Periodic inventory model with controllable lead time where backorder rate depends on protection interval

Article history: Received October 2 2013 Received in revised format November 7 2013 Accepted January 1


Introduction
In the recent inventory control system, modern enterprises realize the importance of managing inventory efficiently to run the system profitably.A renowned Just-in-time (JIT) philosophy emphasizes on the advantages and benefits associated with reducing the lead time.Lead time is a topic of interest in most of the inventory systems.Generally, it is assumed that lead time is prescribed (Deterministic and Stochastic) and which therefore is not subject to control.Tersine (1982) suggested that order preparation, order transit supplier lead time, delivery time and setup time (i.e.preparation time for availability of items) usually constitute the total lead time of the system.The lead-time can be decomposed into several crashing periods for making the present system more effective.In many practical situations, lead time can be reduced at an added crashing cost; in other words it is controllable.By shortening the lead time, we can lower the safety stock, reduce the loss caused by stock out; improve the service level to the customer and increase the competitive ability in business.Many researchers Liao and Shyu (1991), Moon and Choi (1998), Hariga and Ben-Daya (1999) have investigated continuous review inventory models with lead time as a decision variable.
It is generally observed that while shortage occurs, demand can be captured partially.Some customers may prefer their demands to be backordered i.e., some customers whose needs are not urgent can wait for their demands to be satisfied, while others who cannot wait have to fill their demand from another source which is lost sale case.However, certain factors motivate the customer for the backorders out of which price discount from the supplier is the crucial one.To some extent, sufficient price discounts to the customers help the supplier to secure more backorders through negotiation.The supplier could fetch a large number of back order rate with higher price discount.Pan and Hsiao (2001) presented continuous inventory model with backorder discounts and variable lead-time.In this paper, the backorder discount has been also taken as a one of the decision variables.Further, the backorder rate depends on the length of the protection interval (period during which shortages can occur) also.This fact point out that when shortages occur, if longer the length of protection interval is, then, larger the amount of shortages is and obviously, this results in smaller the proportion of customers who can wait their orders to be fulfilled and results in smaller backorder rate.The consideration is 'unsatisfied demand during the shortages can lead to optimal backorder ratio by controlling the price discount and the length of protection interval' which ultimately helps the supplier to minimize his total inventory cost.
In a recent study, Pan and Hsiao (2005) expanded the continuous inventory model by considering the case where lead-time crashing cost is taken as the function of reduced lead-time and ordered quantities.In contrast to the continuous review inventory model, we seek to investigate a periodic review model with back order discounts to accommodate more practical feature of the tangible Inventory systems.The applications of periodic review inventory model can often be found in managing inventory cases such as smaller retailer stores, drugs stores and grocery stores by Taylor (1996).Earlier, Chuang et al. (2004) presented a periodic review inventory model with variable lead-time and reduction of setup cost.Jaggi and Arneja (2010) considered a periodic inventory model with unstable lead-time and setup cost with backorder rate depending on backorder discount only.The main objective of this study is to uncover the benefits associated with reduction of lead time and offering backorder discount where backorder rate is dependent on length of protection interval.Two cases have been discussed for protection interval demand 1. Distribution is known 2. Demand distribution is unknown In this study, an inventory model has been formulated which allows review period, lead time and backorder discount to be optimized with known service level.The lead-time is also controllable and has shown that the significant saving could be obtained by offering suitable backorder discount.

Notations and assumptions
To develop the proposed model, we have used the following notation and assumptions:
The inventory level is reviewed every T units of time.A sufficient quantity is ordered up to the target level R , and the ordering quantity is arrived after L units of time.

2.
The length of the lead-time L does not exceed an inventory cycle time T so that there is never more than a single order outstanding in any cycle.

3.
The target level R = Expected demand during the protection interval + safety stock (SS) i.e.
where A is the safety factor and satisfies   P x R q   , q represent the allowable stock out probability which means service level is defined during the protection interval and is given.

