دقیق تقریبی، و مدل های تکرار شونده عمومی برای مسئله چند محصولی پسر روزنامه فروش با قید بودجه

Because of the increased importance of inventory control/management in today's supply chain environment, we revisit the seminal work of Hadley and Whitin; that is the multi-product Newsboy problem with budget constraint. In this paper, we present exact solution formulae when the demand probability density function is uniform and generic iterative method (GIM), which yields optimum, or near optimum, solution for general continuous density functions of the demand. A salient feature of GIM is that as it progresses, one can compute the error allowing the user to stop when the desired level of accuracy is achieved. Illustrative examples are given in order to show the application of the proposed models.

With today's emphasis on supply chain management and the increased decentralization of manufacturing activities, there is a renewed interest in inventory control analysis. A significant number of articles has been lately published addressing the diverse aspects of the subject matter ranging from the analytical and case oriented works to those that are strategic in nature. Motivated by this interest, we revisit one of the seminal works developed earlier by Hadley and Whitin (1963); that is the Newsboy problem with budget constraint (also known as the capacitated single period problem (SPP)). One of the main reasons of our reconsideration is its importance to the current and rapidly growing practice of outsourcing, where many companies have to procure from domestic as well as global suppliers. Simply described, the Newsboy problem deals with situations where the demand for a commodity is uncertain (random) and those items that are ordered but remain unsold or unused at the end of the cycle become obsolete. Hence, the buyer may incur a cost to dispose of them. On the other hand if the buyer initially decides to buy smaller amounts of these commodities, shortages may occur causing loss of revenue. As one can see, the question becomes how to determine the quantity to be ordered to minimize the costs incurred. Answering this is the main objective of the classical Newsboy model. When companies are outsourcing several of their products to outside vendors, budget usually presents a constraint reducing the quantities ordered. An example that comes to mind is in fashion industry. To place their orders, retailers have to attend shows long before the season for which the apparel is intended. Naturally, they do not know the exact amount of demand for the different fashion lines. Additionally, their budgets are limited and have to be allocated, optimally, among these competing lines. Hadley and Whitin (1963) have developed a numerical approach based on dynamic programming to solve such a problem, that is a multi-product problem with budget constraint. Nevertheless, they report that when the number of items to be procured is large, the dynamic programming approach becomes rather difficult. Therefore, the problem could become intractable. Addressing some of these difficulties is our intent. In this paper, we introduce models for solving the multi-product Newsboy problem with budget constraint. The developed models here yield an exact solution to the problem when the demand is uniformly distributed and near optimal solution when the demand is distributed otherwise. In the latter case of general demand distribution functions, one of the developed models is a generic iterative method (GIM) that converges rapidly toward the optimal solution and provides an estimate for the error at each iteration. This paper is organized as follows. After this introduction, in Section 2, we give the background of the problem. Then in Section 3, the classical problem is presented. In Section 4, we introduce the models; exact, approximate and iterative. Section 5 is dedicated to numerical examples. In Section 6, we present concluding remarks.

The paper proposes models for solving the classical multi-product Newsboy problem with budget constraint. The models augment those existing in today's literature by adding efficient as well as easy to apply procedure to obtain the optimum, or near optimum, lot size for each product. The models can consider random demands that may have different probability distribution functions. In case of uniform probability density functions, the formulae developed for the optimum size is exact, while those for the exponential are approximate with known values of error. For other probability distributions, a GIM has been introduced. Despite its iterative nature, it is easy to apply and fast in reaching good results if not optimum ones. In addition to being general enough to include various distributions of the demand for each of the products, GIM provides the error level at each iteration (the error is expressed as % of deviation between the utilized budget and that which is available). Testing GIM for a wide range of scenarios, demands and number of items, has yielded less than 5% error in only two iterations, and less than 1% in three iterations. The models introduced here, particularly GIM, lay the groundwork for future development in this arena. That is to say, other scenarios of the multi-product constrained Newsboy problem can be more efficiently solved utilizing extensions of the approach presented here.