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\begin{document}

\title{\textbf{Optimal replenishment policy for perishable items with
stock-dependent selling rate and capacity constraint}}
\author{Tsu-Pang Hsieh\thanks{%
Graduate School of Management Sciences, Aletheia University, Tamsui, Taipei,
Taiwan, 251, R.O.C.} and Chung-Yuan Dye\setcounter{footnote}{2}\thanks{%
Department of Business Administration, Shu-Te University, Yen Chau,
Kaohsiung, Taiwan, 824, R.O.C.}\setcounter{footnote}{-1}\thanks{%
Email: tsupang@gmail.com (T.-P. Hsieh)} }
\maketitle

\begin{abstract}
\indent In this paper, a deterministic inventory model is developed for
deteriorating items with stock-dependent demand and finite shelf/display
space. Furthermore, we allow for shortages and the unsatisfied demand is
partially backlogged at the exponential rate with respect to the waiting
time. We provide solution procedures for finding the maximum total profit
per unit time. In a specific circumstance, the model will reduce to the case
with no shortage. Further, we use numerical examples to illustrate the model.

\bigskip

\noindent \textbf{Keywords}: perishable items, stock-dependent demand,
capacity constraint, partial backlogging
\end{abstract}

\newpage

\setlength {\baselineskip} {1.5 \initiallineskip}

\section{Introduction}

\indent In the last several years, the influence of displayed stock-level on
customers has been recognized by many marketing researchers and
practitioners. High inventories might stimulate demand for a variety of
reasons. For example, tall stacks of a product can promote visibility, thus
kindling latent demand. A large inventory might also signal a popular
product, or provide consumers an assurance of high service levels and future
availability. Having many units of a product on hand also permits a retailer
to disperse the product across multiple locations on the sales floor,
thereby potentially capturing additional demand. Researchers like Levin et
al. (\citeyear{Leiven}) and Silver and Peterson (\citeyear{Silver}) observed
the functional relationship between the demand and the on-display stock
level. Due to this fact, various functional forms of the stock-dependent
demand rate were assumed to analyze inventory control policies under
realistic situations. Baker and Urban (\citeyear{Baker}) considered a
power-form inventory-level-dependent demand rate, which would decline along
with the stock level throughout the entire cycle. Datta and Pal (%
\citeyear{Datta}) modified the model of Baker and Urban (\citeyear{Baker})
by assuming that the stock-dependent demand rate was down to a given level
of inventory, beyond which it is a constant. Gupta and Vart (\citeyear{Gupta}%
) assumed that the demand rate was a function of initial stock level. Mandal
and Phaujdar (\citeyear{Mandal}) then developed on deteriorating inventory
model in the case of deterministic demand rate that depended linearly on the
instantaneous stock level.

Due to various uncertainties, the occurrence of shortages in inventory is a
natural phenomenon. Hence, Urban (\citeyear{Urban2}) extended Baker and
Urban's (\citeyear{Baker}) model to allow shortages, in which unsatisfied
demand is backlogged at a fixed fraction of the constant demand rate.
Padmanabhan and Vrat (\citeyear{PadVrat}) developed an inventory model in
which the backlogging rate depends upon the total number of customers in the
waiting line (\textit{i.e.}, the amount of the negative inventory level).
Therefore, the more the amount of demand backlogged, the smaller the demand
to accept backlogging would be. Their definition of backlogging rate,
however, seems to be inappropriate when customers do not know how many
buyers waiting before him/her. When there is a shortage, most customers only
concern on the duration he/she has to wait.

During the shortages period, often some customers are conditioned to a
shipping delay, and may be willing to wait for a short time, while other
will leave for another seller because of urgent need. Therefore, the length
of the waiting time for the next replenishment is the main factor for
deciding whether the backlogging will be accepted or not. To reflect this
phenomenon, Abad (\citeyear{Abad1, Abad2, Abad3}) discussed a pricing and
lot-sizing problem for a product with a variable rate of deterioration,
allowing shortages and partial backlogging. The backlogging rate depends on
the time to replenishment -- the longer customers must wait, the greater the
fraction of lost sales. Since Abad (\citeyear{Abad1}) proposed two specific
examples of impatient functions--the exponential rate and the hyperbolic
rate with respect to waiting time, these two functions have been used to
model backordering in several studies, for example, Papachristos and Skouri (%
\citeyear{Pap00}), Teng et al. (\citeyear{Teng02}), Skouri and Papachristos (%
\citeyear{Skouri03}), San Jose et al. (\citeyear{San06}) and Dye (%
\citeyear{Dye07}).

