4.1. Introduction
Depending upon different demand process assumptions, inventory control model can be broadly separated into two types, deterministic and probabilistic models. In the probabilistic model, assumptions about the probability density function of the demand process are usually made. Various probability density functions have been applied to model demands for establishing inventory management policy.
Every probability density function has its own advantages and disadvantages for application in its field. In most situations, when we only know the mean and variance of demand in the process but do not know what the real probability density function in the demand process is; we would assume the probability density function is a normal distribution for the inventory model. But Scarf in 1958 [18] addressed the newsboy problem where the mean and the variance of the demand are the only known variables without any further assumptions about the form of the distribution of the demand. He also had expressed a closed form for the maximization profile and the optimal order quantity.
If we use a traditional inventory model, we think there will be a higher total cost because we usually assume the demand to be independent. Many publications assumed normality and independence in the inventory model. And there are others
independence by Bagchi et al. (1982) [2], Kottas and Lau (1970, 1980) [13] [17], Lau and Wang (1986) [18], Ray (1980, 1989) [22] [23], and Van Ness and Stevenson (1983) [27]. In practice, demands show a tendency to be autocorrelated.
As we mentioned before, the demands in the manufacturing industry follow fractional Brownian motion. After Mandelbrot and Ness’s reporting, fBm has been used in different fields like hydrology, economics, astronomy, electronics, geophysics, finance and so forth. Many scientists, engineers and statisticians have used fBm to re-model their existing phenomena and problems. However, fBm is not used in inventory, quality control and other fields.
Moreover, Hurst proposed an ideal reservoir based upon the given record of observed discharges from the Great Lake of the Nile Basin; this whole idea is a suitable application in inventory management. For instance, the lead time demands are like the discharges of the river and the market is like the Nile lake. How much product should we make in order to meet the demand? The capacity management plays an important role in manufacturing. And our objective is to achieve fewer stocks and to gain maximum profit.
In order to reach our goal, we should construct our inventory model by fractional Brownian motion. Now we consider a fixed replenishment lead time, (Q, r) inventory model with known mean and variance. Then we use fBm to describe the lead time demand process and try to find a different solution.
4.2. Notations and assumptions
The notations below will be used in this thesis.
HC yearly per unit inventory holding cost UC unit cost
SC shortage cost VC variable cost
x unit time demand during lead time t
average value of x t
t fixed replenishment lead time H Hurst exponent, 0H1
CH a positive constant from fractional Brownian motion X demand during the lead time t i
q allowable stockout probability during the lead time ti ( )
B r the expected shortage at the end of the period cycle N present stock at reorder point
fraction of the demand backordered during the stockout period ,
0,1 fixed penalty cost per unit stockout
0 marginal profit (i.e., cost of lost demand) per unit Decision variables
Q order quantity r reorder point k safety factor
In order to clearly define the scope of the model, the following assumptions are made.
(1) The demand of fixed lead time X is composed of a sequence of demands x . t
0.5H1. When H0.5 , x is a random variable with a normal t distribution.
(3) We assume each item has the same fixed replenishment lead time. On the other hand, all t are the same.
(4) In original EOQ model, it is not permitted to be stockout. But we broke the assumption for pressing close to truth in this research. The symbol q represents the allowable stockout probability during t and the symbol presents the fraction of the fixed lead-time demand backordered during the stockout period.
So 1- is the ratio of the lost sales from the stockout.
(5) k is the safety factor where k statisfies P X( r) q and q represents the allowable stockout probability during t. Additionally, the safety stock equals the safety factor k multiplied by the standard deviation of the lead time demand. So the reorder point r is the expected demand during a fixed lead time plus the safety stock.
(6) As 0.5H1 there exits “persistence”, that is, an increasing or decreasing trend in the past implies an increasing or decreasing trend in the future for all processes. When 0H0.5, we have “anti-persistence”. In these situations an increase or decrease in the past implies a decrease or increase in the future. For process with H0.5, it becomes a common normal distribution demand inventory model.
4.3. Continuous-review System
In this inventory model, t is assumed to be the time lag and the fixed
demand of fixed replenishment lead time X is composed of a sequence of demands
(0 1) the fraction of the backordered demand during the stockout period, the expected number of backorders per cycle is B r( ) and the expected demand lost sales per cycle is thus (1) ( )B r . Hence, the expected shortage cost per period cycle [ (1 ) o] ( )B r is composed of the expected penalty cost B r( ) and the expected profit loss o(1) ( )B r .
The expected net inventory levels that can be derived (Ravindran et al., 1987) [21]
immediately before and after the order Q arrives are rt (1 ) ( )B r and
The expected annual cost function in this model formulation is identified to four separate cost components by unit cost (UC), reordering cost (RC), holding cost (HC)
=UC× Q (3)
Reordering cost component: reorder cost (RC) × number of orders places (1)
= RC (4)
(8) Because the unit cost component is fixed and not influenced by the order quantity, we can only concentrate on the other terms which form the variable cost (VC) with We want to find the order quantity that maximizes the expected profit against the worst possible distribution of the demand with knowing only the mean and variance of the lead time demand. So we consider any cumulative distribution, which is called G, of the lead time demand. We let denote the class of G’s cumulative density functions with finite mean t and variance CHt2H.
Since the distribution G of X is unknown, it is desirable to minimize the total expected annual cost against the worst possible distribution .
In order to resolve the problems derived from the unknown distribution and
Proposition : Gallego and Moon showed that for any G
2 2
( ) 1[ ( ) ( )]
2
H
B r C tH r t r t (10)
1 2
( ) ( 1 )
2
H
B r C tH k k (11) Thus, using inequality (11), the objective function (9) is reduced to minimize
( , ) [ (1 ) ( )] [ (1 ) ] ( )
Hence, we can take the partial derivatives of VC Q k( , ) with respect to k, we obtain and then taking the second derivative for k,
3
It is clear that for any given safety factor k, we have
3 setting the formulation (16) equal to zero. From Eq. (17), we can get
*2
From Eq. (18), we can get the optimal safety factor k, and the decision variables order quantity Q and reorder point r will be obtained from the same equation.
4.4. Numberical example
4.4.1. Parameters and assumption
In order to exhibit solution by our model under the fractional Brownian motion, we use MATLAB to simulate the fBm process. We adopt the Cholesky algorithm for
Hurst exponents as the demand of market. For the value of the H exponent with 0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 0.8, and 0.9 , we have eight cases. According to eight kinds of simulation of demands with H, we develop different inventory policies. We compare our results with a traditional model with H0.5, and then we would see which ones are better. We think our model is built by adding the fBm concept; it will be more suitable for real market. Therefore, it should allow fewer stocks to be stored and also to avoid being out-of-stock.
According to H0.5, the lead time demand is of normal distribution and the fixed replenishment lead time demand in the inventory model is reduced to an independent normal variable. Hence, the optimal order quantity is
* 2 {D RC [ 0(1 )] C tH H ( )}k marginal profit per unit, order ratio 0.6. The result of the solution is summarized in Table 6.
Table 6
*Percent = (Total_VC2 - Total_VC1) / Total_VC2
*Q1 column= optimal order quantity by using our fBm inventory model
*r1 column = reorder quantity by using our fBm inventory model
*k1 column = safety factor by using our fBm inventory model
*VC1 column =variable cost by using our fBm inventory model
*VC2 column = variable cost by using normal formula with H=0.5. (Eq. 19,20)
Table 6 summarizes the results of the computation in MATLAB. The percent in
the first right column shows that the eight values of H 0.5 differ from H 0.5, we have greater performance and better improvement. It could prove the times when we assume demand under fBm is better than normal distribution.