A Study on Iterative Algorithm for Stochastic Distribution Free Inventory Models

and Applied Analysis 3 We know that√1 + x > x such that (k n+1 +k n )/(√1 + k 2 n+1 + √1 + k n ) − 1 < 0. Hence, we obtain the following lemma. Lemma 1. If k n+1 > k n , then Q n+2 < Q n+1 . One recalls (10); then one evaluates that


Introduction
Paknejad et al. [1] developed inventory models where lead time and defective rate are constant.Wu and Ouyang [2] generalized their results to include crashable lead time, and defective items are random variables that follow a probabilistic distribution with known mean and derivation.For the distribution free inventory model, Wu and Ouyang [2] used the minimax approach of Moon and Gallego [3] to study the minimum problem for an upper bound for the expected average cost.Wu and Ouyang [2] mentioned that the optimal order quantity and safety factor can be derived by iterative algorithm.Tung et al. [4] developed an analytical approach to prove that the optimal solutions for the order quantity and safety factor exist and are unique.Moreover, they claimed that iterative algorithm cannot be operated by the formulas in Wu and Ouyang [2].
Tung et al. [4] offered an analytical proof to prove the existence and uniqueness of the optimal solution for the inventory model of Wu and Ouyang [2].During their derivations, they found two upper bounds and one lower bound, and then they used numerical examination to compare these two upper bounds to decide the minimum upper bound.They only studied the first derivative system such that they only considered the interior minimum.Moreover, they examined the iterative algorithm in Wu and Ouyang [2] to claim that the formulas in Wu and Ouyang [2] cannot be used to locate the optimal order quantity and safety factor.In this paper, we will show that the formulas in Wu and Ouyang [2] after two modifications are workable for the iterative algorithm.

Review of Previous Results
To be compatible with the results of Wu and Ouyang [2] and Tung et al. [4], we use the same notation and assumptions.We study the paper of Tung et al. [4].They considered the stochastic inventory model of Wu and Ouyang [2] with crashable lead time, defective items, and minimax approach for distribution free demand with the following objective function: Abstract and Applied Analysis for  ∈ [  ,  −1 ], where   (, , ) is a least upper bound of (, , ).Wu and Ouyang [2] derived that   (, , ) is a concave function of  with  ∈ [  ,  −1 ]; so, the minimum must occur at boundary point   or  −1 .
To simplify the expression, we will use  instead of   or  −1 as Tung et al. [4].For   (, , ), Wu and Ouyang [2] computed the first partial derivatives with respect to  and , separately.

Our Revision for Tung et al. [4]
When running the iterative process, after we derive   , we then plug it into (3) to find a relation of   as To abstractly handle this problem, we assume that Under the assumption of Tung et al. [4], it yields that with  3 = ℎ(1 − ()), and  4 = ( +  0 (1 − )) to imply that From ( 5) and ( 6), and with   > 1, we obtain that and then squaring both sides, we find that to show that there is a unique   that can be derived by ( 3).The assertion of Tung et al. [4] that (3) can not be used to execute the iterative process can be improved.

The Proof for the Convergence of the Proposed Three Iterated Sequences
In this section, we will prove that the two iterative sequences proposed for ( 2) and (10) indeed converge.We combine ( 6) and (10) to derive that For later purpose, we rewrite the iterative process based on (2) as follows: where Based on (12), we derive that We know that √ 1 +  2 >  such that ( +1 +   )/(√1 +  2 +1 + √1 +  2  ) − 1 < 0. Hence, we obtain the following lemma.
One rewrites (16) as follows: that is equivalent to One can cancel out the common factor  +1 −   > 0 from the previous inequality and still preserve the same direction of the inequality sign.Consequently, one tries to show that One knows that 2   +1 −   −  +1 =   ( +1 − 1) +  +1 (  − 1) > 0, owing to the condition of  +1 >   > 1.