CAMWA接受

這篇Paper在2009年9月七日投稿，2010年7月15日修改後馬上接受。速度還算OK～～不過比較鮮的是盡然要我加上3篇風馬牛不相干的參考文獻。直覺的在sciencedirect查了一下"deteriorating items"，得到的結果也頗耐令人尋味～最後，如果你需要這個粒子群演算法(particle swarm optimization) 的Mathematica程式，Email給我。

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\begin{document}
\begin{frontmatter}
\title{\textbf{A particle swarm optimization for solving joint pricing and lot-sizing problem
with fluctuating demand and unit purchasing cost}}
\author{Chung-Yuan Dye\corref{cor1}\cortext[cor1]{Corresponding author}}
\ead{chungyuandye@gmail.com}
\address{Department of Business Management, Shu-Te University, Yen Chao, Kaohsiung
824, Taiwan, R.O.C.}
\author{Tsu-Pang Hsieh\corref{cor2}\cortext[cor2]{Corresponding author}} \ead{tsupang@gmail.com} \address{Department of Business Management, Aletheia University, Tamsui, Taipei, 251, Taiwan, R.O.C.}
\begin{abstract}
In this paper, we extend the classical economic order quantity
model to allow for not only a function of price-dependent and
time-varying demand but also fluctuating unit purchasing cost. The
joint replenishment problem is subject to continuous decay and
general partial backlogging rate. The objective is to find the
optimal replenishment number, time scheduling and periodic selling
price to maximize the discounted total profit. An effective search
procedure is provided to find the optimal solution by employing
the properties derived in this paper and particle swarm
optimization 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.
\end{abstract}
\begin{keyword}  Pricing, Inventory, Partial backlogging, Particle swarm optimization, Fluctuating cost, Inflation
\end{keyword}
\end{frontmatter}
\newpage
\setlength {\baselineskip} {1.5 \initiallineskip}
\section{Introduction}
In many inventory systems, the deterioration of goods is a
realistic phenomenon. It is well known that certain products such as
medicine, volatile liquids, blood bank, food stuff and many others, decrease
under deterioration (vaporization, damage, spoilage, dryness and so on)
during their normal storage period. As a result, while determining the
optimal inventory policy of that type of products, the loss due to
deterioration can not be ignored. The fundamental result in the development of economic order quantity model with deteriorating items is that  Ghare and Schrader \cite{Ghare63} who established the classical no-shortage inventory model with a constant rate of decay. Base on the Ghare and Schradr's \cite{Ghare63}  model, researchers including  Wu et al. \cite{Wu2006}, Huang and Yao \cite{Huang06},  Huang and Liao \cite{Huang08}, Mishra and Mishra \cite{Mishra08},  Maity and Maiti \cite{Maity08}, Geetha and Uthayakumar \cite{Geetha2010}  and Chang et al.  \cite{Chang2010}  developed economic order quantity models that focused on deteriorating items.
Furthermore, in classical inventory models, the economic order quantity (EOQ)
formula is assumed to be a constant demand rate and a fixed unit
purchasing cost, and therefore offers ease of use and application.
In reality, the demand may vary with product life cycle duration.
The assumption of a constant demand rate is usually valid in the
maturity stage. In literature, the demand rate had been well
approximated by specific forms to indicate the stage of a product
in its life cycle. As Goyal and Giri \cite{GoyalGiri01} pointed
out, most of the time-varying demand inventory models considered
either linearly
increasing/decreasing demand (i.e., $f\left( t\right) =a+bt$, with $a>0$, $%
b\neq 0$) or exponentially increasing/decreasing demand (i.e.,
$f\left( t\right) =ae^{bt}$, with $a>0$, $b\neq 0$) patterns. We
refer the reader to their references therein for more details.
Recently, Chen, Liao and Wen \cite{Chen07ESWA}, Chen, Hung and
Weng \cite{Chen07a, Chen07b} dealt with the inventory model under
the demand function following the product-life-cycle shape over a
fixed time horizon.