4.
The lead-time L consists of n mutually independent components.The i th component has a minimum duration a i and normal duration b i , and a crashing cost per unit time c i .Arranging c i such that for the convenience.Since it is clear that the reduction of lead-time should be first on component 1 because it has the minimum unit crashing cost, and then component 2, and so on.

5.
Let 0 and i L be the length of lead time with components 1,2,…,i crashed to their minimum duration, then i L can be expressed as , , i= 1,…,n and the lead time crashing cost per cycle ( )  C L is given as . (Ouyang et al., 1996).

6.
Assuming that a fraction  (0 ≤  ≤1) of the demand during the stock out period can be backordered so the remaining fraction 1   is lost.The backorder rate β is variable and is in proportion to the price discount x  offered by the supplier per unit and the protection interval.Thus 0 ( ) 0 where 0 , here our model is different from the previous models.(Pan & Hsiao, 2005).

Mathematical Model
We have assumed that the protection interval demand X has a p. d. f. f x with finite mean ( where A is already defined. As Ouyang and Chaung (1999) proposed the periodic review model where the expected net inventory at the beginning of the period is Therefore, the expected net inventory at the end of the period is which gives the expected holding cost per year approximately . (1 Now, the expected stock out cost per year is is the expected demand shortage at the end of cycle i.e., ( Now the objective is to minimize the total expected annual cost ( EAC ) which is the sum of = Ordering cost + Stockout cost + Holding cost + Lead-time crashing cost (1) Also, we have assumed that the backorder rate  depends on the backorder price discount x  and protection interval ( , where A is safety factor. The Eq. ( 1) can be written as Here two cases arise for distribution of lead time demand i.e.
a. Normal distribution b.Unknown distribution

Lead time demand with normal distribution
In this section, we have assumed that the probability distribution of protection interval demand X has a normal distribution with mean ( So, the expected shortages occurring at the end of the cycle is given by and  are the standard normal p. d. f. and c. d. f., respectively. Therefore, Eq. ( 2) is reduced to It can be checked that for fixed T and x  , So, for fixed ( , , )   T L x  , the minimum total expected annual cost will occur at the end points of the interval   , 1 L L i i .On the other hand, for a given value of    will occur at the point ( , )   T x  that satisfy ( , , ) 0 This can be written as where Since it is difficult to obtain the solution for T and x  explicitly as the evolution of Eq. ( 5) and Eq. ( 6) need the value of each other.As a result, we must establish the following iterative algorithm to find the optimal ( , ) Step 1 For each L i , 0,1,2,..., 5), using numerical search technique, evaluate T i .
(b) Substitute the value of T i , in Eq. ( 6) to obtain the value of x i  .

Compare x i
 and 0  .
Step 2 For each ( T i , x i  , L i ), Compute the corresponding expected annual cost EAC (T i , x i  , L i ), from Eq. (3) .Go to step 3.
Step 3 Find min 0,1,2,..., Hence ( * T , * x  , * L ) is the optimal solution and the optimal target level is * Theoretically, for given K, D, h, 0  , 0  ,  and each L i (i = 0, 1, 2, ... n), from Eq. ( 5) and Eq. ( 6), we can obtain optimal values of T and x  , then the corresponding total expected annual cost can be found.Thus, the minimum total expected annual cost could be obtained when the lead-time demand is normally distributed.

Lead time demand with unknown distribution
If the lead time demand does not follow normal distribution or the probability distribution is unknown with first two moments, then the solution can be obtained by minimax approach.Since the probability distribution of X is unknown, we cannot find the exact value of ( . Now we use a minimax distribution free procedure to solve min 0, 0, 0 , we need the following proposition to shorten the problem.

