Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie The gardener problem with reservation policy and discount Seyedeh Sara Sadralsharifi a , Seyed Hamid Reza Pasandideh a , Seyed Taghi Akhavan Niaki b, ⁎ , Mohammad Hossein Nahavandian c a Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran b Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran c Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran ARTICLE INFO Keywords: Gardener problem Reservation policy Discount rate Tight constraint Nonlinear programming ABSTRACT The Newsboy problem has always been an important issue in inventory management. The multi-product newsboy problem with random yield and budget constraint named as the Gardener Problem is one of the novels and popular extensions of the newsboy problem. Different from the existing studies, this paper presents a multi- product gardener problem with reservation policy. Moreover, a discount rate is offered to those customers who are willing to make reservations. In addition to the demand from the original customers, extra demand is in- cluded in the model due to the motivation received by the discount rate. A solution algorithm namely the multi- product gardener problem with reservation policy (MGPRA) is proposed to obtain the optimal order quantities and discount rate in order to maximize the expected profit when the yield and demand are uniformly distributed. This solution algorithm is based on Newton’s method and Lagrangian multipliers and solves the problem in unconstrained, constrained and tightly constraint cases. Examples are given to show not only can the MGPRA solve the problem under the constraint budget, but also it is able to solve the problem under tightly budget constraint, efficiently. In addition to the examples provided, the application of the multi-product gardener problem with reservation policy can obtain greater expected profit than the multi-product gardener problem without a reservation policy. 1. Introduction The single period problem (SPP) is one of the most important pro- blems in today’s complex inventory management environments. Single- period commodities, such as newspapers, milk, magazines, flowers, Christmas trees, pop CDs, and so on, are common in daily life. Depending on the type of products the selling period can vary from one day to one year, the expiration dates cause the products to be salvaged or thrown away at the end of the period, and hence these products are usually expensive/cheap at the beginning/end of the selling period. Therefore, it is an important task for decision makers to determine the order quantity at the beginning of the period to maximize the total profit. The classic single-period problem with random demand com- monly referred to as the newsboy or newsvendor problem plays a central role at the conceptual foundations of stochastic inventory theory with vast application in revenue management and supply chain management. In order to connect the current work and to provide the necessity of performing this research in continuation to the literature of the single- period problem, the readers are referred to two comprehensive reviews of the single-period-problem performed by Gallego and Moon (1993) and Khouja (1999), where the latter suggested eleven categories of needed contributions (extensions). In addition, many researchers in- vestigated several stochastic multi-product newsboy problems under different constraints. For instance, Hadley and Whitin (1963) presented a multi-product problem with a single budget constraint. Nahmias and Schmidt (1984) proposed a multiproduct newsboy problem with sto- chastic demands subject to a linear and deterministic constraint on space or budget. Lau and Lau (1995) extended the newsboy problem to handle general demand distributions and provided an efficient solution for a multi-product multi-constraint newsboy problem. Erlebacher (2000) studied a multi-product newsboy problem with one capacity constraint. He proved the optimality of the order quantities for two special cases and proposed heuristics for a few specific probability distribution functions. Vairaktarakis (2000) developed some minimax regret formulations for singly constrained multi-product newsboy pro- blem. Moon and Silver (2000) modeled fixed ordering costs in the multi-product newsvendor problem with a budget constraint and https://doi.org/10.1016/j.cie.2018.06.021 Received 12 December 2016; Received in revised form 5 May 2018; Accepted 15 June 2018 ⁎ Corresponding author. E-mail addresses: sara.68sadr@gmail.com (S.S. Sadralsharifi), shr_pasandideh@khu.ac.ir (S.H.R. Pasandideh), niaki@sharif.edu (S.T.A. Niaki), nahavandian@aut.ac.ir (M.H. Nahavandian). Computers & Industrial Engineering 123 (2018) 82–102 Available online 18 June 2018 0360-8352/ © 2018 Elsevier Ltd. All rights reserved. T