Lot-Sizing Decisions under Trade Credit Depending on the Ordering Quantity in Fuzzy Annual Demand

Problem statement: In reality, there are of various products such as l iquids, volatile, medicines and materials, in which the rate of deter ioration is very large. However, little attention h as been mentioned what the effects of deterioration oc cur in most inventory systems. Also, the loss of th e deterioration should not be neglected as an importa nt issue. Approach: In this study, we proposed the problem of determining the Economic Order Quantity (EOQ) for exponentially deteriorating items under the conditions of permissible delay in paymen ts. Recent researches revealed that the payment for the product must be made forthwith when the order q uantity is less than the allowable delay in payments. The fixed trade credit period was permitt ed, or else. Therefore, we utilized the fuzzy theor y to implement more considerable answers via numerica l examples which were provided to illustrate the results clearly at the end of study. Results: This study discussed the optimal replenishment cyc le time for an exponentially deteriorating product under co nditions of permissible delay in payments to take the order quantity into account. Conclusion/Recommendations: This study presented another fuzzy environment inventory model which accounts for the p rmissible delay in payments offered by supplier to the retailer. From the results, it can be concluded that another point of view to decision makers in uncertainty.


INTRODUCTION
There are of various products such as liquids, volatile, medicines and materials, in which the rate of deterioration is very large. However, most inventory systems are implicit assumption without considering the effects of deterioration and, the loss due to deterioration should not be neglected. In this study, we assume that items have the exponential distribution for the time to deterioration.
To stimulate output, suppliers are often willing to extended payment privileges to retailers which are quite prevalent in some industries nowadays. Such credit policies may be applied as an alternative to price discounts to induce larger orders. It is indicated that [1][2][3][4] , such policies are not thought to provoke competitors to reduce their prices and thus introduce lasting price reductions.
Practically, a few of pharmaceutical companies and agricultural machinery manufacturers offer a larger credit period for larger amount of purchase rather than giving some discount on unit price.
Moreover, there are two categories, presented by Khouja and Mehrez [5] , investigated the effect of supplier credit policies: (1) one is that credit terms are independent of the order quantity and (2) the other is that credit terms are linked to the order quantity. In the further case, suppliers always use favorable credit terms to encourage customers for higher ordering quantities. In other words, the favorable credit terms apply only at large order quantities and are used in place of quantity discounts. In this study, we fully focus on the latter case.
The retailer has to decide whether it is worth to alter the regular ordering pattern to exploit other opportunities. As a result, this study incorporates both Hwang and Shinn and Khouja and Mehrez [3,5] . In addition, the basic decision problem for the retailer can be modeled as the design of the cost-minimizing ordering strategies. The model incorporates the fuzziness of annual demand that use Yao and Wu's [6] ranking method for fuzzy number to defuzzy and then illustrate the solution procedure of the proposed algorithm to find the corresponding optimal solution.

Assumptions:
• Replenishments are instantaneous with a known and constant lead time • No shortages are allowed • The inventory system involves only one item • The supplier proposes a certain credit period and sales revenue generated during the credit period is deposited in an interest-bearing account with rate I. At the end of the period, the credit is settled and the retailer starts paying the capital opportunity cost for the items in stock with rate R (R≥1) • Inventory is depleted not only by demand but also by deterioration. Deterioration follows an exponential distribution with parameter. (From Ghare and Schrader, the deterioration rate is a constant fraction of the on-hand inventory.) • If Q W < , the delay in payments is not permitted.

RESULTS
Two numerical examples which provide more detailed information are presented in this part. The number marked with " * " is the optimal solution in crisp sense. Then, the relative variation between fuzzy case and crisp case for the quantity at which the delay in payments is permitted and the optimal replenishment cycle time can be measured respectively as follows: The results of the example 1 and 2 are shown in Table 1 and 2.
This study discusses the optimal replenishment cycle time for an exponentially deteriorating product under conditions of permissible delay in payments to take the order quantity into account. If Q W < , the delay in payment is not permitted [7] . Or else, the fixed trade period M is permitted. There are two cases (i) M>W*and (ii) M≤W* to be explored. Theorem 1 gives the solution procedure to find T* when M>W*. Theorem 2 gives the solution procedure to find T* when M≤W*. Numerical examples are given to illustrate Theorem 1 and 2 and the effect of the inventory deteriorating rate on the optimal cost and the replenishment time. Furthermore, an algorithm to find the optimal replenishment cycle time is presented. The proposed algorithm indicates that T* can be one of T 1 *, T 2 *, T 3 * and W*. According to the convexity of TVC i (T) (i = 1,2,3), the Newton's method can be used to locate T i * for all i = 1,2,3. Finally, if W 0 = , from the viewpoint of the cost, Hwang and Shinn [3] can be treated as a special case of this study. When the deterioration is ignored, Eq. 4 a-c and 6a and b are reduced to Khouja and Mehrez [5] .
The part of Fuzzy, please see the results in Fig. 1-4.

Definition:
For any a and 0∈R, define the signed distance (Kaufmann and Gupta, Zimmermann, Yao and Wu) from a to 0 as d 0 (a, 0) = a. If a>0, a is on the right hand side of origin 0 and the distance from a to 0 is d 0 (a, 0) = a. If a < 0, a is on the left hand side of origin 0 and the distance from a to 0 is -d 0 (a, 0) = -a. This is the reason why d 0 (a, 0) = a is called the signed distance from a to 0 [6,8,9] .
Further, because of the 1-level fuzzy point 1 0 , ɶ is mapping to the real number 0, the signed distance of [A L (α)α, A U (α)α] to 1 0 ɶ , can be defined as: Thus, since the above function is continuous on 0≦α≦1 for A ∈ Ω ɶ , we can use the following equation to define the signed distance of A ɶ to 1 0 ɶ , as follows.

CONCLUSION
Fundamentally, in this study, we suggest another fuzzy environment inventory model which accounts for the permissible delay in payments offered by the supplier to the retailer, is mainly focused on uncertain environment.
The contributions of this study and the approach we take to solve the problem are significant because: • This study brings attention to the trade credit that is of major importance in the operations of enterprises in many economics and which depends on the order quantity that has been left out of studies in inventory theory that deal with quantity discounts • The focus on permissible delay in payments depending on the ordering quantity, that is, besides stimulates the much needed research in this area; it provides a useful model for many organizations that use the decision rule to improve their total operating cost in reality • It is clear from this study that the retailer can assess their monetary impact to find the optimal ordering policy under realistic conditions linking marketing as well as operations management concerns Finally, future research in this direction should also incorporate more real-world cases, more supply chain stages or echelons considerations and attempt to more refined solution methodologies.