話說，專家系統（Expert Systems With Applications ）真的是研究生的好朋友！
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\begin{document}
\title{\textbf{Pricing and lot-sizing policies for deteriorating items with
partial backlogging under inflation}}
\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 Management, Shu-Te University, Yen Chao, Kaohsiung,
Taiwan, 824, R.O.C.}\setcounter{footnote}{-1}\thanks{%
Corresponding author: Dr. Tsu-Pang Hsieh, E-mail: tsupang@gmail.com} } }
\maketitle
\begin{abstract}
\setlength {\baselineskip} {2.0 \initiallineskip} In this paper, we develop
an inventory lot-size model for deteriorating items under inflation using a
discounted cash flow (DCF) approach over a finite planning horizon. We allow
not only a multivariate demand function of price and time but partial
backlogging. In addition, selling price is allowed for periodical upward and
downward adjustments. The objective is to find the optimal lot size and
periodic pricing strategies so that the net present value of total profit
could be maximized. By using the properties derived from this paper and the
Nelder--Mead algorithm, we provide a complete search procedure to find the
optimal selling price, replenishment number and replenishment timing for the
proposed model. At last, numerical examples are used to illustrate the
algorithm.
\noindent \textbf{Keyword:} Pricing, Lot-Sizing, Deteriorating Items,
Partial Backlogging, Nelder--Mead Algorithm
\end{abstract}
\newpage \setlength {\baselineskip} {2.0 \initiallineskip}
\section{Introduction}
In the past 30 years, analysis of the inventory model allowing the constant
demand rate to vary with time over a finite time horizon has extended the
field of inventory control in practice. In the earlier period, researchers
had discussed different demand patterns fitting the stage of product life
cycle. Resh \textit{et al.} (\citeyear{Resh76}) and Donaldson (%
\citeyear{Donaldson77}) considered the situation of linearly time-varying
demand and established an algorithm to determine the optimal number of
replenishments and timing. Barbosa and Friedman (%
\citeyear{Barbosa78,
Barbosa79}) adjusted the lot-size problem for the cases of different
increasing power-form demand functions and declining demand patterns,
respectively. Henery (\citeyear{Henery79}) further generalized the demand
rate by considering a log-concave demand function increasing with time.
Following the approach of Donaldson, Dave (\citeyear{Dave89}) developed an
exact replenishment policy for an inventory model with shortages. Yang
\textit{et al.} (\citeyear{Yang02}) extended Barbosa and Friedman's (%
\citeyear{Barbosa78}) model to allow for shortages. To characterize the more
practical situation, Chen \textit{et al.} (\citeyear{Chen07AMC, Chen07CMA})
dealt with the inventory model under the demand function following the
product-life-cycle shape. They employed the Nelder--Mead algorithm to solve
the mixed-integer nonlinear programming problem and determined the optimal
number of replenishment and the optimal replenishment time points.
In real life, the deterioration phenomenon is observed on inventory items
such as fruits, vegetables, pharmaceuticals, volatile liquids, and others.
By considering this phenomenon occurring during the holding period, Dave and
Patel (\citeyear{Dave81}) integrated time-proportional demand and a constant
rate of deterioration in the inventory model where shortages were prohibited
over the finite planning horizon. Sachan (\citeyear{Sachan84}) further
extended the model to allow for shortages. Later, Hariga (\citeyear{Hariga96}%
) generalized the demand pattern to any log-concave function. Researchers
including Mudereshwar (\citeyear{Muder88}), Goswami and Chaudhuri (%
\citeyear{Gos91}), Goyal, Morin and Nebebe (\citeyear{Goyal92}), Benkherouf (%
\citeyear{Benk95}), Chakrabarti and Chaudhuri (\citeyear{Chak97}), and
Hariga and Alyan (\citeyear{Hariga97}) developed economic order quantity
models that focused on deteriorating items with time-varying demand and
shortages.
However, the above inventory models unrealistically assume that during
stockout period all demand is either backlogged or lost. In reality, some
customers are willing to wait until replenishment, especially when the
waiting period is short, while others are more impatient and go elsewhere.
To reflect this phenomenon, Abad (\citeyear{Abad96}) provided two sets of
time-proportional backlogging rates: (i) linear time-proportional
backlogging rate and (ii) exponential time-proportional backlogging rate.
