Keywords

1 Introduction

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 (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).

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 [1] and Karlin [2] 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 [3]. Application of the Bayesian approach to the censored demand case is given in [4, 5]. Parameter estimation is first considered in [6] and recent developments are reported in [7, 8]. Liyanage and Shanthikumar [9] propose the concept of operational statistics and apply it to a single period newsvendor inventory control problem.

Within the non-parametric approach, either the empirical distribution or the bootstrapping method (e.g. see [10]) can be applied with the available demand data to obtain an inventory control policy.

Conceptually, it is useful to distinguish between “new-sample” inventory control, “within-sample” inventory control, and “new-within-sample” inventory control.

For the new-sample inventory control process, the data from a past sample of customer demand are used to make a statistical decision on a future time period for the same inventory control process.

For the within-sample inventory control process, the problem is to make a statistical decision on a future time period for the same inventory control process based on early data from that sample of customer demand.

For the new-within-sample inventory control process, the problem is to make a statistical decision on a future time period for the inventory control process based on early data from that sample of customer demand as well as on a past data sample of customer demand from the same process.

In this paper, we consider the case of the within-sample inventory control process, where it is known that the underlying distribution function of the customer demand belongs to a parametric family of distribution functions. However, unlike in the Bayesian approach, we do not assume any prior knowledge on the parameter values.

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 available statistical information as completely as possible in order to improve statistical decisions for inventory control problems under parametric uncertainty.

Assumptions. The customer demand at the \(i\)th period represents a random variable \(Y_{i},i\in \{1, {\ldots }, m\)}. For the cumulative customer demand, \(X\), it is assumed that the random variables

$$\begin{aligned} X_1 =Y_1 ,\ldots ,X_k =\sum _{i=}^k {Y_i ,\ldots ,X_l } =\sum _{i=1}^l {Y_i ,\ldots ,X_m =\sum _{i=1}^m {Y_i } } \end{aligned}$$
(1)

represent the order statistics (\(X_{1} \le {\ldots } \le X_{m}\)) from the exponential distribution with the probability density function

$$\begin{aligned} f_\sigma (x)=\frac{1}{\sigma }\exp \left( {-\frac{x}{\sigma }} \right) ,\quad x\ge 0, \quad \sigma >0, \end{aligned}$$
(2)

and the probability distribution function

$$\begin{aligned} F_\sigma (x)=1-\exp \left( {-\frac{x}{\sigma }} \right) . \end{aligned}$$
(3)

Theorem 1

Let \(X_{1} \le \ldots \le X_{k}\) be the first k ordered observations (order statistics) in a sample of size m from a continuous distribution with some probability density function \(f_{\theta }(x)\) and distribution function \(F_{\theta }(x)\), where \(\theta \) is a parameter (in general, vector). Then the conditional probability density function of the lth order statistics \(X_{l} (1 \le k < l \le m)\) given \(X_{k }= x_{k}\) is

$$\begin{aligned} g_\theta (x_l |x_k )&= \frac{(m-k)!}{(l-k-1)!(m-l)!}\left[ {\frac{F_\theta (x_l )-F_\theta (x_k )}{1-F_\theta (x_k )}} \right] ^{l-k-1}\nonumber \\&\quad \times \left[ {1-\frac{F_\theta (x_l )-F_\theta (x_k )}{1-F_\theta (x_k )}} \right] ^{m-l}\frac{f_\theta (x_l )}{1-F_\theta (x_k )}\nonumber \\&=\frac{(m-k)!}{(l-k-1)!(m-l)!}\sum _{j=0}^{l-k-1} {\left( {{\begin{array}{c} {l-k-1} \\ j \\ \end{array} }} \right) (-1)^{j}} \left[ {\frac{1-F_\theta (x_l )}{1-F_\theta (x_k )}} \right] ^{m-l+j}\nonumber \\&\quad \times \frac{f_\theta (x_l )}{1-F_\theta (x_k )}\nonumber \\&=\frac{(m-k)!}{(l-k-1)!(m-l)!}\sum _{j=0}^{m-l} {\left( {{\begin{array}{c} {m-l} \\ j \\ \end{array} }} \right) (-1)^{j}} \left[ {\frac{F_\theta (x_l)-F_\theta (x_k )}{1-F_\theta (x_k )}}\right] ^{l-k-1+j}\nonumber \\&\quad \times \frac{f_\theta (x_l )}{1-F_\theta (x_k )} \end{aligned}$$
(4)