Companies have recognized that besides maximizing profit, customer
satisfaction plays an important role for getting and keeping a successful
position in a competitive market. Common measures for customer service in
inventory literature are the non-stockout probability per replenishment
cycle. With a lost sale, the customer's demand for the item is lost and
presumably filled by a competitor. It can be considered as the loss of
profit on the sales. Ignoring the stockout cost from the total profit leads
to the overrated profit and less customer satisfaction. Therefore,
information about the backlogging rate during the waiting time is required
and it is of considerable interest to understand the trade-off between the
investment in inventory and the service level. Recently, Dye and Ouyang (%
\citeyear{Dye}) amended Padmanabhan and Vrat's (\citeyear{PadVrat}) partial
backlogging model by adding the cost of lost sales in the profit function
and using Abad's linear time-proportional backlogging rate. They established
a unique optimal solution to the problem in which building up inventory has
a negative effect on the profit. Later, Chang et al. (\citeyear{Teng})
complemented Dye and Ouyang's model for the situation that building up
inventory has a positive impact on the profit by limiting the capacity of
shelf space. Wu et al. (\citeyear{Wu06}) considered the problem for
non-instantaneous deteriorating items with stock-dependent demand to
determine the lot size at the hyperbolic backlogging rate. However, the
exponential backlogging rate has not been used to model the inventory
problem for deteriorating items with stock-dependent demand.

Hence, the main purpose of this paper is to determine the optimal
replenishment policy for deteriorating items with stock-dependent demand. In
the model, shortages are allowed, and the rate of backlogged demand
decreases exponentially as the waiting time for the next replenishment
increases. In addition, the capacity of shelf space is assumed to be finite.
The rest of the paper is organized as follows. In the next section, the
notation and assumptions related to this study are presented. In Section 3,
we develop the criterion for finding the optimal solution for the
replenishment schedule, and prove that the optimal replenishment policy not
only exists but is unique. In the last two sections, numerical examples are
used to illustrate the procedure of solving the model and concluding remarks
are provided.

\section{Notation and Assumptions}

\subsection{Notation}

\indent To develop the mathematical model of inventory replenishment
schedule, the notation adopted in this paper is as below:

\leftmargini=30mm \leftmarginii=35mm \leftmarginiii=51mm

\begin{enumerate}
\item[$A=$] the replenishment cost per order

\item[$c=$] the purchasing cost per unit

\item[$s=$] the selling price per unit, where $s>c$

\item[$Q=$] the ordering quantity per cycle

\item[$B=$] the maximum inventory level per cycle, i.e., the initial
inventory level.

\item[$i=$] the carrying cost rate, the cost of having one dollar of the
item tied up in inventory per unit time

\item[$c_{2}=$] the shortage cost per unit per unit time

\item[$c_{3}=$] the cost of lost sales (i.e., goodwill cost) per unit

\item[$t_{1}=$] the length of period during which the inventory level
reaches zero, where $t_{1}\geq 0$

\item[$t_{2}=$] the length of period during which shortages are allowed,
where $t_{2}\geq 0$

\item[$T=$] the length of the inventory cycle, hence $T=t_{1}+t_{2}$

\item[$I_{1}\left( t\right) =$] the level of positive inventory at time $t$,
where $0\leq t\leq t_{1}$

\item[$I_{2}\left( t\right) =$] the level of negative inventory at time $t$,
where $t_{1}\leq t\leq t_{1}+t_{2}$

\item[$TP\left( t_{1},t_{2}\right) =$] the total profit per unit time
\end{enumerate}

\subsection{Assumptions}

\indent In addition, the following assumptions are imposed:

\leftmargini=5mm

\begin{enumerate}
\item Replenishment rate is infinite, and lead time is zero.

\item The time horizon of the inventory system is infinite.

