Vol. 45, pp. 346 363 Littlewood, K. (1972). Forecasting and Control of Passenger Bookings. In: *Proceedings of the Airline Group International Federation of Operational Research Societies*, Vol. 12, pp. 95–117 Liyanage, L.H. & Shanthikumar, J.G. (2005). A Practical Inventory Control Policy Using

Nahmias, S. (1994). Demand Estimation in Lost Sales Inventory Systems. *Naval Res. Logist*.,

Nechval N. A. & Purgailis, M. (2010e). Improved State Estimation of Stochastic Systems via a New Technique of Invariant Embedding. In: *Stochastic Control*, Chris Myers (ed.), pp.

Nechval, N. A. & Vasermanis, E. K. (2004). *Improved Decisions in Statistics*. SIA "Izglitibas

Nechval, N. A. & Nechval, K. N. (1999). Invariant Embedding Technique and Its Statistical Applications. In: *Conference Volume of Contributed Papers of the 52nd Session of the International Statistical Institute*, Finland, Helsinki: ISIInternational Statistical Institute,Availqable from http://www.stat.fi/isi99/procee-dings/arkisto/varasto/

Nechval, N. A.; Berzins, G.; Purgailis, M. & Nechval, K.N. (2008). Improved Estimation of State of Stochastic Systems via Invariant Embedding Technique. *WSEAS Transactions on* 

Nechval, N. A.; Nechval, K. N. & Purgailis, M. (2011b). Statistical Inferences for Future Outcomes with Applications to Maintenance and Reliability. In: *Lecture Notes in* 

Distribution. *Management Sci*., Vol. 31, pp. 1150 1160

Classes. *Operations Research*, Vol. 41, pp. 127–137

Destination. *Transportation Science*, Vol. 24, pp. 193–204

Acquisition of Information. *Oper. Res*., Vol. 50, pp. 517527

Operational Statistics. *Oper. Res. Lett*., Vol. 33, pp. 341348

167193. Publisher: Sciyo, Croatia, India

*Mathematics*, Vol. 7, pp. 141 159

Inventory Control. *Transportation Science*, Vol. 21, pp. 63 –73

**6. References** 

861

123 127

*Sci*., Vol. 6, pp. 231 258

Vol. 41, 739 757

soli", Riga

nech0902.pdf
