Optimal Policies for a Finite-Horizon Production Inventory Model

The model treated in this paper is based on the following assumptions:

1. the planning horizon is finite;
2. a single item is considered;
3. products are assumed to experience deterioration while in stock;
4. shortages are not permitted;
5. initial inventory at the beginning of planning horizon is zero, also the inventory depletes to zero at the end of the planning horizon;
6. the demand function is strictly positive;

We will initially look at a single period $i$ (production cycle), say, starting at time $t_{i-1}$ and ending at time $t_{i}$, $i = 1,2...$. Some of the notations used in the model are as follows:

$H$: the total planning horizon,
$p$: the constant production rate,
$D(t)$: the demand rate at time $t, 0 \text { < } D(t) \text { < } p$,
$\alpha$: constant deteriorating rate of inventory items with $\alpha \text { > } 0$,
$t_{i}^{p}$: the time at which the inventory level reaches it is maximum in $i$th production cycle,
$K$: set up cost for the inventory model,
$c_{1}$: the cost of one unit of the item with $c_{1} \text { > }  0$,
$c_{h}$: carrying cost per inventory unit held in the model per unit time,
$TC$: total system cost during $H$.

Figure 1 shows the changes of the level of stock for a typical production period.

Figure 1 The changes of inventory levels of various components of the model for a typical production batch.

Let $I(t)$ be the level of stock at time $t$. The change in period $i$, the level of inventory, may be described by the following differential equation $i, i=1,2, \ldots$,

$I^{\prime}(t)=p-D(t)-\alpha I(t), \quad t_{i-1} \leq t \text { < } t_{i}^{p}, \quad I(t) \text { ↓ } 0 \quad \text { as } t \text { ↓ } t_{i-1}$.   (2.1)

The solution to (2.1) is given by

$I(t)=e^{-\alpha t} \int_{t_{i-1}}^{t} e^{\alpha u}\{p-D(u)\} d u, \quad t_{i-1} \leq t \text { < } t_{i}^{p}$,   (2.2)

$I^{\prime}(t)=-D(t)-\alpha I(t), \quad t_{i}^{p} \leq t \text { < } t_{i}, \quad I(t) \uparrow 0 \text { as } t \uparrow t_{i}$.   (2.3)

The solution to (2.3) is given by 

$I(t)=e^{-\alpha t} \int_{t}^{t_{i}} e^{\alpha u} D(u) d u, \quad t_{i}^{p} \leq t \text { < } t_{i}$.   (2.4)

The total costs, (excluding the setup cost) for period $i$, which consist of holding cost and deterioration cost are given by

$\begin{gathered} c_{h} \int_{t_{i-1}}^{t_{i}^{p}} e^{-\alpha t}\left[\int_{t_{i-1}}^{t} e^{\alpha u}\{p-D(u)\} d u\right] d t \\ \quad+c_{h} \int_{t_{i}^{p}}^{t_{i}} e^{-\alpha t}\left\{\int_{t}^{t_{i}} e^{\alpha u} D(u) d u\right\} d t \\ \quad+c_{1}\left\{\int_{t_{i-1}}^{t_{i}^{p}} p d u-\int_{t_{i-1}}^{t_{i}} D(u) d u\right\} \end{gathered}$ (2.5)

We will call models with this cost OHD models. It is possible to consider instead of (2.5) the form

$\begin{aligned} &c_{h} \int_{t_{i-1}}^{t_{i}^{p}} e^{-\alpha t}\left[\int_{t_{i-1}}^{t} e^{\alpha u}\{p-D(u)\} d u\right] d t \\ &\quad+c_{h} \int_{t_{i}^{p}}^{t_{i}} e^{-\alpha t}\left\{\int_{t}^{t_{i}} e^{\alpha u} D(u) d u\right\} d t+c_{1} \int_{t_{i-1}}^{t_{i}^{p}} p d u \end{aligned}$  (2.6)

which considers only holding and purchasing costs, where the expression $c_{1} \int_{t_{i-1}}^{t_{i}^{p}} p d u$ represents the purchasing cost. We call this OHP models.