Proof

From the marginal density function of \(X_{k}\) and the joint density function of \(X_{k}\) and \(X_{l}\), we have the conditional density function of \(X_{l}\), given that \(X_{k}=x_{k}\), as

$$\begin{aligned} g_\theta (x_l |x_k )=g_\theta (x_l ,x_k )/g_\theta (x_k ). \end{aligned}$$
(5)

This ends the proof.

Corollary 1.1

The conditional probability distribution function of X\(_{l}\) given \(X_{k}=x_{k}\) is

$$\begin{aligned}&P_\theta \left\{ {X_l \le x_l |X_k =x_k } \right\} \nonumber \\&\quad =1-\frac{(m-k)!}{(l-k-1)!(m-l)!}\times \sum _{j=0}^{l-k-1} \left( {{\begin{array}{c} {l-k-1} \\ j \\ \end{array} }} \right) \frac{(-1)^{j}}{m-l+1+j}\nonumber \\&\qquad \; \times \left[ {\frac{1-F_\theta (x_l )}{1-F_\theta (x_k )}} \right] ^{m-l+1+j} \nonumber \\&\qquad \;=\frac{(m-k)!}{(l-k-1)!(m-l)!}\sum _{j=0}^{m-l} {\left( {{\begin{array}{c} {m-l} \\ j \\ \end{array} }} \right) }\frac{(-1)^{j}}{l-k+j} \left[ {\frac{F_\theta (x_l )-F_\theta (x_k )}{1-F_\theta (x_k )}} \right] ^{l-k+j}. \end{aligned}$$
(6)

Corollary 1.2

Let \(X_{1} \le \ldots \le X_{k}\) be the first k ordered observations (order statistics) in a sample of size m from the exponential distribution (2). Then the conditional probability density function of the lth order statistics \(X_{l} (1 \le k < l \le m\)) given \(X_{k }= x_{k}\) is

$$\begin{aligned} g_\sigma (x_l |x_k )&=\frac{1}{\mathrm{B}(l-k,(m-l+1)}\sum _{j=0}^{l-k-1} {\left( {{\begin{array}{c} {l-k-1} \\ j \\ \end{array} }} \right) (-1)^{j}} \nonumber \\&\qquad \times \frac{1}{\sigma }\exp \left( {-\frac{(m-l+1+j)(x_l -x_k )}{\sigma }} \right) \nonumber \\&=\frac{1}{\mathrm{B}(l-k,(m-l+1)}\sum _{j=0}^{m-l} {\left( {{\begin{array}{c} {m-l} \\ j \\ \end{array} }} \right) (-1)^{j}}\nonumber \\&\qquad \times \frac{1}{\sigma } \left[ {1-\exp \left( {-\frac{x_l -x_k }{\sigma }} \right) } \right] ^{l-k-1+j} \exp \left( {\frac{x_l -x_k }{\sigma }} \right) , \end{aligned}$$
(7)

and the conditional probability distribution function of the lth order statistics \(X_{l}\) given \(X_{k }= x_{k}\) is

$$\begin{aligned} P_\sigma \left\{ {X_l \le x_l |X_k =x_k } \right\}&= 1-\frac{1}{\mathrm{B}(l-k,(m-l+1)}\sum _{j=0}^{l-k-1} {\left( {{\begin{array}{c} {l-k-1} \\ j \\ \end{array} }} \right) }\nonumber \\&\quad \times \frac{(-1)^{j}}{m-l+1+j}\exp \left( {-\frac{(m-l+1+j)(x_l -x_k )}{\sigma }} \right) \nonumber \\&=\frac{1}{\mathrm{B}(l-k,(m-l+1)}\sum _{j=0}^{m-l} {\left( {{\begin{array}{c} {m-l} \\ j \\ \end{array} }} \right) } \nonumber \\&\quad \times \frac{(-1)^{j}}{l-k+j}\left[ {1-\exp \left( {-\frac{x_l -x_k }{\sigma }} \right) } \right] ^{l-k+j}. \end{aligned}$$
(8)