Chang and Dye (\citeyear{Dye99}) developed a finite time horizon EOQ model
in which the proportion of customers who would like to accept backlogging is
the reciprocal of a linear function of the waiting time. Concurrently,
Papachristos and Skouri (\citeyear{Papa00}) established a partially
backlogged inventory model by assuming that the backlogging rate decreases
exponentially as the waiting time increases. Recently, Teng \textit{et al.} (%
\citeyear{Teng02}) and Chern \textit{et al.} (\citeyear{Chern08}) extended
the fraction of unsatisfied demand backordered to any decreasing function of
the waiting time up to the next replenishment.
Since price is viewed as an important vehicle to influence demand in most of
the business environment, many researchers are led to investigate inventory
models with a price-dependent demand. Cohen (\citeyear{Cohen77}) jointly
determined the optimal replenishment cycle and price for deteriorating items
with the demand rate dependent linearly on the selling price. Abad (%
\citeyear{Abad96,
Abad01, Abad03, Abad08}) incorporated the demand rate described by any
convex decreasing function of the selling price into the inventory model,
taking a general rate of deterioration and a variable backlogging rate. Ho
\textit{et al.} (\citeyear{Ho08}) and Chang \textit{et al.} (%
\citeyear{Chang09}) presented the iso-elastic demand in an integrated
supplier-buyer inventory model under the condition of a permissible delay in
payment, respectively. The aforementioned studies assume firms have no
pricing power to change the selling price periodically and adopt the fixed
price policy. As opposed to the conventional fixed price policy, Chen and
Chen (\citeyear {Chen04}) presented an inventory model for deteriorating
items with a multivariate demand function of price and time, taking account
of the effects of inflation and time discounting over multiperiod planning
horizon. However, the integer length of replenishment cycle is within
certain limits due to the procedure using dynamic programming techniques.
It is noted that the literature about a finite time horizon inventory model
rarely considers the cases with periodic adjustments of price. In this
paper, we investigate the replenishment policies for deteriorating items
with partial backlogging by considering a multivariate demand function of
price and time. The fraction of unsatisfied demand backordered is any
decreasing function of the waiting time up to the next replenishment. In
addition, the selling price is allowed periodical upward and downward
adjustments and the time value of money is taken into consideration. The
objective of the inventory problem here is to determine the number of
replenishments, the selling price per replenishment cycle, the timing of the
reorder points and the shortage points. Following the properties derived
from this paper, we provide a complete search procedure to find the optimal
solutions by employing the Nelder--Mead algorithm. Several numerical
examples are used to illustrate the features of the proposed model. At last,
we make a summary and provide some suggestions for future research.
\section{Assumptions and Notation}
\subsection{Assumptions}
\indent \indent The mathematical model of the inventory replenishment
problem is based on the following assumptions:
\begin{enumerate}
\item The planning horizon of the inventory problem here is finite and is
taken as $H$ time units.
\item Lead time is zero.
\item The initial inventory level is zero.
\item A constant fraction of the on-hand inventory deteriorates per unit of
time and there is no repair or replacement of the deteriorated inventory.
\item Shortages are allowed. The fraction of shortages backordered is a
decreasing function $\beta (x)$, where $x$ is the waiting time up to the
next replenishment, and $0\leq \beta (x)\leq 1$ with $\beta (0)=1$. Note
that if $\beta (x)=1$ (or $0$) for all $x$, then shortages are completely
backlogged (or lost).
\end{enumerate}
\subsection{Notation}
\leftmargini=20mm \leftmarginii=30mm \leftmarginiii=46mm
\begin{itemize}
\item[$\protect\theta =$] the deterioration rate.
\item[$c_{f}=$] the fixed purchasing cost per order.
\item[$p_{i}=$] the selling price per unit (a decision variable) in the $i$%
th replenishment cycle, defined in the interval [0, $p_{u}$], where $p_{u}$
is the upper bound.
