In the real market, as unsatisfied demands occur, the longer the length of lead time is, the smaller the proportion of backorder would be. In order to make up for the inconvenience and even the losses of royal and patient customers, the supplier may offer a backorder price discount to secure orders during the shortage period. Also, ordering policies determined by conventional inventory models may be inappropriate for the situation in which an arrival lot contains some defective items. To compensate for the inconvenience of backordering and to secure orders, the supplier may offer a price discount on the stockout item. The purpose of this study is to explore a coordinated inventory model including defective arrivals by allowing the backorder price discount and ordering cost as decision variables. There are two inventory models proposed in this paper, one with normally distributed demand and another with distribution free demand. A computer code using the software Matlab 7.0 is developed to find the optimal solution and present numerical examples to illustrate the models. The results in the numerical examples indicate that the savings of the total cost are realized through ordering cost reduction and backorder price discount.
In real life, the occurrence of shortage in an inventory system is phenomenon. Under the most market behaviors, we can often observe that many products of famous brands or fashionable goods such as certain brand gum shoes, hifi equipment, and clothes may lead to a situation in which customers may prefer to wait for backorders while shortages occur. Besides the product, the image of selling shop is one of the potential factors that will motivate the customers intention of backorders. To establish the good image and enhance customers’ loyalty, the selling shop could invest in upgrading the servicing facilities, maintaining the high quality of selling products and spending money on advertisement. Other endeavors, such as mailing the greeting card and providing free gift, can also be done to establish a good relationship with their customers. We note that such type of activities is not certainly free. Naturally, an extraadded cost must be spent for these efforts. Further it is expected to have a result to reduce the shortage cost of lostsales and the total expected annual cost. Under the situation, for a vendor, how to control an appropriate length of lead time to determine a target value of backorder rate so as to minimize the relevant inventory cost and increase the competitive edge in business is worth discussing. As a result by shortening the lead time, we can lower the safety stock, reduce the stockout loss, and improve the service level to the customers so as to increase the competitive edge in business. The issue of lead time reduction has received a lot of interest in recent years. With such characteristics, researchers have modified traditional inventory models to incorporate the implementation of lead time concepts.
Liao and Shyu [
As unsatisfied demands occur, practically, we can often observe that some customers may prefer their demands to be backordered, and some may refuse the backorder case. There is a potential factor that may motivate the customers’ desire for backorders. The factor is an offering of a backorder price discount. In general, provided that a supplier could offer a backorder price discount on the stockout item by negotiation to secure more backorders, it may make the customers more willing to wait for the desired items. In other words, the bigger the backorder price discount, the bigger the advantage to the customers, and, hence, a larger number of backorder ratios may be the result. Pan and Hsiao [
In reality all manufacturing industries try to produce products with acceptable quality but in the long run it is difficult to produce perfect quality items due to various causes like machine breakdowns, labor problems, and shortages of raw materials. In the classical inventory model, it is implicitly assumed that the quality level is fixed at an optimal level; that is, all items are assumed to have perfect quantity. However, in the real production environment, it can often be observed that there are defective items being produced due to imperfect processes. The defective items must be rejected, repaired, reworked, or, if they have reached the customer, refunded. In all cases, substantial costs are incurred. Paknejad et al. [
However, to the best of our knowledge, there exists no literature considering the collaborative inventory system in a supply chain with defective arrival units and backorder price discount. Therefore, the paper focuses on establishing collaborative inventory system under the above said concepts. Therefore, the proposed model further fits a more general inventory feature in many reallife situations. In this paper we develop an inventory model including defective arrivals by allowing the backorder price discount as a decision variable. It is assumed that the supplier may offer a backorder price discount to the patient customers with outstanding orders during the shortage period to secure customer orders. Furthermore, the inventory lead time can be shortened at an extra crashing cost and ordering cost can also be reduced by capital investment. We discuss two models, namely, one with normally distributed demand and the other with generally distributed demand. For each model, we develop a separate computational algorithm with the help of the software Matlab 7.0 to find the optimal ordering strategy.
The remainder of this paper is organized as follows. Section
The proposed model is developed based on the following assumptions and notations.
The notation is summarized in the following:
We assume the following assumptions to develop our model.
Inventory is continuously reviewed. Replenishments are made whenever the inventory level (based on the number of nondefective items) falls to the reorder point
An arrival may contain some defective items. We assume that the number of defective items in an arriving order of size
The reorder point
The lead time
The components of lead time are crashed one at a time starting with component 1 (because it has the minimum unit crashing cost), then component 2, and so on.
Let
The supplier makes decisions in order to obtain profits. Therefore, if the price discount
During the stockout period, the backorder ratio
Upon an arrival order lot
Inspection is nondestructive and errorfree.
We assume that the lead time demand
In this model, we consider the ordering cost
We assume that the lead time demand
To solve this problem, taking the first order partial derivatives of
By examining the second order sufficient conditions, for fixed
For fixed
For
Refer to Appendix
In order to examine the effects of
Once we have optimal solution
In practical problems, the probability distribution of the lead time demand information is often quite limited. A decision maker can identify the mean value and variance of the lead time demand. But the exact probability distribution may be unknown. Due to insufficient information, the expected shortage
For this purpose, we need the following proposition which was asserted by Gallego and Moon [
For any
Substituting
Substituting (
For
Refer to Appendix
In order to examine the effects of
Once we have the optimal solution
In order to illustrate the proposed solution procedure and the effects of ordering cost reduction by investment as well as lostsales rate reduction through backorder price discount, let us consider an inventory item with the same data as in Pan and Hsiao [
Lead time data.
Lead time component  Normal duration  Minimum duration  Unit crashing cost 