\item The demand rate function, $D\left( t\right) $, is deterministic and is
a function of instantaneous stock level $I\left( t\right) $; the functional $%
D\left( t\right) $ is given by:%
\begin{equation*}
D(t)=\left\{
\begin{array}{ll}
\alpha +\beta I\left( t\right) , & 0\leq t\leq t_{1} \\
& \\
\alpha , & t_{1}\leq t\leq t_{1}+t_{2}%
\end{array}%
\text{\ }\right.
\end{equation*}%
where $\alpha $ and $\beta $\ are non-negative constants.

\item The distribution of time to deterioration of the items follows
exponential distribution with parameter $\theta $ (i.e. constant rate of
deterioration).

\item The retail outlets have limited shelf space. It is $W$ units, where $%
W\geq B$.

\item Shortages are allowed. We adopt the concept used in Abad (%
\citeyear{Abad1,Abad2}), i.e., the unsatisfied demand is backlogged, and the
fraction of shortages backordered is $e^{-\delta x}$, where $x$ is the
waiting time up to the next replenishment and $\delta $ is a positive
constant.
\end{enumerate}

\section{Mathematical formulation}

\begin{figure}[tph]
\centering \includegraphics[width=0.9\textwidth]{backlog.eps}
\caption{Graphical representation of inventory system}
\label{Stocklevel}
\end{figure}

Using above assumptions, the inventory level follows the pattern depicted in
Fig. \ref{Stocklevel}. To establish the total profit function, we consider
the following time intervals separately, $\left[ 0,t_{1}\right] $ and $\left[
t_{1},t_{1}+t_{2}\right] $. During the interval $\left[ 0,t_{1}\right] $,
the inventory is depleted due to the combined effects of demand and
deterioration. Hence the inventory level is governed by the following
differential equation:%
\begin{equation}
\frac{\text{d}I_{1}(t)}{\text{d}t}=-\alpha -\beta I_{1}\left( t\right)
-\theta I_{1}\left( t\right) ,0<0$. c="\left(">0, & \text{if }t_{2}\in \left[ 0,\widetilde{t}_{2}\right) , \\
\ & \\
<0,>0, & \text{if }t_{2}\in \left( \widetilde{t}_{2},\infty \right) ,%
\end{array}%
\right.
\end{equation*}%
which implies $Z\left( t_{2}\right) $ is strictly decreasing in the interval
$\left[ 0,\widetilde{t}_{2}\right] $ and strictly increasing in the interval
$\left[ \widetilde{t}_{2},\infty \right) $. Besides, $Z\left( t_{2}\right) $
has the minimum value at $\widetilde{t}_{2}$. Then, we have the following
result.

\noindent \textbf{Theorem 1. }\textit{For }$\beta s-\left( i+{\beta +\theta }%
\right) c\geq 0$\textit{, we have}

\begin{enumerate}
\item[(a)] \textit{If }$Z\left( 0\right) >0$\textit{\ and }$Z(\widetilde{t}%
_{2})<0$\textit{,>0$\ and $Z(\widetilde{t}_{2})<0$,>0$\textit{\ and }$Z\left(
\widetilde{t}_{2}\right) \geq 0$\textit{, then the optimal solution is }$%
\left( t_{1}^{\ast \ast },t_{2}^{\ast \ast }\right) =(\widehat{t}_{1},\infty
)$\textit{.}

\item[(c)] \textit{If }$G(\widehat{t}_{2})>0$\textit{\ and }$Z\left(
\widetilde{t}_{2}\right) <0$\textit{, beta ="0.25$," theta ="0.05$," delta ="1$," a="\$250$/per" i="0.35$," c="\$5$/per" alpha ="600$" s="\$15$/per" w="100$" c="0.5$," alpha ="300$" s="\$15$/per" w="700$" c="0.5$" alpha ="600$" s="\$10$/per" w="700$" c="-0.75$," alpha ="600$" s="\$7$/per" w="100$" c="-1.5$," width="0.45\textwidth]{opt1}" width=" 0.45\textwidth]{opt2}" width=" 0.45\textwidth]{opt3}" width=" 0.45\textwidth]{opt4}" width="0.45\textwidth]{sen1-1}" width=" 0.45\textwidth]{sen2-1}" width=" 0.45\textwidth]{sen3-1}" width=" 0.45\textwidth]{sen4-1}">\widetilde{t}_{2}$%
, Newton-Raphson Method fails to produce a solution satisfying the
sufficient condition for the maximality problem of $TP\left(
t_{1},t_{2}\right) $ and it will converge to a saddle point.