Corollary 1.3

If \(l = k + 1\),

$$\begin{aligned} g_\sigma (x_{k+1} |x_k )&=(m-k)\frac{1}{\sigma }\exp \left( {-\frac{(m-k)(x_{k+1} -x_k )}{\sigma }} \right) \nonumber \\&=(m-k)\sum _{j=0}^{m-k-1} {\left( {{\begin{array}{c} {m-k-1} \\ j \\ \end{array} }} \right) (-1)^{j}}\nonumber \\&\quad \times \frac{1}{\sigma }\left[ {1-\exp \left( {-\frac{x_{k+1} -x_k }{\sigma }} \right) } \right] ^{j}\exp \left( {\frac{x_{k+1} -x_k }{\sigma }} \right) , \end{aligned}$$
(9)

and

$$\begin{aligned} P_\sigma \left\{ {X_{k+1} \le x_{k+1} |X_k =x_k } \right\}&=1-\exp \left( {-\frac{(m-k)(x_{k+1} -x_k )}{\sigma }} \right) \nonumber \\&=(m-k)\sum _{j=0}^{m-k-1} {\left( {{\begin{array}{c} {m-k-1} \\ j \\ \end{array} }} \right) } \frac{(-1)^{j}}{1+j} \nonumber \\&\quad \times \left[ {1-\exp \left( {-\frac{x_{k+1} -x_k }{\sigma }} \right) } \right] ^{1+j}, \quad 1\le k\le m-1. \end{aligned}$$
(10)

Corollary 1.4

If \(l = k + 1\) and \(Y_{k+1}=X_{k+1}-X_{k}\), then the probability density function of \(Y_{k+1}, k\in \{1, {\ldots }, m -1\}\), is given by

$$\begin{aligned} g_\sigma (y_{k+1} )=\frac{m-k}{\sigma }\exp \left( {-\frac{(m-k)y_{k+1} }{\sigma }} \right) ,\quad y_{k+1} \ge 0, \end{aligned}$$
(11)

and the probability distribution function of \(Y_{k+1}\) is given by

$$\begin{aligned} G_\sigma \left\{ {y_{k+1} } \right\} =1-\exp \left( {-\frac{(m-k)y_{k+1} }{\sigma }} \right) . \end{aligned}$$
(12)

Theorem 2

Let \(X_{1} \le \ldots \le X_{k}\) be the first k ordered observations (order statistics) in a sample of size m from the exponential distribution (2), where the parameter \(\sigma \) is unknown. Then the predictive probability density function of the lth order statistics \(X_{l} (1 \le k < l \le m\)) is given by

$$\begin{aligned} g_{s_k } (x_l |x_k )&=\frac{k}{\mathrm{B}(l-k,(m-l+1)}\sum _{j=0}^{l-k-1} {\left( {{\begin{array}{c} {l-k-1} \\ j \\ \end{array} }} \right) (-1)^{j}}\nonumber \\&\quad \times \left[ {1+(m-l+1+j)\frac{x_l -x_k }{s_k }} \right] ^{-(k+1)}\frac{1}{s_k }, \quad x_l \ge x_k , \quad s_k >0, \end{aligned}$$
(13)

where

$$\begin{aligned} S_k =\sum _{i=1}^k {X_i +(m-k)X_k } \end{aligned}$$
(14)

is the sufficient statistic for \(\sigma \), and the predictive probability distribution function of the lth order statistics \(X_{l}\) is given by

$$\begin{aligned} P_{s_k } \left\{ {X_l \le x_l |X_k =x_k } \right\}&=1-\frac{1}{\mathrm{B}(l-k,(m-l+1)}\sum _{j=0}^{l-k-1} \left( \begin{array}{c} {l-k-1} \\ j \\ \end{array}\right) \nonumber \\&\quad \times \frac{(-1)^{j}}{m-l+1+j} \left[ {1+(m-l+1+j)\frac{x_l -x_k }{s_k }} \right] ^{-k}. \end{aligned}$$
(15)