Note that since the function $I$ is continuous at $t_{i}^{p}$, we have

$e^{-\alpha t_{i}^{p}} \int_{t_{i-1}}^{t_{i}^{p}} e^{\alpha u}\{p-D(u)\} d u=e^{-\alpha t_{i}^{p}} \int_{t_{i}^{p}}^{t_{i}} e^{\alpha u} D(u) d u$,   (2.7)

or

$\int_{t_{i-1}}^{t_{i}^{p}} e^{\alpha u}\{p-D(u)\} d u=\int_{t_{i}^{p}}^{t_{i}} e^{\alpha u} D(u) d u$,   (2.8)

then

$p \int_{t_{i-1}}^{t_{i}^{p}} e^{\alpha u} d u=\int_{t_{i-1}}^{t_{i}} e^{\alpha u} D(u) d u$,   (2.9)

or

$\frac{p}{\alpha}\left(e^{\alpha t_{i}^{p}}-e^{\alpha t_{i-1}}\right)=\frac{p}{\alpha} e^{\alpha t_{i-1}}\left\{e^{\alpha\left(t_{i}^{p}-t_{i-1}\right)}-1\right\}=\int_{t_{j-1}}^{t_{i}} e^{\alpha u} D(u) d u$,   (2.10)

$e^{\alpha\left(t_{i}^{p}-t_{i-1}\right)}-1=\frac{\alpha}{p} e^{-\alpha t_{i-1}} \int_{t_{i-1}}^{t_{i}} e^{\alpha u} D(u) d u$ ).  (2.11)

Therefore,

$t_{i}^{p}-t_{i-1}=\frac{1}{\alpha} \log \left\{1+\frac{\alpha}{p} e^{-\alpha t_{i-1}} \int_{t_{i-1}}^{t_{i}} e^{\alpha u} D(u) d u\right\}$.   (2.12)

Lemma 2.1. The expression of the cost in (2.5) is equal to

$\left(c_{h}+\alpha c_{1}\right)\left[\frac{p}{\alpha^{2}} \log \left\{1+\frac{\alpha}{p} e^{-\alpha t_{i-1}} \int_{t_{i-1}}^{t_{i}} e^{\alpha u} D(u) d u\right\}-\frac{1}{\alpha} \int_{t_{i-1}}^{t_{i}} D(u) d u\right]$.   (2.13)

Proof. Applying integration by parts, we get that (2.5) reduces to

$\begin{aligned} &\frac{c_{h}}{\alpha} \int_{t_{i-1}}^{t_{i}^{p}}\left\{1-e^{\alpha\left(u-t_{i}^{p}\right)}\right\}\{p-D(u)\} d u \\ &\quad+\frac{c_{h}}{\alpha} \int_{t_{i}^{p}}^{t_{i}}\left\{e^{\alpha\left(u-t_{i}^{p}\right)}-1\right\} D(u) d u \\ &\quad+c_{1} \int_{t_{i-1}}^{t_{i}^{p}} p d u-c_{1} \int_{t_{i-1}}^{t_{i}} D(u) d u \end{aligned}$   (2.14)

or

$\begin{aligned} &\frac{c_{h}}{\alpha}\left\{p \int_{t_{i-1}}^{t_{i}^{p}} d u-\int_{t_{i-1}}^{t_{i}^{p}} D(u) d u-p \int_{t_{i-1}}^{t_{i}^{p}} e^{\alpha\left(u-t_{i}^{p}\right)} d u+\int_{t_{i-1}}^{t_{i}^{p}} e^{\alpha\left(u-t_{i}^{p}\right)} D(u) d u\right\} \\ &\quad+\frac{c_{h}}{\alpha}\left\{\int_{t_{i}^{p}}^{t_{i}} e^{\alpha\left(u-t_{i}^{p}\right)} D(u) d u-\int_{t_{i}^{p}}^{t_{i}} D(u) d u\right\} \\ &\quad+c_{1} \int_{t_{i-1}}^{t_{i}^{p}} p d u-c_{1} \int_{t_{i-1}}^{t_{i}} D(u) d u \end{aligned}$   (2.15)

