A Benders decomposition approach to product location in carousel storage systems

In this paper we discuss the problem of locating items within carousel bins in order to minimize the average carousel rotational distance per retrieval. We consider two cases: (1) a single two-dimensional carousel and (2) a group of two one-dimensional carousels. The corresponding problems are formulated as mixed-integer programs. In the second problem we apply concepts from Markovian performance evaluation methods to write the objective functions in a simple linear form. We also define a set of uniqueness constraints that significantly reduces the size of the solution feasibility space. We then apply Benders decomposition algorithm to solve both problems. We develop a closed form solution for the dual subprobem and present numerical results to show the efficiency of the proposed solution methodology.

[1]  Graham K. Rand,et al.  Decision Systems for Inventory Management and Production Planning , 1979 .

[2]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[3]  H. Sherali,et al.  Benders' partitioning scheme applied to a new formulation of the quadratic assignment problem , 1980 .

[4]  Stanley E. Griffis,et al.  Failure to deliver? Linking online order fulfillment glitches with future purchase behavior , 2011 .

[5]  G. Pólya,et al.  Inequalities (Cambridge Mathematical Library) , 1934 .

[6]  A. M. Geoffrion Generalized Benders decomposition , 1972 .

[7]  J. F. Benders Partitioning procedures for solving mixed-variables programming problems , 1962 .

[8]  Arie Segev,et al.  Optimal Arrangements of Cartridges in Carousel Type Mass Storage Systems , 1994, Comput. J..

[9]  Elkafi Hassini,et al.  One-Dimensional Carousel Storage Problems: Applications, Review and Generalizations , 2009, INFOR Inf. Syst. Oper. Res..

[10]  Patrick P. Bergmans Minimizing Expected Travel Time on Geometrical Patterns by Optimal Probability Rearrangements , 1972, Inf. Control..

[11]  L. Kaufman,et al.  An algorithm for the quadratic assignment problem using Bender's decomposition , 1978 .

[12]  Elkafi Hassini,et al.  A two-carousel storage location problem , 2003, Comput. Oper. Res..

[13]  Robert Raeside,et al.  Zone shapes in class based storage and multicommand order picking when storage/retrieval machines are used , 1992 .

[14]  Chak-Kuen Wong,et al.  Algorithmic Studies in Mass Storage Systems , 1983, Springer Berlin Heidelberg.

[15]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[16]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[17]  Francesco Maffioli,et al.  Note on Combinatorial Optimization with Max-Linear Objective Functions , 1993, Discret. Appl. Math..

[18]  Raymond G. Vickson,et al.  Optimal storage locations in a carousel storage and retrieval system , 1996 .

[19]  Golgen Bengu An optimal storage assignment for automated rotating carousels , 1995 .

[20]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[21]  Elkafi Hassini Storage space allocation to maximize inter-replenishment times , 2008, Comput. Oper. Res..

[22]  H. P. Williams,et al.  Model Building in Mathematical Programming , 1979 .

[23]  R. Durrett Essentials of Stochastic Processes , 1999 .