Proof

Using the technique of invariant embedding [11, 22], we reduce (7) to

$$\begin{aligned}&g_{\sigma } (x_l |x_k )=\frac{1}{\mathrm{B}(l-k,(m-l+1)}\sum _{j=0}^{l-k-1} {\left( {{\begin{array}{c} {l-k-1} \\ j \\ \end{array} }} \right) (-1)^{j}}\nonumber \\&\qquad \qquad \qquad \;\times v\exp \left( {-\frac{(m-l+1+j)(x_l -x_k )}{s_k }v} \right) \frac{1}{s_k } =g_{s_k } (x_l |x_k ,v), \end{aligned}$$
(16)

where

$$\begin{aligned} V=S_k /\sigma \end{aligned}$$
(17)

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

$$\begin{aligned} f(v)=\frac{1}{\Gamma (k)}v^{k-1}\exp (-v),\quad v\ge 0. \end{aligned}$$
(18)

Then

$$\begin{aligned} g_{s_k } (x_l |x_k )=E\{g_{s_k } (x_l |x_k ,v)\} =\int \limits _0^\infty {g_{s_k } (x_l |x_k ,v)f(v)dv} . \end{aligned}$$
(19)

This ends the proof.

Corollary 2.1

If \(l = k + 1\),

$$\begin{aligned} g_{s_k } (x_{k+1} |x_k )=k(m-k)\left[ {1+(m-k)\frac{x_{k+1} -x_k }{s_k }} \right] ^{-(k+1)}\frac{1}{s_k }, \end{aligned}$$
(20)

and

$$\begin{aligned} P_{s_k } \left\{ {X_{k+1} \le x_{k+1} |X_k =x_k } \right\} =1-\left[ {1+(m-k)\frac{x_{k+1} -x_k }{s_k }} \right] ^{-k}, \end{aligned}$$
(21)

Corollary 2.2

If \(l=k\) + 1 and \(Y_{k+1}=X_{k+1}-X_{k}\), then the predictive probability density function of \(Y_{k+1}, k\in \{1, \ldots , m -1\}\), is given by

$$\begin{aligned} g_{s_k } (y_{k+1} )=k(m-k)\left[ {1+(m-k)\frac{y_{k+1} }{s_k }} \right] ^{-(k+1)}\frac{1}{s_k }, \quad y_{k+1} \ge 0, \end{aligned}$$
(22)

and the predictive probability distribution function of \(Y_{k+1}\) is given by

$$\begin{aligned} G_{s_k } (y_{k+1} )=1-\left[ {1+(m-k)\frac{y_{k+1} }{s_k }} \right] ^{-k}. \end{aligned}$$
(23)

3 Inventory Control Models

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,

\(c_{1}\)= holding cost per held unit during the period,

\(c_{2}\) = penalty cost per shortage unit during the period,

\(g_{\sigma }(y_{k+1})\) = probability density function of customer demand, \(Y_{k+1}\), during the (\(k+1\))th period,

\(\sigma \) = scale parameter,

\(u\) = order quantity,

\(q\) = inventory on hand before an order is placed.

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:

  1. 1.

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

  2. 2.

    No setup cost is incurred.

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 \(Y_{k+1}\le u\), the quantity \(u-Y_{k+1}\) is held during the (\(k+1\))th period. Otherwise, a shortage amount \(Y_{k+1}-u\) will result if \(Y_{k+1}> u\). Thus, the cost per the (\(k+1\))th period is

$$\begin{aligned} C(u)=\left\{ {\begin{array}{l} \displaystyle c_1 \frac{u-Y_{k+1} }{\sigma }\quad \text{ if }\ Y_{k+1} \le u, \\ \displaystyle c_2 \frac{Y_{k+1} -u}{\sigma }\quad \text{ if }\ Y_{k+1} >u. \\ \end{array}} \right. \end{aligned}$$
(24)

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

$$\begin{aligned} E_\sigma \{C(u)\}=\frac{1}{\sigma }\left( c_1 \int \limits _0^u (u-y_{k+1})g_\sigma (y_{k+1})dy_{k+1} +c_2 \int \limits _u^\infty {(y_{k+1} -u)g_\sigma (y_{k+1} )dy_{k+1} } \right) . \end{aligned}$$
(25)

