یک مکمل برای "A مدل جبری کلی برای بهینه سازی تصمیمات موجودی در زنجیره تامین چند شرکتی چند مرحله ای متمرکز یا غیرمتمرکز"

In this supplement, a number of integrated models under the integer multiplier coordination mechanism is generalized by allowing complete backorders penalized by not only linear but also fixed costs for some/all retailers. The optimal solution to such a three-stage generalized model is algebraically derived, from which optimal expressions for some well-known models are deduced. In addition, an expression for sharing the coordination benefits is modified, and a numerical example for illustrative purposes is presented. A ready future research work involving extension of the generalized model is suggested to conclude the supplement.

In the inventory/production literature, some researchers (such as Wu and Ouyang, 2003, Wee and Chung, 2007 and Chung and Wee, 2007) have constructed their models under the assumption of allowing complete backorders penalized by a linear (i.e. time-dependent) shortage cost. The main purpose of the supplement is to build a further generalized model incorporating complete backorders penalized by not only a linear but also a fixed shortage cost, and solve it algebraically. As a result, we can deduce and solve such special models as Khouja, 2003, Cárdenas-Barrón, 2007, Ben-Daya and Al-Nassar, 2008 and Seliaman and Ahmad, 2009, and Leung, 2009, Leung, 2010a and Leung, 2010b. In addition, with appropriate assignments as in Section 5 of Leung (2010a), we can also deduce and solve other special models: Yang and Wee, 2002, Wu and Ouyang, 2003, Wee and Chung, 2007 and Chung and Wee, 2007. An extensive review of different optimization approaches used in inventory theory can be found in Cárdenas-Barrón (2010).

The main contribution of the supplement to the literature is to establish the n-stage (n = 2, 3, 4,…) model, which is more general than that of Leung (2010b), by allowing complete backorders penalized by both linear and fixed costs for some/all retailers. A ready extension of our model that warrant future research endeavors in this field is: Using complete and perfect squares, we can solve the integrated model of a n-stage multi-firm supply chain either for an equal-cycle-time, or an integer multiplier at each stage, with both linear and fixed shortage costs for a fixed ratio partial backordering allowed for some/all retailers.