Moreover, the assumption of a fixed unit purchasing cost does not
reflect the situation where inflation rate is high or the
situation where price increase or decrease is expected. With the
advances in technology and global division of labor, the unit cost
of high-tech products might drop due to new products introduction.
In the personal computer (PC) industry, Lee et al. \cite{Lee00}
showed components' price declines constantly over the product life
cycle. Under an exponential cost decrease but a constant demand
rate, Khouja and Park \cite{Khouja03} analyzed the problem of
optimizing the lot size with equal length for the entire horizon.
Teunter \cite{Teunter05} then developed a net present value
formulation of Khouja and Park's \cite{Khouja03}  model, and derived a simple
modified EOQ formula. Khouja and Goyal \cite{Khouja06} relaxed the
restriction of equal length to allow varying cycle times.
Besides the continuous decrease of purchasing cost, in real-life
situations, the gasoline price or raw material price may be going
up constantly. When the cost of purchases as a percentage of sales
is often substantial, it is necessary to include fluctuating
purchasing cost for the inventory system. Khouja et al.
\cite{Khouja05} developed the joint replenishment problem to
analyse the effect of continuous unit purchasing cost decrease or
increase on the optimal ordering frequencies. In contrast to the
traditional EOQ model, Teng and Yang \cite{Teng04} assumed that
both the demand function and the unit purchasing cost are
fluctuating with time, which are more general than increasing,
decreasing, and log-concave functions. Teng et al. \cite{Teng05}
then provided an easy-to-use algorithm to find the optimal
replenishment number and schedule for completely shortages. Teng
and Yang \cite{Teng07} further allowed for time-varying purchasing
cost and generalized holding cost over a finite-planning horizon.
From the business competitiveness standpoint, maximizing profit
plays an important role for getting and keeping a successful
position in a competitive market. However, the above inventory
models subject to decreasing or increasing unit purchasing cost
are developed to minimize the total relevant cost. To achieve
profit maximization,
Chen and Chen \cite{Chen04} presented an
inventory model for a deteriorating item with a multivariate
demand function of price and time but a fixed unit purchasing
cost. Their model is solved by dynamic programming techniques to
adjust the selling price upward or downward periodically. Chang et
al. \cite{Chang06} established an inventory model for a retailer
to determine its optimal selling price, replenishment number and
replenishment schedule. They also assume a fixed unit purchasing
cost, but the existence and uniqueness of the maximum solution is
obtained under the same selling price per cycle. Similarly, when
the unit purchasing cost is fluctuating with time, a decision
maker needs to adjust its pricing strategy, but the joint pricing
and replenishment policy is seldomly discussed.
In this paper, we assume that unit purchasing cost is positive and
fluctuating with time. We investigate the replenishment policies
for a deteriorating item with partial backlogging by considering a
multivariate demand function of price and time and the effect of
discount rate over multiperiod planning horizon. 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 for periodical upward and downward
adjustments. 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 search method based on particle swarm
optimization 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}
The mathematical model in this paper is developed on the basis of the following assumptions and notations:
\subsection{Assumptions}
\begin{enumerate}
\item A single item is considered with a constant rate of deterioration over a known and finite planning horizon of length $H$.
\item The replenishment occurs instantaneously at an infinite rate.
\item There is no repair or replacement of deteriorated units during the planning horizon. The items will be withdrawn from warehouse immediately as they become deterioration.
\item Shortages are allowed in all cycles and each cycle starts with shortages.
\item 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[$n=$] The number of replenishment cycles  during the planning horizon (a decision variable).
\item[$\theta =$] the deterioration rate.
\item[$r=$] the discount rate.
\item[$c_{f}=$] the ordering cost per order.
\item[$c_{v}\left( t\right) =$] the unit purchasing cost at time $t$, where $c_{v}\left( t\right)$ is a positive and continuous function of time in
the planning horizon.
