The gardener problem with reservation policy and discount

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 included 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]  L. Abdel-Malek,et al.  Separable programming/duality approach to solving the multi-product Newsboy/Gardener Problem with linear constraints , 2013 .

[2]  Hau L. Lee,et al.  Production Control in Multistage Systems with Variable Yield Losses , 1988, Oper. Res..

[3]  G. Gallego,et al.  The Distribution Free Newsboy Problem: Review and Extensions , 1993 .

[4]  K. Sadeghy,et al.  Stability of Thixotropic Fluids in Pipe Flow , 2017 .

[5]  Liang-Hsuan Chen,et al.  A newsboy problem with a simple reservation arrangement , 2009, Comput. Ind. Eng..

[6]  Wallace J. Hopp,et al.  A simple linear heuristic for the service constrained random yield problem , 2002 .

[7]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[8]  Steven J. Erlebacher.  OPTIMAL AND HEURISTIC SOLUTIONS FOR THE MULTI‐ITEM NEWSVENDOR PROBLEM WITH A SINGLE CAPACITY CONSTRAINT , 2000 .

[9]  Athanassios S. Fokas,et al.  A semi-analytical numerical method for solving evolution and elliptic partial differential equations , 2009 .

[10]  Ali R. Ansari,et al.  A semi-analytical iterative technique for solving nonlinear problems , 2011, Comput. Math. Appl..

[11]  Hau L. Lee,et al.  Lot Sizing with Random Yields: A Review , 1995, Oper. Res..

[12]  C. P. Schmidt,et al.  An efficient heuristic for the multi‐item newsboy problem with a single constraint , 1984 .

[13]  Yigal Gerchak,et al.  The Structure of Periodic Review Policies in the Presence of Random Yield , 1990, Oper. Res..

[14]  Liang-Hsuan Chen,et al.  A multiple-item budget-constraint newsboy problem with a reservation policy , 2010 .

[15]  M. Khouja.  The single-period (news-vendor) problem: literature review and suggestions for future research , 1999 .

[16]  W. Shih.  Optimal inventory policies when stockouts result from defective products , 1980 .

[17]  Layek Abdel-Malek,et al.  An analysis of the multi-product newsboy problem with a budget constraint , 2005 .

[18]  G. Baranenkov,et al.  Problems in mathematical analysis , 1964 .

[19]  Abdul Raouf,et al.  On the Constrained Multi‐item Single‐period Inventory Problem , 1993 .

[20]  Gerald Keller,et al.  One-Period Order Quantity Strategy With Uncertain Match Between The Amount Received And Quantity Requisitioned , 1986 .

[21]  A. Lau,et al.  The newsstand problem: A capacitated multiple-product single-period inventory problem , 1996 .

[22]  Yehuda Bassok,et al.  Random Yield and Random Demand in a Production System with Downward Substitution , 1999, Oper. Res..

[23]  Larry R. Taube,et al.  An inventory model with random replenishment quantities. , 2010 .

[24]  Phillip E. Pfeifer,et al.  The economic value of using advance booking of orders , 1994 .

[25]  Fikri Karaesmen,et al.  Multi-product newsvendor problem with value-at-risk considerations , 2009 .

[26]  Layek Abdel-Malek,et al.  The capacitated newsboy problem with random yield: The Gardener Problem , 2008 .

[27]  Steven M. Shugan,et al.  Advance Pricing of Services and Other Implications of Separating Purchase and Consumption , 2000 .

[28]  Edward A. Silver,et al.  The multi-item newsvendor problem with a budget constraint and fixed ordering costs , 2000, J. Oper. Res. Soc..

[29]  Moutaz Khouja,et al.  Linking advertising and quantity decisions in the single-period inventory model , 2003 .

[30]  Yves Dallery,et al.  A comprehensive analysis of the Newsvendor model with unreliable supply , 2007, OR Spectr..

[31]  J. J. D. Dana,et al.  Advance-Purchase Discounts and Price Discrimination in Competitive Markets , 1998 .

[32]  Layek Abdel-Malek,et al.  Exact, approximate, and generic iterative models for the multi-product Newsboy problem with budget constraint , 2004 .

[33]  George L. Vairaktarakis,et al.  Robust multi-item newsboy models with a budget constraint , 2000 .

[34]  A. Lau,et al.  The multi-product multi-constraint newsboy problem: Applications, formulation and solution , 1995 .