On the single item multi-supplier system with variable lead-time, price-quantity discount, and resource constraints

Abstract

In this paper, we propose a mixed integer approach for solving a single item multi-supplier problem with variable lead-time, price-quantity discount, and resource constraints. This problem is important in the field of inventory control. However, to the best of the authors’ knowledge, no work has been done for the problem. The proposed method not only can easily determine the optimal order quantity from multi-supplier system with different price-quantity discount policies, but also allows resource constraints to be added by inventory decision-maker as deemed appropriate in real-world situations.

Introduction

In manufacture industry, especially, materials purchasing from outside suppliers can represent a large percentage of the total operating costs [24]. Thus, these companies must be able to make profitable buying decision under price-quantity discount (PQD), variable lead-time and resource limitations (e.g., capital and space limitations and so on) for gaining their competitive advantages. The PQD is the reduction of purchase price from the suppliers due to large order (or quantities) being replaced, which is usually represented as a decreasing step function of the size of the delivery lot. To solve this problem, traditional approaches had used multiple-step process to obtain only local optimal solutions [1], [5], [6], [8], [11], [15].

Lead-time is defined as: the time interval from the purchasing is done to it is physically on the shelf for satisfying demands [14]. In many situations, lead-time can be shortened by adding extra crashing cost for saving stock costs [2], [10], [14], [16], [17], [19], [20], [21]. These conventional methods treat lead-time as an unconstrained non-linear problem without any constraints and then use derivatives to find the optimal solutions. However, their methods are not suitable for real-world situations due to resource constraints are omitted in their models.

In the few decades, most inventory control models assume that an inventory item is replenished from a single supplier or two suppliers [3], [7], [9], [12], [13], [17], [22], [23], [25] until an acquisition policy for multi-supplier condition was proposed by Rosenblatt et al. [24]. Based on their model, the decision-maker can easily determine the optimal order quantity from which suppliers. However, there are three defects in their model: (i) A complicated heuristic procedure is required in the solution process. (ii) It is not able to solve the multi-supplier system with variable lead-time, PQD, and resource constraints problem. (iii) Problem arises when constraints are added in their model. (iv). A given combination of d_i and Q_i, where d_i is the average periodic quantity order from supplier i and Q_i is the order quantity from supplier i, may not be executable because the number of orders placed with a buyer is not an integer [24]. Recently, Chang and Chang [4] proposed a new method for inventory model with variable lead-time and price-quantity discount, but the single item multi-supplier situations are still not considered in their model. To the best of the authors’ knowledge, there is no work done on the single item multi-supplier system with variable lead-time, PQD and resource constraints optimally.

In the paper, we propose a mixed integer model for solving this problem. Three advantages of the proposed model are as follows: (i) A new model is derived for the single item multi-suppler system with variable lead-time, PQD, and resource constraints. (ii) The global optimal solution obtained by propose model is better than the local optimal solution obtained by complicated heuristic procedure in traditional methods. (iii) Constraints can easily be added by inventory decision-maker as deemed appropriate in real-world situations.