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

$$\begin{aligned} \frac{1}{\sigma }\left( {c_1 \int \limits _0^u {g_\sigma } (y_{k+1} )dy_{k+1} -c_2 \int \limits _u^\infty {g_\sigma } (y_{k+1} )dy_{k+1} } \right) =0 \end{aligned}$$
(26)

or

$$\begin{aligned} c_1 P_\sigma \{Y_{k+1} \le u\}-c_2 (1-P_\sigma \{Y_{k+1} \le u\})=0 \end{aligned}$$
(27)

or

$$\begin{aligned} P_\sigma \{Y_{k+1} \le u\}=\frac{c_2 }{c_1 +c_2 }. \end{aligned}$$
(28)

It follows from (11), (12), (25), and (28) that

$$\begin{aligned} u^{{*}}=\frac{\sigma }{m-k}\ln \left( {1+\frac{c_2 }{c_1 }} \right) \end{aligned}$$
(29)

and

$$\begin{aligned} E_\sigma \{C(u^{{*}})\}&=\frac{1}{\sigma }\left( {c_2 E_\sigma \{Y_{k+1} \}-(c_1 +c_2 )\int \limits _0^{u^{{*}}} {y_{k+1} g_\sigma (y_{k+1} )dy_{k+1} } } \right) \nonumber \\&=\frac{c_1 }{m-k}\ln \left( {1+\frac{c_2 }{c_1 }} \right) . \end{aligned}$$
(30)

Parametric uncertainty. Consider the case when the parameter \(\sigma \) is unknown. To find the best invariant decision rule \(u^{\mathrm{{BI}}},\) we use the invariant embedding technique [11–22] to transform (24) to the form, which is depended only on the pivotal quantities \(V, V_{1}\), and the ancillary factor \(\eta \).

Transformation of \(C(u)\) based on the pivotal quantities \(V\), \(V_{1}\) is given by

$$\begin{aligned} C^{(1)}(\eta )=\left\{ {\begin{array}{l} c_1 (\eta V-V_1 ) \quad \text{ if }\ V_1 \le \eta V, \\ \\ c_2 (V_1 -\eta V)\quad \text{ if }\ V_1 >\eta V, \\ \end{array}} \right. \end{aligned}$$
(31)

where

$$\begin{aligned} \eta =\frac{u}{S_k }, \end{aligned}$$
(32)
$$\begin{aligned} V_1 =\frac{Y_{k+1} }{\sigma }\sim g(v_1 )=(m-k)\exp [-(m-k)v_1 ], \quad v_1 \ge 0. \end{aligned}$$
(33)

Then \(E\{C^{(1)}(\eta )\}\) is expressed as

$$\begin{aligned} E\{C^{(1)}(\eta )\}=\int \limits _0^\infty {\left( {c_1 \int \limits _0^{\eta v} {(\eta v-v_1 } )g(v_1 )dv_1 +c_2 \int \limits _{\eta v}^\infty {(v_1 -\eta v)g(v_1 )dv_1 } } \right) } f(v)dv. \end{aligned}$$
(34)

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

$$\begin{aligned} \int \limits _0^\infty {v\left( {c_1 \int \limits _0^{\eta v} {g(v_1 )dv_1 -c_2 \int \limits _{\eta v}^\infty {g(v_1 )dv_1 } } } \right) } f(v)dv=0 \end{aligned}$$
(35)

or

$$\begin{aligned} \frac{\int \limits _0^\infty {vP(V_1 \le \eta v)f(v)dv} }{\int \limits _0^\infty {vf(v)dv} }=\frac{c_2 }{c_1 +c_2 }. \end{aligned}$$
(36)