\item[$p_{i}=$] the selling price per unit (a decision variable) in the $i$%
th replenishment cycle, defined in the interval $\left[ 0,p_{u}\right] $.
\item[$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   and $A\left( p_{i}\right) $ is any
non-negative, continuous, convex, decreasing function of the selling price
in $\left[ 0,p_{u}\right] $.
\item[$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}=$] the unit cost of lost sales.
\item[$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), $i=1,2,\ldots ,n-1$.
\end{itemize}
\noindent As a result, the decision problem here has $3n$ decision variables.
\section{Model Formulation}
\begin{figure}[hptb]
\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}<\cdots<&0.  \label{tppipi} \end{eqnarray} and \begin{eqnarray} \frac{\partial ^{2}TP(\mathbf{p}|n,\mathbf{t},\mathbf{s})}{\partial p_{i} \partial p_{j} }=0, \text{ }i\neq j,\text{ }i=1,2,\ldots ,n,\text{ }j=1,2,\ldots ,n.\text{ } \end{eqnarray}  As the discussion above,  the Hessian matrix at the stationary point $\left( p_{1}^{\text{opt}},p_{2}^{\text{opt} },\ldots ,p_{n}^{\text{opt} }\right) $, denoted by $\mathbf{p}^{\text{opt}}$, is given by \begin{equation*} \mathbf{H}=\left[ \begin{array}{cccc} \frac{\partial ^{2}TP(\mathbf{p}|n,\mathbf{t},\mathbf{s})}{\partial p_{1}^{2}} & 0 & {\ 0} & {\ 0} \\ 0 & \frac{\partial ^{2}TP(\mathbf{p}|n,\mathbf{t},\mathbf{s})}{\partial p_{2}^{2} } & {\ 0} & {\ 0} \\ {\ \vdots } & {\ \vdots } & {\ \ddots } & {\ \vdots } \\ {\ 0} & {\ 0} & {\ 0} & \frac{\partial ^{2}TP(\mathbf{p}|n,\mathbf{t},\mathbf{s})}{\partial p_{n}^{2}}% \end{array}% \right] . \end{equation*}% We can see that the diagonal elements of $\mathbf{H}$ are all negative and off-diagonal elements are all zero, and thus, the Hessian matrix $\mathbf{H}$ at point $\mathbf{p}^{\text{opt}}$ is negative definite and $\mathbf{p}^{\text{opt}}$ represents a global maximum point.   From the analysis carried out so far, it is easy to see that $p_{i}^{\text{opt}}=p^{\ast}_{i}$ can be written as a function of $\mathbf{t}$ and $\mathbf{s}$, and this result reduces the $3n-1$ dimensional problem of finding the optimal pricing and schedule to a $2n-1$ dimensional problem as follows:  \begin{center} \begin{tabular}{ll} Maximize & $TP(\mathbf{t},\mathbf{s}|n)$ \\ &  \\ subject to& $cv(t_{i})<10^{-5}$ k="\text{iter}_{\text{max}}$," k="k+1$" n="\text{round" n="1$" leftmargini="20mm" leftmarginii="10mm" leftmarginiii="46mm">TP(k-1)$. Set $n^{\ast }=k$ and stop.
\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{Computational results}
\subsection{Numerical examples}
To illustrate the results, let us 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.
\noindent \textbf{Example 1.} We first redo the same example of Chen and
Chen \cite{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 $, $c_{s}=0.5$, $\theta =0.2$, $H=12$, $r=0.02$. Besides, we
assume that the time-dependent backlogging rate is $c_{v}(t)=e^{0.01t}$, $%
\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 $TP(5)=474.1$, $TP(6)=487.4$ and $%
TP(7)=487.2$. %$TP(8)=478.428$.
