The capacitated newsboy problem with random yield: The Gardener Problem Layek Abdel-Malek a,Ã , Roberto Montanari b , Diego Meneghetti b a Department of Industrial and Management Systems Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA b Dipartimento di Ingegneria Industriale, Universita ` degli Studi di Parma, Viale delle Scienze, 43100 Parma, Italy article info Article history: Received 10 December 2006 Accepted 12 May 2008 Available online 24 May 2008 Keywords: Newsboy Inventory control Perishable commodity Supply chain Optimization abstract This publication is the third in a series of articles by Abdel-Malek et al. [2004. Exact, approximate, and generic iterative models for the multi-product newsboy with budget constraint. International Journal of Production Economics 91, 189–198] and Abdel-Malek and Montanari [2005. An analysis of multi-product newsboy problem with a budget constraint. International Journal of Production Economics 97, 296–307], which appeared in IJPE addressing issues regarding the newsboy models. In this paper, models are developed to extend the existing ones to cover random yield scenarios. We designate this type of models as the Gardener Problem. The models are based on the application of Lagrange multipliers, Leibniz’s rule and Newton’s method to obtain the optimum solution for the considered random yield and probabilistic demand situations. The developed methodologies are applicable to general probability distribution functions. Examples are given to show the applicability of the developed approach to different probability distributions for both the supply and the demand as well as for interdependent yield structures. & 2008 Elsevier B.V. All rights reserved. 1. Introduction In today’s global economy, we have become increas- ingly dependent on complex and intricate supply chains in the various industrial sectors. A key factor in supply chain management is inventory control. The newsboy model introduced by Hadley and Whitin has proven to be one of the most important tools in managing inventories in the present environment, see Denardo (2001). It is known that the newsboy dilemma is in ordering the optimum number of copies of a newspaper or a collection of them for the next day’s distribution in order to minimize cost or maximize revenue. In some instances, the newsboy is bound by some constraints such as budget, and space. In Hadley and Whitin’s (1963) models as well as in most extensions that followed, it is assumed that the newsboy gets the exact amounts ordered. Nevertheless, in other situations that is not the case. More clearly, the amounts received may vary from those ordered. Also, as in the original newsboy model the demand for the product is random. We name this extension as the Gardener Problem. We consider a gardener who would like to divide available acreage among a number of possible crops for the coming season. The demand for each of the crops is random and the yield depends on the weather conditions of the season. The gardener’s objective is to optimize his planting strategy and divide the available land (resource) among a list of possible crops. There are many similar situations in practice to that of the gardener. Among them are, these encountered in supply chains, particularly those of global nature where the buyers may opt for e-bidding as opposed to long-term partnerships. In the former situation, the buyer does not have knowledge of the pro- duct’s quality that is received compared with the latter when the buyer knows with some certainty the percentage Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijpe Int. J. Production Economics ARTICLE IN PRESS 0925-5273/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2008.05.003 Ã Corresponding author. Tel.: +1973 596 3648; fax: +1973 596 3652. E-mail address: malek@njit.edu (L. Abdel-Malek). Int. J. Production Economics 115 (2008) 113–127