\item[$f\left( t,p_{i}\right) =$] the demand rate at time $t$ and price $%
p_{i}$ with $f\left( t,p_{i}\right) =g\left( t\right) A\left( p_{i}\right) $%
, where $g\left( t\right) $ is a positive and continuous function of time in
the planning horizon $(0,H]$ and $A\left( p_{i}\right) $ is any
non-negative, continuous, convex, decreasing function of the selling price
in [0, $p_{u}$].
\item[$c_{v}=$] the purchasing cost per unit.
\item[$c_{h}=$] the inventory holding cost per unit per unit time.
\item[$c_{s}=$] the backlogging cost per unit per unit time due to shortages.
\item[$c_{l}=$] the unit cost of lost sales.
\item[$n=$] the number of replenishments over [0, $H$] (a decision variable).
\item[$t_{i}=$] the $i$th replenishment time (a decision variable), $%
i=1,2,\ldots ,n$.
\item[$s_{i}=$] the time at which the inventory level reaches zero in the $i$%
th replenishment cycle (a decision variable), $i=1,2,\ldots ,n$.
\end{itemize}
\noindent As a result, the decision problem here has $3n$ decision variables.
\section{Model Formulation}
\begin{figure}[tph]
\centering \includegraphics[height=4in]{StockLevel.eps}
\caption{Graphical Representation of Inventory System}
\end{figure}
According to the notations and assumptions mentioned above, the behavior of
inventory system at any time can be depicted in Fig. 1. From Fig. 1, it can
be seen that the depletion of the inventory occurs due to the combined
effects of the demand and the deterioration during the interval $\left[
t_{i},s_{i}\right) $ of the $i$th replenishment cycle. Hence, the variation
of inventory with respect to time can be described by the following
differential equation:%
\begin{equation}
\frac{\text{d}I(t)}{\text{d}t}=-f\left( t,p_{i}\right) -\theta I\left(
t\right) ,\text{ }t_{i}
\textit{\ for }$i=1,$\textit{\ }$2,\ldots ,n-1$\textit{.}}
\item[(b)] {\textit{If }$g\left( t\right) $\textit{\ is decreasing with
respect to }$t$\textit{, then the shortage periods and the inventory periods
are getting larger with respect to the number of replenishment, i.e.,}$\
t_{i}-s_{i-1}
%for }$i=1,$\textit{\ }$2,\ldots ,n-1$\textit{.}}
%
%\item[(b)] {\textit{If }$g\left( t\right) $\textit{\ is decreasing with
%respect to }$t$\textit{, then }$\ s_{i}-s_{i-1}
other hand, If $g\left( t\right) $\ is decreasing with respect to $t$, $\
s_{i}^{\text{0}}-s_{i-1}^{\text{0}}
\item[\textbf{Step 3}] If $TP(n)>TP(n-1)$, then compute $TP(n+1)$, $TP(n+2)$%
, \ldots , until we find $TP(k)>TP(k+1)$. Set $n^{\ast }=k$ and stop.
\end{itemize}
\section{Numerical examples}
To illustrate the results, apply the proposed algorithms to solve the
following numerical examples. Algorithms 1 and 2 are implemented on a
personal computer with Intel Core 2 Duo under Mac OS X 10.5.6 operating
system using Mathematica version 7. Algorithm 1 is terminated if the
standard deviation, the values of the function being minimized, at the
vertices of the simplex is less than the 0.001.
\noindent \textbf{Example 1.} We first redo the same example of Chen and
Chen (\citeyear{Chen04}) while considering the time increasing demand. $%
f\left( t,p \right) =\left( 300-120p \right)e^{0.06t}$, $c_{f}=40$, $%
c_{h}=0.02 $, $c_{s}=0.5$, $\theta =0.2$, $c_{v}=1$, $H=12$,
$r=0.02$. Besides, we assume that the time-dependent backlogging
rate is $\beta \left( x\right) =e^{-0.2x} $ and take $c_{l}=0.6$.
By applying (\ref{neoq}), we obtain the estimated number of
replenishments $n=6$. Then, applying the Algorithm 1 and 2, we get
$TP(5)=553.94$, $TP(6)=569.32$, $TP(7)=570.53$ and $TP(8)=562.61$.
