Abstract and Applied Analysis

Volume 2013 (2013), Article ID 251694, 3 pages

http://dx.doi.org/10.1155/2013/251694

Research Article

A Study on Iterative Algorithm for Stochastic Distribution Free Inventory Models

Jennifer Lin

Department of Transportation Logistics & Marketing Management, Toko University, Chiayi County 61363, Taiwan

Received 27 September 2012; Accepted 13 February 2013

Academic Editor: Somyot Plubtieng

Copyright © 2013 Jennifer Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We studied the iterative algorithm in Tung et al. (2010) to find out that their assertion is questionable. We derived two new relations between safe factor and order quantity so that we can execute the iterative algorithm proposed by Wu and Ouyang (2001). We have proved that three generated sequences indeed converge to provide a theoretical validation for their iterative procedure.

1. 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.

2. 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:

for , where is a least upper bound of . Wu and Ouyang [2] derived that is a concave function of with ; so, the minimum must occur at boundary point or . To simplify the expression, we will use instead of or as Tung et al. [4]. For , Wu and Ouyang [2] computed the first partial derivatives with respect to and , separately.

Using and , Wu and Ouyang [2] derived that

where , and

Wu and Ouyang [2] claimed that the optimal solution can be obtained by the iterative method. In Tung et al. [4], they assumed that to plug it into (2) to derive , and then they plugged it into (3) to derive a relation for as

Tung et al. [4] mentioned that, in (4), they only derived the value for such that they can not use (3) to execute the iterative process. In this paper, we will show that (3) can be improved to operate the iterative process.

3. 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 , and to imply that

From (5) and (6), and with , 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.

4. 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 and with and .

Based on (12), we derive that

We know that such that . Hence, we obtain the following lemma.

Lemma 1. If , then .

One recalls (10); then one evaluates that

We need the next lemma for our future proof.

Lemma 2. If , then .

After cross-multiplication of (14), one finds that if and only if

that is,

One rewrites (16) as follows:

that is equivalent to

One can cancel out the common factor from the previous inequality and still preserve the same direction of the inequality sign.

Consequently, one tries to show that

One knows that , owing to the condition of .

From the iterative procedure, , which is plugged it into (2) to derive , and then we plug into (11) to obtain that .

Owing to , we apply Lemma 1 to obtain .

We may simplify (6) as follows:

where and with and .

Using , by (20), and we have , applying Lemma 2, we know that .

Repeating the previous argument, we derived that is an increasing sequence and is a decreasing sequence and bounded below by zero such that must converge, and then sequence converges. Finally, the sequence converges too.

5. Numerical Example to Support Our Proof

We will use the same numerical examples in Wu and Ouyang [2] and Tung et al. [4] to illustrate our findings in Section 5. For the precious space of this journal, please refer to Wu and Ouyang [2] for the detailed data, and we refer to the findings of Tung et al. [4] to consider the representative case of and to demonstrate the convergence of the proposed two sequences and with our auxiliary sequence . We list the numerical results in Table 1.

Table 1: For and , the convergence of proposed three sequences.

0
0

1
1.925382
271.444019
8.884377

2
2.433960
179.201796
13.328831

3
2.490503
171.872873
13.886532

4
2.495817
171.206597
13.939601

5
2.496308
171.145237
13.944509

6
2.496353
171.139579
13.944962

7
2.496357
171.139060
13.945003

8
2.496357
171.139014
13.945007

9
2.496357
171.139014
13.945007

If we observe Table 1, then the sequence increases to its least upper bound and the sequence decreases to its greatest lower bound as described by our proposed analysis. Our finding is consistent with Tung et al. [4] in which they mentioned that for and , .

References

M. J. Paknejad, F. Nasri, and J. F. Affisco, “Defective units in a continuous review (s, Q) system,” International Journal of Production Research, vol. 33, pp. 2767–2777, 1995.

K. S. Wu and L. Y. Ouyang, “(Q, r, L) Inventory model with defective items,” Computers and Industrial Engineering, vol. 39, no. 1-2, pp. 173–185, 2001.

I. Moon and G. Gallego, “Distribution free procedures for some inventory models,” Journal of Operational Research Society, vol. 45, no. 6, pp. 651–658, 1994.

C.-T. Tung, Y.-W. Wou, S.-W. Lin, and P. Deng, “Technical note on inventory model with defective items,” Abstract and Applied Analysis, vol. 2010, Article ID 878645, 8 pages, 2010.