Therefore, the optimal number of replenishments is 6, and the optimal
pricing and time schedule are shown in Table 1. The behavior of inventory
system over the planning horizon and the convergence result of PSO
algorithms for optimal solution are depicted in Fig. \ref{example1} and \ref%
{iter:a}, respectively.
\begin{table}
\centering
\begin{threeparttable}[hptb]
\caption{Optimal pricing and time schedule for Example 1}
\begin{tabular}{rrrrrrrr}
\hline
\multicolumn{1}{c}{$i$} & \multicolumn{1}{c}{$p_i$} & \multicolumn{1}{c}{$cv (t_i)$} & \multicolumn{1}{c}{$t_i$} & \multicolumn{1}{c}{$s_i$} & \multicolumn{1}{c}{$Q_i$} & \multicolumn{1}{c}{$LS_i$~\tnote{a}} & \multicolumn{1}{c}{$LI_i$~\tnote{b}} \\ \hline
1 & 1.8630 & 1.0067 & 0.6643 & 2.2946 & 137.4 & 0.6643 & 1.6303 \\
2 & 1.8688 & 1.0296 & 2.9202 & 4.4519 & 90.8 & 0.6256 & 1.5317 \\
3 & 1.8750 & 1.0517 & 5.0443 & 6.4886 & 73.4 & 0.5924 & 1.4443 \\
4 & 1.8812 & 1.0731 & 7.0520 & 8.4184 & 67.8 & 0.5634 & 1.3664 \\
5 & 1.8876 & 1.0937 & 8.9559 & 10.2523 & 67.2 & 0.5375 & 1.2964 \\
6 & 1.8940 & 1.1137 & 10.7670 & 12.0000 & 68.9 & 0.5147 & 1.2330\\ \hline
\end{tabular}%
\begin{tablenotes}
\footnotesize
\item[a] $LS_i = t_i -s_{i-1}$
\item[b] $LI_i = s_i -t_i$
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{figure}[hptb]
\centering \includegraphics[height=2.5in]{inventorylevelplot1.eps}
\caption{Graphical Representation of Inventory System for Example 1}
\label{example1}
\end{figure}
\noindent \textbf{Example 2.} In this example, we redo an inventory
situation proposed by Teng and Yang \cite{Teng04} while considering $%
f\left( t,p\right) =(5-0.005p) e^{0.98t}$, $\beta \left( x\right) =e^{-0.2x}
$, $c_{f}=250$, $c_{h}=40$, $c_{s}=50$, $c_{l}=500$, $\theta =0.08$, $%
c_{v}(t)=200+20e^{-2t}$, $H=4$ and $r=0.02$. By applying
(\ref{neoq}), we obtain the estimated number of replenishments
$n=6$. Then, apply the
Algorithms 1 and 2 to get $TP(5)=35195.7$, $TP(6)=35217.4$ and $%
TP(7)=35163.8 $. %$TP(8)=35063.4$.
Therefore, the optimal number of replenishments is 6, and the optimal
pricing and time schedule is shown in Table 3. The behavior of inventory
system over the planning horizon and the convergence result of PSO
algorithms for optimal solution are depicted in Fig. \ref{example2} and \ref%
{iter:b}, respectively.