Therefore, the optimal number of replenishments is 7, and the
optimal pricing and replenishment policy is shown in Table 1. The
behavior of inventory system over the planning horizon is depicted
in Fig. 1.
\begin{table}[tph]
\caption{Optimal pricing and replenishment schedule for Example 1}
\label{ex1}
\begin{center}
\begin{tabular}{cccc}
\hline
$i$ & $p_i$ & $t_i$ & $s_i$ \\ \hline
1 & 1.8448 & 0.5938 & 2.0208 \\
2 & 1.8385 & 2.5686 & 3.9141 \\
3 & 1.8338 & 4.4251 & 5.7114 \\
4 & 1.8289 & 6.1884 & 7.4092 \\
5 & 1.8246 & 7.8635 & 9.0212 \\
6 & 1.8205 & 9.4418 & 10.5462 \\
7 & 1.8170 & 10.9428 & 12.0000 \\ \hline
\end{tabular}%
\end{center}
\end{table}
\begin{figure}[tph]
\centering \includegraphics[height=2.5in]{inventorylevelplot1.eps}
\caption{Graphical Representation of Inventory System for Example 1}
\end{figure}
\noindent \textbf{Example 2.} In this example, we redo an inventory
situation proposed by Chen \textit{et al.} (\citeyear{Chen07CMA}) while
considering the deterioration and partial backlogging. $c_{f}=50$, $c_{h}=5$%
, $c_{s}=7$, $c_{v}=10$, $H=1$, $r=0.02$. $f\left( t,p\right) =\left(
5000-150p\right) \frac{1}{B\left( 3,2\right) }t^{3-1}\left( H-t\right) ^{2-1}
$. By considering the deterioration and partial backlogging, we assume that
the deterioration rate is $\theta =0.08$ and the time-dependent backlogging
rate is $\beta \left( x\right) =e^{-0.2x}$. Besides, we take $c_{l}=6$.
Applying (\ref{neoq}), we obtain the estimated number of replenishments $n=8$%
. Then, applying the Algorithm 1 and 2, we get $TP(8)=19435.40$, $%
TP(7)=19438.41$, $TP(6)=19426.04$ and $TP(5)=19390.90$. Therefore,
the optimal number of replenishments is 7, and the optimal pricing
and replenishment policy is shown in Table 2. The behavior of
inventory system over the planning horizon is depicted in Fig. 2.
\begin{table}[tph]
\caption{Optimal pricing and replenishment schedule for Example 2}
\label{ex5}
\begin{center}
\begin{tabular}{cccc}
\hline
$i$ & $p_i$ & $t_i$ & $s_i$ \\ \hline
1 & 21.8919 & 0.2152 & 0.3086 \\
2 & 21.7980 & 0.3610 & 0.4340 \\
3 & 21.7767 & 0.4749 & 0.5395 \\
4 & 21.7667 & 0.5756 & 0.6359 \\
5 & 21.7639 & 0.6697 & 0.7302 \\
6 & 21.7691 & 0.7644 & 0.8312 \\
7 & 21.7997 & 0.8682 & 1.0000 \\ \hline
\end{tabular}%
\end{center}
\end{table}
\begin{figure}[tph]
\centering \includegraphics[height=2.5in]{inventorylevelplot3.eps}
\caption{Graphical Representation of Inventory System for Example 2}
\end{figure}
\noindent \textbf{Example 3.} $f\left( t,p\right) =30000p^{-2}\left(
100-15t\right) $, $\beta \left( x\right) =1/\left( 1+x\right) $, $c_{f}=250$%
, $c_{h}=40$, $c_{s}=80$, $c_{l}=120$, $\theta =0.08$, $c_{v}=200$, $H=4$, $%
r=0.02$. By applying (\ref{neoq}), we obtain the estimated number of
replenishments $n=4$. Then, apply the Algorithms 1 and 2 to get $%
TP(3)=7941.30$, $TP(4)=8026.90$, $TP(5)=7994.21$ and
$TP(6)=7897.58$. Therefore, the optimal number of replenishments
is 4, and the optimal pricing and replenishment policy is shown in
Table 3. The behavior of inventory system over the planning
horizon is depicted in Fig. 3.