The proposed model can be extended in several ways. For instance, we could
extend the deterministic demand function to stochastic demand patterns. The
demand could also be generalized as a function of the price and stock level.
Furthermore, we could generalize the model to allow for permissible delay in
payments.

\section*{Acknowledgements}

\indent \indent The authors would like to thank the editor and anonymous
reviewers for their valuable and constructive comments, which have led to a
significant improvement in the manuscript. This research was partially
supported by the National Science Council of the Republic of China under
Grant NSC-97-2221-E-156-005 and NSC-97-2221-E-366-006-MY2.

\section*{Appendix A}

\noindent \textbf{The proof of Theorem 1\null.}

\noindent (a) First, we consider the interval $\left[ 0,\widetilde{t}_{2}%
\right] $. Because $Z\left( t_{2}\right) $ is strictly decreasing in the
interval $\left[ 0,\widetilde{t}_{2}\right] $, and on condition that $%
Z\left( 0\right) >0$ and $Z(\widetilde{t}_{2})<0$,>0$ for $t_{2}<\widetilde{t}_{2}$, we obtain% \begin{equation*} \left. \frac{\text{d}^{2}TP\left( t_{2}\right) }{\text{d}t_{2}^{2}}% \right\vert _{t_{2}=t_{2}^{\ast }}=\frac{1}{(\widehat{t}_{1}+t_{2})^{2}}% \frac{\text{d}Z(t_{2})}{\text{d}t_{2}}<0.>0.
\end{equation*}%
This means that $t_{2}^{\ast \ast }$ is not the optimal solution for our
maximum problem. Hence, to find the optimal solution of $t_{2}$, the
interval $\left[ \widetilde{t}_{2},\infty \right) $ should be excluded from
consideration. This completes the proof. $\square $

\noindent (b) Since $Z\left( t_{2}\right) $ has a global minimum at $%
\widetilde{t}_{2}$, if $Z\left( \widetilde{t}_{2}\right) >0$, then we have $%
Z\left( t_{2}\right) >Z\left( \widetilde{t}_{2}\right) >0$ for all $%
t_{2}\neq \widetilde{t}_{2}$. Therefore, from Eq. (\ref{Zt2}), we obtain
that $\frac{\text{d}TP\left( t_{2}\right) }{\text{d}t_{2}}=\frac{Z\left(
t_{2}\right) }{(\widehat{t}_{1}+t_{2})^{2}}>0$, which implies that a larger
value of $t_{2}$ causes a higher value of $TP\left( t_{2}\right) $. Hence
the maximum value of $TP\left( t_{2}\right) $ occurs at the point $%
t_{2}^{\ast }\rightarrow \infty $. For another case $Z\left( \widetilde{t}%
_{2}\right) =0$, since $\left. \frac{\text{d}TP\left( t_{2}\right) }{\text{d}%
t_{2}}\right\vert _{t_{2}=\widetilde{t}_{2}}=0$ and $TP\left( t_{2}\right) $
is strictly increasing in $(0,\widetilde{t}_{2})$ and $(\widetilde{t}%
_{2},\infty )$, $t_{2}=\widetilde{t}_{2}$ is an inflection point. As a
result, the maximum value of $TP\left( t_{2}\right) $ occurs at the point $%
t_{2}^{\ast }\rightarrow \infty $. This completes the proof. $\square $

\noindent (c) First, we consider $Z\left( 0\right) <0$.>0$. Because $G\left( t_{2}\right) $ is strictly
decreasing in the interval $[0,\widehat{t}_{2}]$, and on condition that $G(%
\widehat{t}_{2})<0$,>0$ for $t_{2}<\widehat{t}_{2}$, we obtain \begin{equation*} \left. \frac{\partial ^{2}TP(t_{1},t_{2})}{\partial t_{2}^{2}}\right\vert _{\left( t_{1},t_{2}\right) =\left( t_{1}^{\ast \ast },t_{2}^{\ast \ast }\right) }=\frac{-\alpha }{t_{1}^{\ast \ast }+t_{2}^{\ast \ast }}\left[ \delta \left( s-c+c_{3}\right) +c_{2}\left( 1-\delta t_{2}^{\ast \ast }\right) \right] e^{-\delta t_{2}^{\ast \ast }}<0,>&0.
\end{eqnarray*}%
As a result, we can conclude that the stationary point $\left( t_{1}^{\ast
\ast },t_{2}^{\ast \ast }\right) $\ is the optimal solution for our maximum
problem. This completes the proof. $\square $