It follows from (32), (34), and (36) that the optimum value of \(\eta \) is given by

$$\begin{aligned} \eta ^{{*}}=\frac{1}{m-k}\left[ {\left( {1+\frac{c_2 }{c_1 }} \right) ^{1/(k+1)}-1} \right] , \end{aligned}$$
(37)

the best invariant decision rule is

$$\begin{aligned} u^{\mathrm{{BI}}}=\eta ^{{*}}S_k =\frac{S_k }{m-k}\left[ {\left( {1+\frac{c_2 }{c_1 }} \right) ^{1/(k+1)}-1} \right] , \end{aligned}$$
(38)

and the expected cost, if we use \(u^\mathrm{{BI}}\), is given by

$$\begin{aligned} E_\sigma \{C(u^{\mathrm{{BI}}})\} =\frac{c_1 (k+1)}{m-k}\left[ {\left( {1+\frac{c_2 }{c_1 }} \right) ^{1/(k+1)}-1} \right] =\frac{c_1 (k+1)u^{\mathrm{{BI}}}}{S_k }=E\{C^{(1)}(\eta ^{{*}})\}. \end{aligned}$$
(39)

It will be noted that, on the other hand, the invariant embedding technique [11–22] allows one to transform equation (25) as follows:

$$\begin{aligned} E_\sigma \{C(u)\}&=\frac{1}{\sigma }\left( {c_1 \int \limits _0^u {(u-y_{k+1} } )g_\sigma (y_{k+1} )dy_{k+1} } \right. \left. {+c_2 \int \limits _u^\infty {(y_{k+1} -u)g_\sigma (y_{k+1} )dy_{k+1} } } \right) \nonumber \\&=\frac{1}{s_k }\left( {c_1 \int \limits _0^u (u-y_{k+1} )v^{2}(m-k)\exp \left( {-\frac{v(m-k)y_{k+1} }{s_k }} \right) \frac{1}{s_k }dy_{k+1}}\right. \nonumber \\&\left. \quad +\,c_2 \int \limits _u^\infty {(y_{k+1} -u)v^{2}(m-k)\exp \left( {-\frac{v(m-k)y_{k+1} }{s_k }} \right) \frac{1}{s_k }dy_{k+1} } \right) . \end{aligned}$$
(40)

Then it follows from (40) that

$$\begin{aligned} E\{E_\sigma \{(C(u)\}\}=\int \limits _0^\infty {E_\sigma } \{C(u)\}f(v)dv=E_{s_k } \{C^{(1)}(u)\}, \end{aligned}$$
(41)

where

$$\begin{aligned} E_{s_k } \{C^{(1)}(u)\}=&\frac{k}{s_k }\left( {c_1 \int \limits _0^u {(u-y_{k+1} } )g_{s_k }^{\bullet } (y_{k+1} )dy_{k+1} } \right. \nonumber \\&\left. \qquad \qquad +\,c_2 \int \limits _u^\infty {(y_{k+1} -u)g_{s_k }^{\bullet } (y_{k+1} )dy_{k+1} } \right) \end{aligned}$$
(42)

represents the expected predictive cost for the (\(k+1\))th period. It follows from (42) that the cost per the (\(k+1\))th period is reduced to

$$\begin{aligned} C^{(2)}(u)=\left\{ {\begin{array}{l} \displaystyle c_1 \frac{u-Y_{k+1} }{s_k /k}\quad \text{ if }\ Y_{k+1} \le u, \\ \displaystyle c_2 \frac{Y_{k+1} -u}{s_k /k}\quad \text{ if }\ Y_{k+1} >u, \\ \end{array}} \right. \end{aligned}$$
(43)

and the predictive probability density function of \(Y_{k+1}\) (compatible with (25)) is given by

$$\begin{aligned} g_{s_k }^{\bullet } (y_{k+1} )=(k+1)(m-k)\left[ {1+(m-k)\frac{y_{k+1} }{s_k }} \right] ^{-(k+2)}\frac{1}{s_k }, \quad y_{k+1} \ge 0. \end{aligned}$$
(44)