\begin{table}
\centering
\begin{threeparttable}[hptb]
\centering
\caption{Optimal pricing and time schedule for Example 2}
\begin{tabular}{rrrrrrrr}
\hline \multicolumn{1}{c}{$i$} & \multicolumn{1}{c}{$p_i$} &
\multicolumn{1}{c}{$cv (t_i)$} & \multicolumn{1}{c}{$t_i$} &
\multicolumn{1}{c}{$s_i$} & \multicolumn{1}{c}{$Q_i$} &
\multicolumn{1}{c}{$LS_i$~\tnote{a}} &
\multicolumn{1}{c}{$LI_i$~\tnote{b}} \\ \hline 1 & 619.023 &
206.100 & 0.5938 & 1.4155 & 5.2 & 0.5938 & 0.8218 \\ 2 & 609.932 &
200.725 & 1.6590 & 2.2426 & 7.9 & 0.2434 & 0.5836 \\ 3 & 606.923 &
200.165 & 2.3980 & 2.8360 & 10.5 & 0.1555 & 0.4379 \\ 4 & 605.339
& 200.055 & 2.9502 & 3.2992 & 12.9 & 0.1143 & 0.3490 \\ 5 &
604.344 & 200.023 & 3.3895 & 3.6789 & 15.2 & 0.0903 & 0.2894 \\ 6
& 603.657 & 200.011 & 3.7533 & 4.0000 & 17.5 & 0.0744 & 0.2467 \\
\hline
\end{tabular}%
\begin{tablenotes}
\footnotesize
\item[a] $LS_i = t_i -s_{i-1}$
\item[b] $LI_i = s_i -t_i$
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{figure}[hptb]
\centering \includegraphics[height=2.5in]{inventorylevelplot2.eps}
\caption{Graphical Representation of Inventory System for Example 2}
\label{example2}
\end{figure}
\noindent \textbf{Example 3.} In this example, we redo an
inventory situation proposed by Chen \textit{et al.}
\cite{Chen07b} while considering the variable purchase cost,
deterioration and partial backlogging. $f\left( t,p\right) =\left(
5000-150p\right) t^{3-1}\left(
H-t\right) ^{2-1}/{\mathcal B}\left( 3,2\right) $, $c_{f}=50$, $c_{h}=5$, $c_{s}=7$, $%
c_{l}=6$, $c_{v}(t)=10 e^{-0.25t}$, $H=1$, $r=0.02$, $\theta =0.08$ and $%
\beta \left( x\right) =1/(1+x)$. Applying (\ref{neoq}), we obtain the
estimated number of replenishments $n=8$. Then, applying the Algorithm 1 and
2, we get $TP(7)=21654.2$, $TP(8)=21680.1$, $TP(9)=21691.9$ and $%
TP(10)=21689.8$. Therefore, the optimal number of replenishments is 9, and
the optimal pricing and time schedule is shown in Table 3. The behavior of
inventory system over the planning horizon and the convergence result of PSO
algorithms for optimal solution are depicted in Fig. \ref{example3} and \ref%
{iter:c}, respectively.
\begin{table}
\centering
\begin{threeparttable}[hptb]
\centering
\caption{Optimal pricing and time schedule for Example 3}
\begin{tabular}{rrrrrrrr}
\hline
\multicolumn{1}{c}{$i$} & \multicolumn{1}{c}{$p_i$} & \multicolumn{1}{c}{$cv (t_i)$} & \multicolumn{1}{c}{$t_i$} & \multicolumn{1}{c}{$s_i$} & \multicolumn{1}{c}{$Q_i$} & \multicolumn{1}{c}{$LS_i$~\tnote{a}} & \multicolumn{1}{c}{$LI_i$~\tnote{b}} \\ \hline
1 & 21.6551 & 9.5927 & 0.1663 & 0.2629 & 98.9 & 0.1663 & 0.0966 \\
2 & 21.4171 & 9.2795 & 0.2991 & 0.3695 & 150.0 & 0.0362 & 0.0704 \\
3 & 21.2898 & 9.0557 & 0.3968 & 0.4613 & 190.2 & 0.0273 & 0.0645 \\
4 & 21.1804 & 8.8596 & 0.4843 & 0.5427 & 209.8 & 0.0231 & 0.0583 \\
5 & 21.0879 & 8.6860 & 0.5635 & 0.6189 & 221.1 & 0.0208 & 0.0554 \\
6 & 21.0057 & 8.5235 & 0.6390 & 0.6947 & 229.4 & 0.0201 & 0.0557 \\
7 & 20.9270 & 8.3680 & 0.7127 & 0.7696 & 218.9 & 0.0180 & 0.0569 \\
8 & 20.8562 & 8.2092 & 0.7893 & 0.8545 & 213.0 & 0.0197 & 0.0652 \\
9 & 20.7954 & 8.0334 & 0.8759 & 1.0000 & 175.0 & 0.0214 & 0.1241 \\ \hline
\end{tabular}%
\begin{tablenotes}
\footnotesize
\item[a] $LS_i = t_i -s_{i-1}$
\item[b] $LI_i = s_i -t_i$
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{figure}[hptb]
\centering \includegraphics[height=2.5in]{inventorylevelplot3.eps}
\caption{Graphical Representation of Inventory System for Example 3}
\label{example3}
\end{figure}
\noindent \textbf{Example 4.} In this example, we consider an inventory
situation with $f\left( t,p\right) =30000p^{-2}\linebreak\times(100-15t)$, $%
c_{f}=250$, $c_{h}=40$, $c_{s}=80$, $c_{l}=120$, $c_{v}(t)=200 e^{-0.05t}$, $%
H=4$, $r=0.02$, $\theta =0.08$ and $\beta \left( x\right) =1/(1+x)$.