\begin{table}[tph]
\caption{Optimal pricing and replenishment schedule for Example 3}
\label{ex4}
\begin{center}
\begin{tabular}{cccc}
\hline
$i$ & $p_i$ & $t_i$ & $s_i$ \\ \hline
1 & 440.8304 & 0.1053 & 0.8338 \\
2 & 444.4457 & 0.9474 & 1.7438 \\
3 & 449.8203 & 1.8689 & 2.7688 \\
4 & 459.0861 & 2.9107 & 4.0000 \\ \hline
\end{tabular}%
\end{center}
\end{table}
\begin{figure}[tph]
\centering \includegraphics[height=2.5in]{inventorylevelplot2.eps}
\caption{Graphical Representation of Inventory System for Example 3}
\end{figure}
\noindent \textbf{Example 4.} In this example, we illustrate the power-form
demand function varying with time which has been presented by Barbosa and
Friedman (\citeyear{Barbosa78}) and Yang \textit{et al.} (\citeyear{Yang02}%
). The same parameter values in Example 3 are used except the power-form
demand function $f\left( t,p\right) =30000p^{-2.2}\left( 10+3t\right) ^{2}$.
By applying (\ref{neoq}), we obtain the estimated number of replenishments $%
n=5$. Then, applying the Algorithm 1 and 2, we get $TP(4)=9078.61$, $%
TP(5)=9141.17$, $TP(6)=9108.96$ and $TP(7)=9019.85$. Therefore,
the optimal number of replenishments is 5, and the optimal pricing
and replenishment policy is shown in Table 4. The behavior of
inventory system over the planning horizon is depicted in Fig. 4.
\begin{table}[tph]
\caption{Optimal pricing and replenishment schedule for Example 4}
\begin{center}
\begin{tabular}{cccc}
\hline
$i$ & $p_i$ & $t_i$ & $s_i$ \\ \hline
1 & 418.725 & 0.2195 & 1.1139 \\
2 & 407.154 & 1.2662 & 1.9900 \\
3 & 400.915 & 2.1123 & 2.7365 \\
4 & 396.788 & 2.8419 & 3.3987 \\
5 & 393.999 & 3.4927 & 4.0000 \\ \hline
\end{tabular}%
\end{center}
\end{table}
\begin{figure}[tph]
\centering \includegraphics[height=2.5in]{inventorylevelplot4.eps}
\caption{Graphical Representation of Inventory System for Example 4}
\end{figure}
The following inferences can be made from the results in Table 1-4. %
\leftmargini=5mm
\begin{enumerate}
\item For the model with time-increasing demand such as Example 1 and 4, the
selling price $p_i$, the length of shortage period ($t_{i}-s_{i-1}$) and the
length of holding period ($s_{i}-t_{i}$) in the $i$th replenishment cycle
decreases over the planning horizon, respectively.
\item For the model with the demand function following the shape of a
product-life-cycle, the selling price, the length of shortage period and the
length of holding period in the $i$th replenishment cycle decreases in the
growth stage of the product life cycle, then increases in the declining
stage of the product life cycle, respectively.
\item For the model with time-decreasing demand such as Example 3, the
selling price, the length of shortage period and the length of holding
period in the $i$th replenishment cycle increases over the planning horizon,
respectively.
\end{enumerate}
\section{Sensitivity analysis}
In this section, we examine the effects of change in the value of $\delta $
(the backlogging parameter of $\beta (x)=e^{-\delta x}$ or $\beta
(x)=1/\left( 1+\delta x\right) $, where $\delta \geq 0$) on the optimal
discount total profit and the optimal number of replenishments. A
sensitivity analysis is performed by considering the same numerical
examples. Seven different values of $\delta $ are adopted. Computed results
are shown in Table 5. The following inferences can be made from the results
in Table 5. \leftmargini=5mm
\begin{enumerate}
\item Increasing the value of $\delta $ will result in a decrease in the
optimal discount total profit.
\item Increasing the value of $\delta $ will result in an increase in the
optimal replenishment times.
\item As the value of $\delta $ decreases, the optimal discount total profit
becomes close to the optimal total profit without shortage.
\item The optimal discount total profit with partial backlogging is more
sensitive to $\delta $ when it's value is small.