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

$$\begin{aligned} E_{s_k } \{C^{(2)}(u)\}&= \frac{k}{s_k }\left( c_1 \int \limits _0^u (u-y_{k+1} )g_{s_k }^{\bullet } (y_{k+1})dy_{k+1}\right. \nonumber \\&\qquad \quad \left. {}+c_2\int \limits _u^\infty {(y_{k+1} -u)g_{s_k }^{\bullet } (y_{k+1} )dy_{k+1} } \right) , \end{aligned}$$
(45)

with respect to \(u\), we obtain \(u^{\mathrm{{BI}}}\) immediately, and

$$\begin{aligned} E_{s_k } \{C^{(2)}(u^{\mathrm{{BI}}})\} =\frac{c_1 (k+1)}{m-k}\left[ {\left( {1+\frac{c_2 }{c_1 }} \right) ^{1/(k+1)}-1} \right] . \end{aligned}$$
(46)

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

$$\begin{aligned} C^{(3)}(\eta )=\left\{ {\begin{array}{l} \displaystyle c_1 k\left( {\frac{u}{s_k }-\frac{Y_{k+1} }{s_k }} \right) \quad \text{ if }\ \frac{Y_{k+1} }{s_k }\le \frac{u}{s_k } \\ \displaystyle c_2 k\left( {\frac{Y_{k+1} }{s_k }-\frac{u}{s_k }} \right) \quad \text{ if }\ \frac{Y_{k+1} }{s_k }>\frac{u}{s_k } \\ \end{array}} \right. =\left\{ {\begin{array}{l} c_1 k(\eta -W) \quad \text{ if }\ W\le \eta \\ \\ c_2 k(W-\eta )\quad \text{ if }\ W>\eta , \\ \end{array}} \right. \end{aligned}$$
(47)

where the probability density function of the ancillary statistic \(W=Y_{k+1}/S_{k}\) (compatible with (25)) is given by

$$\begin{aligned} g^{\circ }(w)=(k+1)(m-k)\left[ {1+(m-k)w} \right] ^{-(k+2)}, \quad w\ge 0. \end{aligned}$$
(48)

Then the best invariant decision rule \(u^{\mathrm{{BI}}}=\eta ^{{*}}S_k ,\) where \(\eta ^{{*}}\) minimizes

$$\begin{aligned} E\{C^{(3)}(\eta )\}=k\left( {c_1 \int \limits _0^\eta {(\eta -w} )g^{\circ }(w)dw} \right. \left. \,{+}\,c_2 \int \limits _\eta ^\infty {(w-\eta )g^{\circ }(w)dw} \right) . \end{aligned}$$
(49)

Comparison of statistical decision rules. For comparison, consider the maximum likelihood decision rule that may be obtained from (29),

$$\begin{aligned} u^{\mathrm{{ML}}}=\frac{{\mathop \sigma \limits ^{\frown } }}{{m - k}}\ln \left( {1+\frac{c_2 }{c_1 }} \right) =\eta _j^{\mathrm{{ML}}} S_k , \end{aligned}$$
(50)

where \({{\mathop \sigma \limits ^{\frown } }}=S_k /k\) is the maximum likelihood estimator of \(\sigma \),

$$\begin{aligned} \eta ^{\mathrm{{ML}}}=\frac{1}{m-k}\ln \left( {1+\frac{c_2 }{c_1 }} \right) ^{1/k}. \end{aligned}$$
(51)

Since \(u^{\mathrm{{BI}}}\) and \(u^{\mathrm{{ML}}}\) belong to the same class,

$$\begin{aligned} \mathbf{C }=\{u:u=\eta S_k \}, \end{aligned}$$
(52)

it follows from the above that \(u^{\mathrm{{ML}}}\) is inadmissible in relation to \(u^{\mathrm{{BI}}}\).

Numerical example. If, say, \(k=1\) and \(c_2 /c_1 =100\), we have that

$$\begin{aligned} \text{ Rel.eff }.\{u^{\mathrm{{ML}}}, u^{\mathrm{{BI}}}, {\sigma }\}={E_\sigma \{C(u^{\mathrm{{BI}}})\}}/{E_\sigma \{C(u^{\mathrm{{ML}}})\}=0.838.} \end{aligned}$$
(53)