Applying (\ref{neoq}), we obtain the estimated number of replenishments $n=5$%
. Then, applying the Algorithm 1 and 2, we get $TP(4)=8606.1$, $TP(5)=8629.3$
and $TP(6)=8572.8$. Therefore, the optimal number of replenishments is 5,
and the optimal pricing and time schedule is shown in Table 4. The behavior
of inventory system over the planning horizon and the convergence result of
PSO algorithms for optimal solution are depicted in Fig. \ref{example4} and %
\ref{iter:d}, respectively.
\begin{table}
\centering
\begin{threeparttable}[hptb]
\centering
\caption{Optimal pricing and time schedule for Example 4}
\begin{tabular}{rrrrrrrr}
\hline
\multicolumn{1}{c}{$i$} & \multicolumn{1}{c}{$p_i$} & \multicolumn{1}{c}{$cv (t_i)$} & \multicolumn{1}{c}{$t_i$} & \multicolumn{1}{c}{$s_i$} & \multicolumn{1}{c}{$Q_i$} & \multicolumn{1}{c}{$LS_i$~\tnote{a}} & \multicolumn{1}{c}{$LI_i$~\tnote{b}} \\ \hline
1 & 431.821 & 198.943 & 0.1060 & 0.7100 & 10.4 & 0.1060 & 0.6040 \\
2 & 418.718 & 191.963 & 0.8203 & 1.4476 & 9.2 & 0.1103 & 0.6274 \\
3 & 406.133 & 184.969 & 1.5626 & 2.2252 & 8.1 & 0.1150 & 0.6625 \\
4 & 393.983 & 177.851 & 2.3474 & 3.0606 & 6.9 & 0.1222 & 0.7132 \\
5 & 383.163 & 170.486 & 3.1932 & 4.0000 & 5.8 & 0.1326 & 0.8068 \\ \hline
\end{tabular}%
\begin{tablenotes}
\footnotesize
\item[a] $LS_i = t_i -s_{i-1}$
\item[b] $LI_i = s_i -t_i$
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{figure}[hptb]
\label{fig:example4} \centering %
\includegraphics[height=2.5in]{inventorylevelplot4.eps}
\caption{Graphical Representation of Inventory System for Example 4}
\label{example4}
\end{figure}
\begin{figure}[ht]
\begin{center}
\mbox{
\subfigure[Example 1]{
\label{iter:a}
\includegraphics[width=0.45\textwidth]{iter1}
}
\subfigure[Example 2]{
\label{iter:b}
\includegraphics[width= 0.45\textwidth]{iter2}
}
}
\mbox{
\subfigure[Example 3]{
\label{iter:c}
\includegraphics[width= 0.45\textwidth]{iter3}
}
\subfigure[Example 4]{
\label{iter:d}
\includegraphics[width= 0.45\textwidth]{iter4}}
}
\end{center}
\caption{Convergence results of PSO algorithm for $TP$}
\label{iter}
\end{figure}
The following inferences can be made from the results in Tables 1-4. %
\leftmargini=5mm
\begin{enumerate}
\item When the demand is increasing with time, the length of shortages
period ($LS_i$) in the $i$th replenishment cycle is decreasing.