$\begin{aligned} =& p\left(\frac{c_{h}}{\alpha}+c_{1}\right)\left(t_{i}^{p}-t_{i-1}\right)-\frac{c_{h}}{\alpha} e^{-\alpha t_{i}^{p}} \int_{t_{i-1}}^{t_{i}^{p}} e^{\alpha u u}\{p-D(u)\} d u \\ &+\frac{c_{h}}{\alpha} e^{-\alpha t_{i}^{p}} \int_{t_{i}^{p}}^{t_{i}} e^{\alpha u} D(u) d u-\frac{c_{h}}{\alpha} \int_{t_{i-1}}^{t_{i}} D(u) d u-c_{1} \int_{t_{i-1}}^{t_{i}} D(u) d u . \end{aligned}$   (2.16)

This is equal, using (2.7), to

$p\left(\frac{c_{h}}{\alpha}+c_{1}\right)\left(t_{i}^{p}-t_{i-1}\right)-\left(\frac{c_{h}}{\alpha}+c_{1}\right) \int_{t_{i-1}}^{t_{i}} D(u) d u$.   (2.17)

Now, the lemma follows from (2.12) and (2.17).

Note that Lemma 2.1 reduced the dependence of the inventory cost in period $i$ from three variables to two variables. This reduction can be significant for an $n$-period model. Let

$R(x, y):=\left(c_{h}+\alpha c_{1}\right)\left[\frac{p}{\alpha^{2}} \log \left\{1+\frac{\alpha}{p} e^{-\alpha x} \int_{x}^{y} e^{\alpha t} D(t) d t\right\}-\frac{1}{\alpha} \int_{x}^{y} D(t) d t\right]$.   (2.18)

Remark 2.2. Let $\alpha \rightarrow 0$, in (2.18), and recall that $\log (1+x)$ may be expanded as

$x-\frac{1}{2} x^{2}+\frac{1}{3} x^{3}-\frac{1}{4} x^{4}+\frac{1}{5} x^{5}+O\left(x^{6}\right)$,   (2.19)

to get that as $\alpha \rightarrow 0$, $R(x, y)$ is equivalent to

$\left(c_{h}+\alpha c_{1}\right)\left[\int_{x}^{y}\left\{\frac{e^{\alpha(t-x)}-1}{\alpha}\right\} D(t) d t-\frac{1}{2 p}\left\{\int_{x}^{y} e^{\alpha(t-x)} D(t) d t\right\}^{2}\right]$,   (2.20)

which leads to the expression

$c_{h}\left[\int_{x}^{y}(t-x) D(t) d t-\frac{1}{2 p}\left\{\int_{x}^{y} D(t) d t\right\}^{2}\right]$.    (2.21)

This expression may be found in Hill, Omar and Smith, and Rau and Ouyang. However, their interest in finding the optimal inventory policy for their model centered around treating special cases for demand rate functions or devising heuristics.

The total inventory costs where $n$ ordered are made may be written as follows:

$\mathrm{TC}\left(t_{1}, \ldots, t_{n}\right)=n K+\sum_{i=1}^{n} R\left(t_{i-1}, t_{i}\right)$,   (2.22)

which is given by (2.18).

The objective now is to find $n$ and $t_{1}, \ldots, t_{n}$ which minimizes $TC$ subject to $t_{0}=0 \text { < } t_{1} \text { < } \cdots \text { < } t_{n}=H$. The problem becomes a mixed integer programming problem. The approach that we will use to solve it is based on a procedure developed by Benkherouf and Gilding. The next section contains the ingredients of the approach.