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

Setup Model (s-S Policy). The present model differs from the one in (24) in that a setup cost \(c\) is incurred. Using the same notation, the total expected cost per the (\(k+1\))th period is

$$\begin{aligned}&\qquad \qquad E_\sigma \{\bar{{C}}(u)\}=c+E_\sigma \{C(u)\} \nonumber \\&=c+\frac{1}{\sigma }\left( c_1 \int \limits _0^u {(u-y_{k+1} } )g_\sigma (y_{k+1} )dy_{k+1} +c_2 \int \limits _u^\infty (y_{k+1} -u)g_\sigma (y_{k+1} )dy_{k+1} \right) . \end{aligned}$$
(54)

As shown above, the optimum value \(u^{*}\) must satisfy (28). Because \(c\) is constant, the minimum value of \(E_\sigma \{\bar{{C}}(u)\}\) must also occur at \(u^{*}\). In this case, \(S=u^{*}\), and the value of \(s ({<}S)\) is determined from the equation

$$\begin{aligned} E_\sigma \{C(s)\}=E_\sigma \{\bar{{C}}(S)\}=c+E_\sigma \{C(S)\},\quad s<S. \end{aligned}$$
(55)

This equation yields another value \(s_{1} ({>}S)\), 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) \(s \le q \le S\); (3) \(q > S\).

Case 1 (\(q < s\)). Because \(q\) is already on hand, its equivalent cost is given by \(E_\sigma \{C(q)\}.\) If any additional amount \(u-q\) (\(u>q)\) is ordered, the corresponding cost given \(u\) is \(E_\sigma \{\bar{{C}}(u)\}\), which includes the setup cost \(c\), and we have

$$\begin{aligned} \mathop {\min }\limits _{u>q} \ E_\sigma \{\bar{{C}}(u)\}=E_\sigma \{\bar{{C}}(S)\}<E_\sigma \{C(q)\}. \end{aligned}$$
(56)

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

Case 2 (\(s \le q \le S\)). In this case, we have

$$\begin{aligned} E_\sigma \{C(q)\}\le \mathop {\min }\limits _{u>q}\ E_\sigma \{\bar{{C}}(u)\}=E_\sigma \{\bar{{C}}(S)\}. \end{aligned}$$
(57)

Thus, it is not advantageous to order in this case and \(u^{*}=q\).

Case 3 (\(q > S\)). In this case, we have for \(u > q\),

$$\begin{aligned} E_\sigma \{C(q)\}<E_\sigma \{\bar{{C}}(u)\}. \end{aligned}$$
(58)

This condition indicates that, as in Case 2, it 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

$$\begin{aligned} \begin{array}{l} \text{ if }\ q<s,\;\text{ order }\;S-q, \\ \text{ if }\ q\ge s,\;\text{ do } \text{ not } \text{ order }. \\ \end{array} \end{aligned}$$
(59)

The optimality of the \(s-S\) policy is guaranteed because the associated cost function is convex.

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

$$\begin{aligned}&\qquad \qquad \quad E_{s_k } \{\bar{{C}}^{(1)}(u)\}=c+E_{s_k } \{C^{(1)}(u)\}\nonumber \\&=c+\frac{k}{s_k }\left( c_1 \int \limits _0^u {(u-y_{k+1} } )g_{s_k }^{\bullet } (y_{k+1} )dy_{k+1} + c_2 \int \limits _u^\infty (y_{k+1} -u)g_{s_k }^{\bullet } (y_{k+1} )dy_{k+1} \right) , \end{aligned}$$
(60)

is considered in the same manner as above.

4 Conclusion and Future Work

In this paper, we develop a new frequentist approach to improve predictive statistical decisions for inventory control problems under parametric uncertainty of the underlying distributions for the cumulative customer demand. Frequentist probability interpretations of the methods considered are clear. Bayesian methods are not considered here. We note, however, that, although subjective Bayesian prediction has a clear personal probability interpretation, it is not generally clear how this should be applied to non-personal prediction or decisions. Objective Bayesian methods, on the other hand, do not have clear probability interpretations in finite samples. For constructing the improved statistical decisions, a new technique of invariant embedding of sample statistics in a performance index is proposed. This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics.

The methodology described here can be extended in several different directions to handle various problems that arise in practice. We have illustrated the proposed methodology for location-scale distributions (such as the exponential distribution). Application to other distributions could follow directly.