Proposition 3.2.1
For any Moreover, the upper bound ( 7) is tight.Then the Eq. ( 2) can be reduced to where As notified in the preceding section, it can be shown that Therefore, the minimum upper bound of the expected total annual cost will occur at the end point of the interval . Therefore, the first order conditions are necessary and sufficient conditions for optimality.Using the first condition of derivatives, we get and Since it is difficult to obtain the exact value of service factor A which depends upon the required service level on the basis of allowable stock out probability q, because the p. d. f. ( ) is unknown.So, the following proposition has been used to find accurate value of A. Therefore, the algorithm to find the optimal review period, lead-time and backorder discount can be established by using the proposition given below: Algorithm 3.2.3 Step 1 For each q , divide the interval Step 2 For each L i ( 0,1, 2,..., ) i n  , perform step (3) and (4).
Step 3 For given A l   , ,..., 0 1 A A A N  , 0,1, 2,..., l N  , using numerical search technique, evaluate i T from Eq. (9) simultaneously.If T i ≥ L i , then go to step (4) otherwise Set T i = L i , and go to step (4).
Step 4 By using T, Calculate the value of i  using the Eq. ( 10).Compare i  and 0  .
Step 5 For each ( T i , x i  , L i ), Compute the corresponding expected annual cost Step 6 Find min ,..., 0, 1 Step 7 Find is the required optimal solution.

Numerical Example
In order to illustrate the solution algorithms, we have considered an inventory system with the following data having data: D=600 units per year, K= $ 200 per order, S= $ 50 per short out,  = 7 units per week, π 0 = $ 150 per unit, h= $ 20 per unit per year, q = 0.2 where A 0 = 0 and A N = 2, N=200.We have started with fixed service level A = 0.8 (i.e.i A = 0.845 and ( ) i A  =0.1120) by checking the table for Silver and Peterson (1985) (p.p. 699-708).The lead-time has three components, which have been shown in Table 1.Table 3 provides the solution with crashing of lead time with normal distribution.Here we observed that the total annual expected cost decreases as the backorder ratio increases since supplier can fetch a large number of backorders by offering the price discount with no loss although with less cost.The optimal inventory results with relevant savings where lead-time have been crashed given in table 4. In table 5, algorithm 2 has been applied for crashing of lead-time for different backorder ratio when demand during the protection interval is unknown.Furthermore, Table 6 listed the optimal result for controllable lead-time with unknown distribution.

Conclusion
In the proposed model, the effect of backorder discount and length of protection interval on backorder rate with the reduction of lead time in periodic review model has been considered.Reduction in lead time plays an important role to run the system profitably as it helps the supplier to reduce the overall cost of the system by reducing the loss caused by shortages and improving the service level to the customers.Further, longer length of the protection interval results as large amount of shortages and obviously small proportion of customers who can wait their orders to be fulfilled which means smaller backorder rate.Thus, the reduction of lead time and backorder discount are two significant factors which help the supplier to increase his backorder rate and to earn more profit.This model jointly optimizes the review period, lead time and backorder discount.Further, we consider both cases of protection interval demand with known distribution and unknown distribution.(2003).Properties of the periodic review (R, T) inventory control policy for stationary.

:::::
Fraction of the demand back ordered during stock out period such as 0 ≤  ≤ 1 0 Upper bound of the backorder rate 0 Marginal Profit (i.e.cost of lost demand) per unit x  Back order price discount offered by the supplier per unit L : Length of lead-Time X : Protection interval demand which has a p. d. f. x f with finite mean ( ) ) for the protection interval ( L T  ) where  denotes the standard deviation of the demand per unit. The class of p. d. f. x f of the protection interval demand with finite mean ( ) upper bound of expected annual cost.
the protection interval demand that has p. d. f. ( )

Table 1
Lead time data Then applying the algorithm 1, crashing has been carried out for lead-time for different backorder ratio and illustrated in Table2.It is observed that by reducing the lead time the total expected cost decreases.

Table 2
Crashing (Normal) of lead time when the protection interval demand is known

Table 3
Optimal Solutions when demand has Normal Distribution

Table 4
Savings (%) Obtained by crashing of lead time with normally distributed demand

Table 5
Crashing of lead time when the protection interval demand is unknown (Minimax)