1  20  6  0.4 
2  20  6  1.2 
3  16  9  5.0 
Hence, the mean of
Suppose that the lead time demand follows a normal distribution. Applying the proposed computational algorithm 1 yields the results shown in Table
Solution procedure of Example









0.2  8  0  83  68  164  75.74  3981.45 
6  5.6  97  72  133  75.67  3792.34  
4  22.4  112  80  114  75.40  3684.56  
3  57.4  121  92  98  75.31  3741.71  


0.4  8  0  85  69  160  75.81  3842.64 
6  5.6  97  72  129  75.41  3780.53  
4  22.4  115  80  108  75.36  3622.36  
3  57.4  121  92  91  75.12  3791.89  


0.6  8  0  85  69  157  75.84  3763.51 
6  5.6  96  72  123  75.38  3692.46  
4  22.4  116  80  106  75.18  3543.28  
3  57.4  121  93  88  75.09  3699.63  


0.8  8  0  88  70  140  75.92  3701.46 
6  5.6  96  73  118  75.21  3600.24  
4  22.4  116  81  103  75.11  3490.32  
3  57.4  122  93  84  75.02  3578.41 
Summary of the optimal solutions of Example

The proposed model 
Model with fixed 
Saving (%)  












0.2  4  112  80  114  75.40  3684.56  4  132  76  4519.96  18.48 
0.4  4  115  80  108  75.36  3622.36  4  132  73  4470.70  18.97 
0.6  4  116  80  106  75.18  3543.28  4  133  70  4420.67  19.85 
0.8  4  116  81  103  75.11  3490.32  4  133  67  4363.56  20.01 
The data is the same as in Example
Solution procedure of Example









0.2  8  0  154  135  160  81.72  5151.23 
6  5.6  140  124  138  81.68  5042.41  
4  22.4  121  120  107  81.41  4832.53  
3  57.4  123  122  94  81.32  4795.67  


0.4  8  0  149  121  151  81.75  4773.43 
6  5.6  136  114  130  81.71  4616.13  
4  22.4  116  112  96  81.46  4501.39  
3  57.4  118  116  82  81.38  4467.81  


0.6  8  0  138  110  143  81.86  4450.58 
6  5.6  121  106  117  81.82  4393.73  
4  22.4  110  102  88  81.56  4235.46  
3  57.4  114  104  76  81.42  4286.82  


