تجزیه و تحلیل مسئله پسر روزنامه فروش با تقاضاهای فازی و تخفیف افزایشی

This paper proposes an analysis method for the single-period (newsboy) inventory problem with fuzzy demands and incremental quantity discounts. In fuzzy environments, the availability of the quantity discount makes the analysis of the associated model more complex. The proposed analysis method is based on ranking fuzzy number 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. To effectively and efficiently find the optimal inventory policy, the proofs of two properties regarding the relative position between the price break and minimums of these nonlinear functions are proposed. The closed-form solutions to the optimal order quantities are also derived. Four cases of a numerical example are solved to demonstrate the validity of the proposed analysis method. It is clear that the proposed methodology is applicable to further cases with different types of quantity discounts and other more complicated cases. More importantly, managerial implications are also provided for decision-makers’ references.

Effective and efficient inventory management is becoming increasingly important for all types of organizations and their supply chains in today’s highly competitive business environment (Arshinder et al., 2008, Krajewski et al., 2010 and Stevenson, 2005). One type of inventory problem often encountered with seasonal or customized products is the newsboy problem, also called the single-period inventory problem or the newsvendor problem (see, e.g., Mostard et al., 2005, Wu et al., 2007, Serel, 2008 and Özler et al., 2009), since only a single procurement is made (Hadley and Whitin, 1963, Tersine, 1988 and Hillier and Lieberman, 2005). A typical example is making a one-period decision regarding how many items of a seasonal good, which cannot be sold the next year, to purchase for the current year. One of the principal factors affecting decisions on order quantity in the newsboy problem is the nature of the demand. In real life, demands are uncertain and need to be estimated from historical data and described by certain probability distributions under traditional (crisp) environments. However, there are cases where the probability distribution of the demand cannot be obtained. For example, the probability distribution of the demand of new products is usually unknown due to lack of historical information. In this case the demands are suitably described subjectively by linguistic terms, such as “high”, “low”, or “approximately equal to a certain amount.” That is, the demands are fuzzy rather than probabilistic. Fuzzy set theory has been applied to inventory problems, with demand uncertainties attributed to fuzziness rather than randomness (see, e.g., Park, 1987, Roy and Maiti, 1997, Chang and Yao, 1998, Lee and Yao, 1998, Yao and Lee, 1996, Yao and Lee, 1999, Chang, 1999, Amid et al., 2009 and Shi and Zhang, 2010). Additional related studies can be found in Mula et al. (2006) which provides a good review. Several scholars have approached the newsboy problem from the standpoint of fuzzy environments. For example, Petrovic et al. (1996) proposed a model for the newsvendor problem where the demand is described by a discrete fuzzy set in which the cost is presented by a triangular fuzzy number. Ishii and Konno (1998) used the fuzziness concept to consider shortage cost in the classical newsvendor problem, although the demand was still stochastic. Kao and Hsu (2002) proposed a model to find the optimal order quantity of the classical single-period problem with fuzzy demand. Li et al. (2002) proposed two models for newsvendor problems that have two types of uncertainties, one of which is randomness and the other is fuzziness. Quantity discounts often occur in practice, though they have not been discussed in the aforementioned articles. In fact, only a few articles have been published on the fuzzy inventory problem with quantity discounts. Lam and Wong (1996) applied fuzzy mathematical programming to solve economic lot-size problems with multiple price breaks, though they did not focus on newsboy problems. On the basis of genetic algorithms and fuzzy simulation, Ji and Shao (2006) proposed a hybrid intelligent algorithm to solve the bi-level newsboy problem with fuzzy demand and discounts, but they did not provide analytical solutions. This paper considers the newsboy problem with incremental quantity discounts and demands being fuzzy numbers, with the objective of minimizing the total cost per unit time. Next, a solution procedure for finding the optimal inventory policy is developed, and the results are illustrated with numerical examples. Finally, discussions and conclusions together are presented.

Newsboy problems have been extensively studied in the literature. This paper investigates the case where quantity discounts are available and the demands are fuzzy rather than stochastic. The idea of this paper is to use 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 position between the price break and the minimums of these nonlinear functions, the best inventory policy is provided, and the derivation of the optimal order quantity is analytically discussed from several cases. Furthermore, four numerical examples are successfully solved to demonstrate the validity of the proposed analysis method. Also noteworthy is that in this paper we prove two properties of the ranking indices of fuzzy total costs with incremental quantity discounts. This eliminates several impossible cases, so that the analyses become more efficient and easier. Although the discussions of this paper are concentrated on the case where demands are triangular fuzzy numbers with a single price break, it is clear that the methodology can be applied straightforwardly to other more complicated cases to find the best inventory policy. In practice there are other discount schemes, such as lot-size all-unit quantity discount and volume based quantity discount. Clearly, the proposed analysis method is based on ranking fuzzy numbers and optimization theory, and is not dependent on particular discount schemes. Therefore, the proposed method is still applicable to the newsboy problem with different discount schemes. In fact, the authors have successfully applied the proposed method to the newsboy problem with lot-size all-unit quantity discount. Comparing the results of these two cases, the results of the incremental discount case obtained in this paper are much more complicated than those of all-unit discount case. It is believed that the results are significantly different under different discount schemes. The analysis of newsboy problems with other different discount schemes is a research task to be explored in the future. Furthermore, the issue of the pricing policy made endogenously by the seller is also an area of future research. Another issue of interest is the difference between using fuzzy numbers and using probability distribution to characterize fuzzy demands. Since the fuzzy demands in this paper have been described by membership rather than probability functions, therefore the given possibility distributions need to be transformed into probability distributions. As Giannoccaro et al. (2003) pointed out, this task is not trivial due to the substantial difference between membership and probability functions (Mizumoto and Tanaka, 1991). The comparison between these two concepts for describing the fuzzy parameters in newsboy problems with different discount schemes is a direction of future research.