A note on “a simple heuristic for maximizing service of carousel storage”

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Abstract

We consider the same carousel storage system as that given by Jacobs et al. We propose a different but simpler approach for the solution, in which the solution is obtained by solving a linear equation. We also propose a simple heuristic that appears to give a feasible solution with more accuracy compared to that of Jacobs et al. Only linear computation time is needed.

Scope and purpose

A carousel storage system is loaded with cases of items of different types. Items are retrieved in groups, in which a group is a certain number of items of each type. The objective is to maximize service of the carousel storage system in such a way that as many groups can be retrieved as possible without running out of items of any type. In this study we propose a different approach that is simpler than the heuristic proposed by Jacobs et al., and our approach also gives a feasible solution with more accuracy.

Introduction

In this study we give a different but simpler approach for the solution of the carousel storage problem studied by Jacobs et al. [1]. We adopt their notation in the problem description. In their paper Jacobs et al. consider the following carousel storage system: There are N fixed-size cases of items in the storage system, and there are j different types of items, but within each case, the items are identical. The storage system is to be stocked with full cases, each containing ci items of type i,1⩽i⩽j. Items will be removed from the system in groups, each consisting of exactly ni items of type i. The problem is to maximize the number of groups that can be removed without encountering any stock outages, by determining appropriate values for mi, the number of type i cases that should be loaded in the storage system. The problem is formally stated as follows:

Given positive integers c1,…,cj,n1,…,nj, and N, find the largest integer g, such that for some nonnegative integers mi we haveg⩽c1m1n1,…,g⩽cjmjnjandi=1jmi=N.

As stated in their paper, when g and mi are not required to be integers, Jacobs et al. obtain the following relaxed ordinary linear programming problem (LP1):

(LP1) Given positive integers c1,…,cj,n1,…,nj, and N, find the largest real number g′, such that for some nonnegative real numbers xiwe haveg′⩽c1x1n1,…,g′⩽cjxjnjandi=1jxi′=N,xi′,g′∈R.

Jacobs et al. also point out that, by assuming that equality is achieved, the following linear programming problem (LP2) has the same solution as that of the relaxed problem (LP1):

(LP2) Given positive integers c1,…,cj,n1,…,nj, and N, find the largest real number g″, such that for some nonnegative real numbers xiwe haveg″=c1x1n1,…,g″=cjxjnjandi=1jxi″=N,xi″,g″∈R.

The heuristic proposed by Jacobs et al. is based on solving the above linear programming problem (LP2).

Section snippets

Heuristic

In this study we solve the linear programming problem (LP2) by a different but simpler approach as follows: As shown in the above problem, since g″=cixi″/ni,1⩽i⩽j, we may rewrite each xi″ in terms of g″ as xi″=wig″,1⩽i⩽j, where wi=ni/ci. Thus, by substituting wig″ for xi″ into equation ∑i=1jxi″=N, we obtain the following simple linear equation:i=1jxi″=g″i=1jwi=N.Hence, we have g″=N/∑i=1jwi, and then solve for xi″ to getxi″=wig″=wil=1jwlN,1⩽i⩽j.Note that the above Eq. (2) gives the same

Illustrated example

We illustrate our heuristic by solving the following problem given by Jacobs et al.: Consider a military clothing issue facility having a carousel storage system capable of holding N=1352 cases. Each new recruit is given 1 jacket (n1=1), 3 T-shirts (n2=3), 1 pair of boots (n3=1), 1 pair of pants (n4=1), 6 pairs of socks (n5=6), 4 pairs of shorts (n6=4). These items come in cases of 6, 30, 6, 6, 120, 120, respectively (i.e. (c1,c2,c3,c4,c5,c6)=(6,30,6,6,120,120)). The problem is to maximize the

Conclusion

In this study we propose a simpler approach using iterative heuristic for the solution of the carousel storage problem considered by Jacobs et al. Our heuristic also takes linear computation time, as observed by Jacobs et al. However, it appears to give a feasible solution with more accuracy than the heuristic proposed by Jacobs et al. The error of the feasible solution is more tightly bounded, and the efficiency of searching for the optimal solution is improved.

Din-Horng Yeh has a Ph.D. from Georgia Institute of Technology, and M.S. and B.S. from National Tsinghua University in Taiwan. He is an associate professor in the Department of Industrial Management at Aletheia University in Taiwan. His main research interest is applied operations research.

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Din-Horng Yeh has a Ph.D. from Georgia Institute of Technology, and M.S. and B.S. from National Tsinghua University in Taiwan. He is an associate professor in the Department of Industrial Management at Aletheia University in Taiwan. His main research interest is applied operations research.

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