Otherwise, the length of shortages period ($LS_i$) is increasing.
\item When the demand is increasing with time, the length of inventory period ($LI_i$) in the $i$th replenishment cycle is decreasing. Otherwise, the length of inventory period ($LI_i$)
is increasing.
\item When the unit purchasing cost is increasing with time, the selling
price $p_i$ in the $i$th replenishment cycle is increasing.
Otherwise, the selling price $p_i$ is decreasing.
\end{enumerate}
\subsection{Sensitivity analyses}
Using the settings in the examples above, we perform the
sensitivity analysis to investigate the effects of several factors
such as ordering cost ($c_f$), holding cost ($c_h$),
shortages cost ($c_s$), cost of lost sale ($c_l$) and deteriorating rate ($%
\theta$) on the optimal discounted total profit ($TP^\ast$) and
the optimal number of replenishments ($n^\ast$). The numerical
results obtained are shown in Table 5, and are portrayed in Fig.
\ref{sen}. The following inferences are made based on the results
in Table 5 and Fig. \ref{sen}.
\begin{enumerate}
\item The discounted total profit increases if $c_f$, $c_h$, $c_s$, $c_l$ or
$\theta$ decreases.
\item When the unit purchasing cost is increasing with time such as Example 1, the discounted total profit is more sensitive on the change in $c_f$
or $\theta$.
\item When the unit purchasing cost is decreasing with time such as Examples 2-4, the discounted total profit is not sensitive on the change in
$c_f$, $c_h$, $c_s$, $c_l$ or $\theta$.
\item The number of replenishments increases as $c_h$, $c_s$, $c_l$ or
$\theta$ increases, while it decreases as $c_f$ increases.
\end{enumerate}
\begin{table}[hptb]
\caption{Sensitivity analysis on $TP^{\ast}$ and $n^{\ast}$ for Examples 1-4}%
\centering \centering
\begin{tabular}{ccccccc}
\hline
& \multirow{2}{*}{Parameters} & \multicolumn{5}{c}{\% change} \\ \cline{3-7}
&  & $-50\%$ & $-25\%$ & $+0\%$ & $+25\%$ & $+50\%$ \\ \hline
& $c_f$ & 623.6 & 549.5 & 487.4 & 434.0 & 384.9 \\
&  & 9 & 7 & 6 & 6 & 5 \\ \cline{3-7}
& $c_h$ & 494.1 & 490.7 & 487.4 & 484.4 & 481.5 \\
&  & 6 & 6 & 6 & 7 & 7 \\ \cline{3-7}
Example 1 & $c_s$ & 515.4 & 499.6 & 487.4 & 479.0 & 472.2 \\
&  & 6 & 6 & 6 & 7 & 7 \\ \cline{3-7}
& $c_l$ & 493.2 & 490.2 & 487.4 & 485.0 & 482.9 \\
&  & 6 & 6 & 6 & 7 & 7 \\ \cline{3-7}
& $\theta$ & 580.3 & 529.0 & 487.4 & 455.7 & 428.0 \\
&  & 5 & 6 & 6 & 7 & 7 \\ \hline
& $c_f$ & 9314.1 & 8935.5 & 8629.3 & 8363.6 & 8121.1 \\
&  & 7 & 6 & 5 & 4 & 4 \\ \cline{3-7}
& $c_h$ & 8958.4 & 8775.4 & 8629.3 & 8496.4 & 8372.1 \\
&  & 4 & 4 & 5 & 5 & 5 \\ \cline{3-7}
Example 2 & $c_s$ & 8647.8 & 8638.1 & 8629.3 & 8621.2 & 8613.9 \\
&  & 5 & 5 & 5 & 5 & 5 \\ \cline{3-7}
& $c_l$ & 8658.6 & 8642.8 & 8629.3 & 8617.5 & 8607.1 \\
&  & 5 & 5 & 5 & 5 & 5 \\ \cline{3-7}
