There are studies regarding monopolistic all-unit Quantity discount but we add the duopoly Bertrand competition under incomplete information with it, which is not treated earlier. Then, a systematic procedure is outlined to compute pay-off functions for the developed game. A real numerical example is given in support of the solution procedure and a sensitivity analysis is carried out to examine the important parameters on the equilibrium quantities. This study provides several interesting insights. For instance, it is approved that if retailers know each other completely or not, under some special conditions, rejecting the offered discount is the strictly dominant strategy adopted by the retailers. By doing sensitivity analysis for a real-world case, it is displayed that for maximizing profit, the company should pay more attention to increasing the demand rather than reducing the costs. It could be seen that a change in the value of all parameters except the fixed ordering cost and the holding cost rate results in changing the discount strategy selected by the retailers. Investigating an incomplete information in comparison with the common knowledge case, it is necessary that retailers obtain more accurate knowledge about their rivals.

The task at hand involves selecting the most suitable supplier(s), determining the optimal lot size, and allocating the total order quantities among the suppliers based on various selection criteria. However, this can become more complex when taking into account Quantity discount offers and transportation selection decisions in the selection and order allocation process. To address this challenge, this paper proposes an integrated approach that combines the Analytic Hierarchy Process (AHP) with a multi-objective mixed integer nonlinear program. The approach is designed for a multi-item, capacitated multi-supplier scenario, where the goal is to select suppliers, determine lot sizes, and allocate orders while taking into account unit Quantity discounts and intermodal freight costs. The proposed approach aims to minimize costs and the percentage of rejected items, while maximizing the purchasing value. To solve this problem, an efficient genetic algorithm with problem-specific operators is utilized.

Joint economic lot sizing (JELS) addresses integrated inventory models in a supply chain. Most of the studies in this field either do not consider the role of the transportation cost in their analysis or consider transportation cost as a fixed part of the ordering costs. In this article, a model is developed to analyze an incremental Quantity discount in transportation cost. Appropriate equations are derived to compute the costs related to the inventory systems in the buyer and vendor sites. Then, a procedure including five steps is proposed to optimize the model and determine the values of the decision variables. To analyze the performance of the incremental discount, the JELS problem is studied in two other states of transportation costs. These states include fixed transportation cost and all-unit Quantity discount. Moreover, some numerical analyses are carried out to show the impact of transportation costs and inventory-related parameters on the system performance. According to the results of the sensitivity analyses, it is observed that all-unit Quantity discount leads to a better performance of the system in comparison with the incremental Quantity discount.

Supplier selection and allocation of optimal order Quantity are two of the most important processes in closed-loop supply chain (CLSC) and reverse logistic (RL). So that providing high quality raw material is considered as a basic requirement for a manufacturer to produce popular products, as well as achieve more market shares. On the other hand, considering the existence of competitive environment, suppliers have to offer customers incentives like discounts and enhance the quality of their products in a competition with other manufacturers. Therefore, in this study, a model is presented for CLSC optimization, efficient supplier selection, as well as orders allocation considering Quantity discount policy. It is modeled using multi-objective programming based on the integrated simultaneous data envelopment analysis–Nash bargaining game. In this study, maximizing profit and efficiency and minimizing defective and functions of delivery delay rate are taken into accounts. Beside supplier selection, the suggested model selects refurbishing sites, as well as determining the number of products and parts in each network’s sector. The suggested model’s solution is carried out using global criteria method. Furthermore, based on related studies, a numerical example is examined to validate it.