سیاست موجودی مطلوب برای مسئله پسر روزنامه فروش فازی با تخفیف در کمیت

Newsboy models have wide applications in solving real-world inventory problems. This paper analyzes the optimal inventory policy for the single-order newsboy problem with fuzzy demand and quantity discounts. The availability of the quantity discount causes the analysis of the associated model to be more complex, and the proposed solution is based on the ranking of fuzzy numbers and optimization theory. By applying the Yager ranking method, the fuzzy total cost functions with different unit purchasing costs are transformed into convex, piecewise nonlinear functions. By proving certain properties of the ranking index of the fuzzy total cost, several possible cases are identified for investigation. After analyzing the relative positions between the price break and the minimums of these nonlinear functions, the optimal inventory policies are provided and closed-form solutions to the optimal order quantities are derived. Several cases of a numerical example are solved to demonstrate the validity of the proposed analysis method. The advantage of using the proposed approach is also demonstrated by comparing it to the classic stochastic approach. It is clear that the proposed methodology is applicable to other cases with different types of quantity discounts and more complicated cases.

In today’s highly competitive business environment, inventory management’s ability to plan and control inventories to meet the competitive priorities is becoming increasingly important in many types of organizations [21] and [34]. One type of inventory problem frequently encountered with seasonal or customized products is the newsboy problem, also called the newsvendor problem or single-period stochastic inventory problem because only a single procurement is made [12] and [27]. Typical practical examples are the dilemmas of making a one-period decision on the quantity of newspapers that a newsboy should buy on a given day or the quantity of seasonal goods that a retailer should purchase for the current year, goods that cannot be sold the next year because of style changes. One of the principal factors in order quantity decisions in the inventory problem is the nature of the demand. In actual applications, demand is uncertain and must be predicted. Notably, there are cases in which the probability distribution of the demand for new products is typically unknown because of a lack of historical information, and the use of linguistic expressions by experts for demand forecasting is often employed. Consequently, the decision maker faces a fuzzy environment rather than a stochastic one in these cases. Fuzzy set theory has been applied to inventory problems with demand uncertainties attributed to fuzziness rather than randomness (for example [3], [4], [5], [7], [8], [9], [10], [24], [25], [26], [28], [30], [31], [37], [38], [41] and [42]). In particular, some scholars have approached the newsboy problem from the standpoint of fuzzy environments. For example, Petrovic et al. [29] proposed a model for the newsboy problem where the demand is described by a discrete fuzzy set and the cost is represented by a triangular fuzzy number. Ishii and Konno [16] used the fuzziness concept to consider the shortage cost in the classic newsboy problem, although the demand was stochastic. Kao and Hsu [19] proposed a model to find the optimal order quantity of the classic newsboy problem with fuzzy demand. Li et al. [25] proposed two models for newsboy problems that have both random and fuzzy uncertainties. Recent work on fuzzy, random newsboy problems includes the studies by Xu and Hu [37] and Hu et al. [13]. There are several other studies on fuzzy newsboy problems. For example, based on fuzzy clustering and fuzzy rules, Cardoso and Gomide [2] proposed a predictive data mining model to predict newspaper demand. In supply chain environments, Xu and Zhai [39] investigated the fuzzy newsboy problem with one manufacturer and one retailer. Hu et al. [13] investigated the fuzzy random newsboy problem with imperfect quality. Dutta and Chakraborty [8] considered the product substitution policy and proposed a fuzzy single-period inventory model with fuzzy demand. These models can be applied to find the optimal order quantity for the classic newsboy problem in fuzzy environments. However, there are many practical situations that are worth further investigation. In particular, quantity discounting (see, for example [43]), which is a price reduction for large orders offered to customers to induce them to buy in larger quantities, is a situation often encountered in practice. However, relatively few papers have been published on the fuzzy inventory problem with quantity discounts. Lam and Wong [22] applied fuzzy mathematical programming to solve economic lot-size problems with multiple price breaks, although they did not focus on newsboy problems. Ji and Shao [17] proposed a hybrid intelligent algorithm based on genetic algorithms and fuzzy simulation to solve the bi-level newsboy problem with fuzzy demand and discounts, but they did not provide analytical solutions. The purpose of this study is to find the optimal order quantity for the single-order newsboy problem when the demand is fuzzy and quantity discounts are available. The proposed solution is based on the ranking of fuzzy numbers and optimization theory. This idea is inspired by the concept of Kao and Hsu [19], but the analysis and results are significantly different and more complex because of the nature of quantity discounts. In the following sections, the classic newsboy problem is reviewed and the fuzzy newsboy model with quantity discounts is briefly introduced. Next, the fuzzy total cost functions with different unit purchasing costs are transformed into convex, piecewise nonlinear functions by applying the Yager ranking method. We propose and prove two properties of the ranking index of the fuzzy total cost to identify possible cases for investigation. Following this, after analyzing the relative positions between the price break and minimums of these nonlinear functions, the best inventory policies are provided and closed-form solutions for the optimal order quantities are derived. Several numerical examples are solved to demonstrate the validity of the proposed analysis method. Finally, the conclusions are presented.

This paper investigated the single-order newsboy problem where quantity discounts are available and the demand is fuzzy rather than stochastic. Our approach used the Yager ranking method to transform the fuzzy total cost functions with different unit purchasing costs into convex, piecewise nonlinear functions. According to the relative positions between the price break and the minimums of these nonlinear functions, the optimal inventory policy was provided and the derivation of the optimal order quantity was analytically discussed for several cases. Furthermore, 27 cases of a numerical example were solved to demonstrate the validity of the proposed analysis method. Another significant aspect is that we proved two properties of the ranking indices of the fuzzy total cost with or without quantity discounts. This advancement eliminates several impossible cases, which simplifies the analyses. Although the discussions in this paper are concentrated on problems with a demand described by a trapezoidal fuzzy number, single order opportunity, single price break, and all-units discount, it is clear that this methodology can be applied to find the optimal inventory policy in other more complicated cases.