& $\theta$ & 8740.6 & 8684.3 & 8629.3 & 8575.4 & 8522.6 \\
&  & 4 & 5 & 5 & 5 & 5 \\ \hline
& $c_f$ & 21960.1 & 21810.4 & 21691.9 & 21581.5 & 21482.9 \\
&  & 12 & 10 & 9 & 8 & 8 \\ \cline{3-7}
& $c_h$ & 21816. & 21749.6 & 21691.9 & 21640.4 & 21597.7 \\
&  & 8 & 9 & 9 & 10 & 10 \\ \cline{3-7}
Example 3 & $c_s$ & 21715.7 & 21701.6 & 21691.9 & 21681.6 & 21673.3 \\
&  & 9 & 9 & 9 & 10 & 10 \\ \cline{3-7}
& $c_l$ & 21710.4 & 21701.5 & 21691.9 & 21682.2 & 21673.8 \\
&  & 9 & 9 & 9 & 9 & 10 \\ \cline{3-7}
& $\theta$ & 21707.3 & 21698.9 & 21691.9 & 21684. & 21675.2 \\
&  & 9 & 9 & 9 & 9 & 10 \\ \hline
& $c_f$ & 36007.3 & 35580.3 & 35217.4 & 34898.1 & 34600.4 \\
&  & 8 & 7 & 6 & 5 & 5 \\ \cline{3-7}
& $c_h$ & 35581.7 & 35380.0 & 35217.4 & 35075. & 34943.6 \\
&  & 5 & 5 & 6 & 6 & 6 \\ \cline{3-7}
Example 4 & $c_s$ & 35255.4 & 35235.5 & 35217.4 & 35200.1 & 35184.4 \\
&  & 6 & 6 & 6 & 6 & 6 \\ \cline{3-7}
& $c_l$ & 35302.1 & 35256.2 & 35217.4 & 35181.1 & 35155.0 \\
&  & 6 & 6 & 6 & 6 & 6 \\ \cline{3-7}
& $\theta$ & 35350.0 & 35280.6 & 35217.4 & 35152.5 & 35096.1 \\
&  & 5 & 6 & 6 & 6 & 6 \\ \hline
\end{tabular}%
\end{table}
\begin{figure}[hptb]
\begin{center}
\mbox{ \subfigure[Example 1]{ \label{sen:a}
\includegraphics[width= 0.45\textwidth]{sen1}
} \subfigure[Example 2]{ \label{sen:b}
\includegraphics[width= 0.45\textwidth]{sen2}
} } \mbox{ \subfigure[Example 3]{ \label{sen:c}
\includegraphics[width= 0.45\textwidth]{sen3}
} \subfigure[Example 4]{ \label{sen:d}
\includegraphics[width= 0.45\textwidth]{sen4}}
}
\end{center}
\caption{Effects of $c_f$, $c_h$, $c_s$, $c_l$, and
$\protect\theta$ on the discounted total profit} \label{sen}
\end{figure}
\section{Concluding Remarks}
In this paper, an inventory control problem is developed with a deteriorating item, partial backordering of shortages, a time-varying item purchase cost, and a deterministic demand rate that depends on both time and price. The PSO algorithm is used to search for the optimal replenishment strategy, and the price within each replenishment cycle.   In contrast to the classical fixed selling price policy under fixed unit purchasing cost, the pricing policy in this model is more flexible by changing price upward or downward periodically. Consequently, the model is more suitable for managers to plan marketing strategies to address the challenges that their products are likely to face. Furthermore, from the convergence results, the PSO algorithm offers acceptable efficiency and accurate search capability.
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 price  and stock-dependent demand patterns.
Finally, we could generalize the model to allow for quantity discounts,
finite capacity and others.
\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
NSC-97-2221-E-366-006-MY2.
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