مسئله پسر روزنامه فروش با تنظیم رزرو ساده

Newsboy problems have always been an important issue in inventory management. A number of methods have been proposed to solve such problems based on different objectives or considerations. Different from the existing studies, this paper presents a newsboy model with a simple reservation arrangement by introducing the willingness rate, represented as the function of the discount rate, into the models. Mathematical models are developed, and the solution procedure is derived for determining the optimal discount rate and the optimal order quantity. Through the numerical example, we demonstrated the varied profits yielded from the models considering the reservation arrangement, depending on the number of consumers who accept the reservation policy. The profits derived are greater than those from the classical newsboy model, due to the consideration of the reservation.

Inventory management is an important task in business operations. The classical newsboy problem deals with the purchasing inventory problem for single-period products, such as perishable or seasonal goods. For such problems, a business manager should determine product order quantity at the beginning of each period. Products cannot be sold (or the price is trivial) in the next period if the order quantity exceeds actual demand; however, there is an opportunity cost of lost profit in the reverse situation. Therefore, the determination of the order quantity is critical in the classical newsboy problem. In the real world cases, a reservation policy is usually adopted by business units for some commodities, including newsboy-type products, since it can reduce the demand uncertainty and therefore profits could increase. The motivation for a consumer to make a reservation may be complicated, including both psychological and economic factors. Regarding the economic factors, a discount could be an important incentive, and in general, the higher the discount the higher the willingness of a customer to make a reservation. In other words, the discount rates have an influence on the sale quantity and therefore the order quantity. However, a higher discount rate could lead to lower profits, or even cause a loss, despite the higher sales. Accordingly, from the business unit’s side, the determination of the optimal order quantity as well as the optimal discount rate will influence the expected profit arising from the solution to a problem. For classical newsboy problems, many researchers have proposed a number of approaches to determine the optimal order quantity in order to deal with the uncertainty of demand (e.g. Anvari, 1987, Fathi and Nuttle, 1987, Hadley and Whitin, 1963, Ismail and Louderback, 1979, Kabak and Schiff, 1978, Khouja and Mehrez, 1996, Lau and Lau, 1988, Lau, 1980, Lau and Lau, 1995, Li et al., 1991, Sankarasubramanian and Kumaraswamy, 1983 and Silver et al., 1998). Almost all of these studies consider the product quantity demand as a random variable to determine the optimal order quantity in order to maximally achieve the anticipated objective, such as expected profit, expected utility, the firm’s market value, and the probability of reaching the expected profit level. In addition, some researchers have extended the classical newsboy problem based on various considerations. For example, considering the products with a long-selling period and highly volatile stochastic demand, Chung and Flynn (2001) introduced reactive production into the classical newsboy problem. Khouja and Robbins (2003) addressed the important effects of advertising on sales, assuming that the mean demand is increasing and concave with respect to the advertising expenditure. Weatherford and Pfeifer (1994) introduced the system of advance booking of orders, and indicated that the optimal discount rate is an important factor to maximize the expected profit. Unlike the existing literature, this research proposes a newsboy problem with a simple reservation arrangement by introducing the willingness rate, represented as a function of the discount rate, into the models. The solution procedure is derived for determining the optimal discount rate and then the optimal order quantity for such a problem. This paper is organized as follows. In Section 2, we briefly introduce the model of the classical newsboy problem. The model with the reservation arrangement is then presented, and the optimal solutions are derived in Section 3. In Section 4, a numerical example is given to demonstrate the feasibility of the proposed model. Finally, conclusions are provided in Section 5.

A newsboy problem with a simple reservation arrangement is presented in this paper. A special discount rate is given to the consumers who make a reservation for the product. The willingness rate is the function of the discount rate to represent the proportion of the consumers having a desire to make a reservation. The model is then formulated to maximize the total expected profit. In the solution procedure, the optimal order quantities for the reservation and the usual sales depend on the willingness rate, when the parameter values, like those in the classical newsboy model, are known. The decision variable of the discount rate in the model is critical, since the willingness rate is the function of a discount rate. The optimal total order quantity in the proposed model is the linear combination of the expected demand quantity and the optimal order quantity in the classical newsboy model, taking the optimal willingness rate as the weight. Through the example with different willingness functions, we demonstrated that various profits are yielded based on different willingness functions. Meanwhile, the profits from the model considering the reservation arrangement are greater than those from the classical newsboy model.