\noindent (b) If $G(\widehat{t}_{2})>0$, since $G\left( t_{2}\right) $ is a
strictly decreasing in the interval $[0,\widehat{t}_{2}]$, we obtain $\frac{%
\partial TP\left( t_{1},t_{2}\right) }{\partial t_{2}}=\frac{G\left(
t_{2}\right) }{\left( t_{1}+t_{2}\right) ^{2}}\geq \frac{G(\widehat{t}_{2})}{%
\left( t_{1}+t_{2}\right) ^{2}}>0$ for any $0\leq t_{2}\leq \widehat{t}_{2}$%
, which implies that for fixed $t_{1}\in \lbrack 0,\widehat{t}_{1}]$, $%
TP\left( t_{1},t_{2}\right) $ is\ a strictly increasing function in $%
t_{2}\in \lbrack 0,\widehat{t}_{2}]$. Therefore, a larger value of $t_{2}$
causes a larger value of $TP\left( t_{1},t_{2}\right) $. It yields $t_{2}=%
\widehat{t}_{2}$. By substituting $t_{2}=\widehat{t}_{2}$ into Eq. (\ref%
{dTPdt1}), we see that Eq. (\ref{dTPdt1}) holds only if $t_{1}=\widehat{t}%
_{1}$. Hence inventory should be displayed to the maximum allowable $W$
units.

However, when $t_{1}=\widehat{t}_{1}$, we consider the function $TP(\widehat{%
t}_{1},t_{2})=TP\left( t_{2}\right) $. From Eqs. (\ref{Zt2}) and (\ref{Gt2}%
), it is not difficult to check that $G(\widehat{t}_{2})=Z(\widehat{t}_{2})$%
. Then, from Theorem 1(b), under the condition $Z\left( \widetilde{t}%
_{2}\right) \geq 0$, we know that $t_{1}^{\ast \ast }=\widehat{t}_{1}$ and
the maximum value of $TP\left( t_{2}\right) $ occurs at the point $%
t_{2}^{\ast \ast }\rightarrow \infty $. This completes the proof. $\square $

\noindent (c) If $G(\widehat{t}_{2})>0$, we have known that $t_{1}^{\ast
\ast }=\widehat{t}_{1}$. Since $G(\widehat{t}_{2})=Z(\widehat{t}_{2})>0$ and
$Z\left( \widetilde{t}_{2}\right) <0$,>0$, then we have $%
G\left( t_{2}\right) >G\left( \widetilde{t}_{2}\right) >0$ for all $%
t_{2}\neq \widetilde{t}_{2}$. Therefore, from Eqs. (\ref{TPdt2}) and (\ref%
{Gt2}), we obtain that $\frac{\partial TP\left( t_{1},t_{2}\right) }{%
\partial t_{2}}=\frac{G\left( t_{2}\right) }{\left( t_{1}+t_{2}\right) ^{2}}%
>0$, which implies that a larger value of $t_{2}$ causes a higher value of $%
TP\left( t_{1},t_{2}\right) $. Hence the maximum value of $TP\left(
t_{1},t_{2}\right) $ occurs at the point $t_{2}^{\ast \ast }\rightarrow
\infty $. For the another case $G\left( \widetilde{t}_{2}\right) =0$, since $%
\left. \frac{\partial TP\left( t_{1},t_{2}\right) }{\partial t_{2}}%
\right\vert _{t_{2}=\widetilde{t}_{2}}=0$ and $TP\left( t_{1},t_{2}\right) $
is strictly increasing in $(0,\widetilde{t}_{2})$ and $(\widetilde{t}%
_{2},\infty )$, $t_{2}=\widetilde{t}_{2}$ is an inflection point. As a
result, the maximum value of $TP\left( t_{1},t_{2}\right) $ occurs at the
point $t_{2}^{\ast \ast }\rightarrow \infty $. This completes the proof. $%
\square $

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\end{thebibliography}

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