0.8  8  0  130  100  139  81.91  4152.40 
6  5.6  115  95  101  81.84  4085.61  
4  22.4  104  93  76  81.58  3923.32  
3  57.4  107  97  68  81.47  3988.54 
Summary of the optimal solutions of Example

The proposed model 
Model with fixed 
Saving (%)  












0.2  4  119  122  94  81.32  4795.67  3  183  72  5697.95  15.84 
0.4  4  115  116  82  81.38  4467.81  3  174  64  5354.01  16.55 
0.6  3  110  102  88  81.56  4235.46  4  163  74  5082.14  16.66 
0.8  3  104  93  76  81.58  3923.32  4  157  67  4810.65  18.45 
From the above examples, it is noted that the savings of joint total expected annual cost are realized through ordering cost reduction and backorder price discount. It is also seen that when the upper bound of the backorder ratio
Furthermore, we examine the performance of distributionfree approach against the normal distribution. If we utilize the solution
Calculation of EVAI.



EVAI  Cost penalty 

0.2  3966.32  3684.56  281.76  1.076 
0.4  3843.43  3622.36  221.07  1.061 
0.6  3752.64  3543.28  209.36  1.059 
0.8  3681.72  3490.32  191.40  1.054 
In practice, this ordering cost reduction model with backorder price discount at a retailer is more matched to real life supply chains. For example, in India, small scale industries spend an extra added cost by means of investment as well as allowing backorder price discount and it is expected to have a result to reduce the shortage cost of lostsales and the joint total annual expected cost. Again the goal of justintime (JIT) from the inventory management standpoint is to produce smalllot sizes with reduced lostsales and order cost. Investing capital in shortening lead time, reducing order cost, and allowing backorder price discount are regarded as the most effective means of achieving this goal and also we can lower the safety stock, reduce the out of stock loss, and improve the customer service level so as to gain competitive advantages in business.
In the real market, as unsatisfied demands occur, the longer the length of lead time is, the smaller the proportion of backorder would be. Under probabilistic demand, inventory shortage is unavoidable. In order to make up for the inconvenience and even the losses of royal and patient customers, the supplier may offer a backorder price discount to secure orders during the shortage period. Also, ordering policies determined by conventional inventory models may be inappropriate for the situation in which an arrival lot contains some defective items. The purpose of this paper is to examine the effect of defective arrivals on an integrated inventory model by allowing the backorder price discount and order cost as decision variables. We first assume that the lead time demand follows a normal distribution and then relax this assumption by only assuming that the first two moments of the lead time demand are given and then solve this inventory model by using minimax distributionfree approach and, for each model, separate efficient computational algorithm is constructed to find the optimal solutions. Our algorithm is easy to use and mathematically sound.
Since the marketing environment is continuously changing, corporate strategy must be adjusted accordingly. If the ordering cost could be reduced effectively and if orders could be secured well, the total relevant cost per unit time could be automatically improved. On the other hand, if the buyer permits seller bigger backorder price discount, then it will be allowed to promote the service level and to reduce total expected annual cost. This paper fulfills the above literature gaps. Since our criterion is supported by the numerical examples, from the practical point of view, it is valid and useful to the competitive business.
By computing the proposed models, specifically, from the results of numerical examples, we observe that a significant amount of savings can be easily achieved. Also it shows that when there is an investing option of improving the system, it is advisable to invest. Computational results reveal the serious impact of inferior supply quality on system cost as well as the sensitivity of the system parameters. This model contributes an application in an inventory system consisting of ordering quantity, inspections, and defective items. The proposed model further fits a more general inventory feature in many reallife situations. In future research on this problem, it would be interesting research topic to deal with the inventory model with a service level constraint.
We want to prove the Hessian matrix of
Then we proceed by evaluating the principal minor determinants of
Consequently, after substituting
We want to prove the Hessian matrix of
Then we proceed to evaluate the principal minor determinants of
The second principal minor determinant of
Consequently, after substituting
Hence, it is concluded that the Hessian matrix
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the anonymous reviewers and the editor for their insightful and constructive comments and helpful suggestions, which have led to a significant improvement in the earlier version of the paper. Best efforts have been made by the authors to revise the paper abiding by the constructive comments of the reviewers.