\end{enumerate}
In addition, from Example 1, when $\delta =0$ (i.e., the model reduces to
the case of complete backlogging), the proposed model here which starts with
shortages and ends without shortages can be compared with the one developed
by Chen and Chen (\citeyear{Chen04}) which starts with an instant
replenishment and ends with shortages. The discount total profit in Example
1 generating $3.44\%$ increment is superior to that of Chen and Chen's (%
\citeyear{Chen04}) applying a time-increasing demand function ($598.5$
versus $578.6$).
\begin{table}[tph]
\caption{Effects of $\protect\delta$ on the discount total profit and the
number of replenishments}
\label{sen1}
\begin{center}
{\footnotesize
\begin{tabular}{ccccccccc}
\hline
& & Complete & & & $\delta $ & & & Without \\ \cline{4-8}
& & backlogging & 0.1 & 0.2 & 0.5 & 0.75 & 1 & shortage \\
& & ($\delta =0$) & & & & & & ($\delta \rightarrow \infty $) \\
\cline{2-9}
\textbf{Example 1} & $TP^{\star }$ & 598.50 & 582.11 & 570.53 & 548.74 &
537.63 & 529.87 & 487.36 \\
& $n^{\star }$ & 6 & 6 & 7 & 7 & 7 & 7 & 8 \\
& & & & & & & & \\ \cline{2-9}
\textbf{Example 2} & $TP^{\star }$ & 19498.2 & 19464.78 & 19438.41 & 19386.90
& 19358.77 & 19339.91 & 19096.5 \\
& $n^{\star }$ & 7 & 7 & 7 & 8 & 8 & 8 & 10 \\
& & & & & & & & \\ \hline
& & Complete & & & $\delta $ & & & Without \\ \cline{4-8}
& & backlogging & 0.5 & 1 & 2.5 & 5 & 10 & shortage \\
& & ($\delta =0$) & & & & & & ($\delta \rightarrow \infty $) \\
\cline{2-9}
\textbf{Example 3} & $TP^{\star }$ & 8377.53 & 8105.98 & 8026.90 & 7953.05 &
7921.16 & 7903.40 & 7839.16 \\
& $n^{\star }$ & 4 & 4 & 4 & 4 & 4 & 4 & 5 \\
& & & & & & & & \\ \cline{2-9}
\textbf{Example 4} & $TP^{\star }$ & 9610.05 & 9249.56 & 9141.17 & 9032.39 &
8976.23 & 8943.59 & 8877.33 \\
& $n^{\star }$ & 4 & 5 & 5 & 5 & 5 & 5 & 6 \\
& & & & & & & & \\ \hline
\end{tabular}%
}
\end{center}
\end{table}
\section{Concluding Remarks}
In this paper, we properly extend the fixed price policy to change selling
prices upward or downward periodically. We allow not only for the
multivariate demand function of price and time but also general partial
backlogging rate. For the mixed-integer nonlinear programming problem with $%
3n$ decision variables, the compute of the approximately accurate estimate
for the optimal number of replenishment significantly reduces computational
complexity. Further, a complete search procedure is provided to find optimal
selling prices and replenishment schedule by employing the Nelder--Mead
algorithm. The proposed algorithm help us determine periodic selling prices
and the optimal replenishment schedule for the cases of demand patterns in
the different stage of the product life cycle. Therefore, this lot size
model can be viewed as an extension of numerous previous models, such as
Hariga (\citeyear{Hariga96}), Teng \textit{et al.} (\citeyear{Teng02}), Yang
\textit{et al.} (\citeyear{Yang02}), Chen and Chen (\citeyear{Chen04}), Chen
\textit{et al.} (\citeyear{Chen07AMC, Chen07CMA}), and Chern \textit{et al.}
(\citeyear{Chern08}).
The proposed model can be extended in several ways. For instance, we may
consider the permissible delay in payments. Also, we could extend the
deterministic demand function to stock-dependent demand patterns. Finally,
we could generalize the model to allow for quantity discounts, finite
capacity and others.
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\end{document}
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這個期刊好投還是不好投啊 審稿期間多長?
我想投看看 給個建議吧
丁吉峰
網路上盛傳它是研究生的好朋友！我本來不信，試了一下，果不其然！不過審稿時間就不一定，我之前國科會的老師曾經有一篇三天就接受。我這篇則快一年，但是完全沒修改，直接接受！
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