Although the picking of items may make up as much as 60% of all labor activities in a warehouse and may account for as much as 65% of all operating expenses, many order picking problems are still not well understood. Indeed, usually simple rules of thumb or straightforward constructive heuristics are used in practice, even in state-of-the-art warehouse management systems, however, it might well be that more attractive algorithmic alternatives could be developed. We address one such a fundamental materials handling problem: the batching of orders in a parallel-aisle warehouse so as to minimize the total traveling time needed to pick all items. Many heuristics have been proposed for this problem, however, a fundamental analysis of the problem is still lacking. In this paper, we first address the computational complexity of the problem. We prove that this problem is NP-hard in the strong sense but that it is solvable in polynomial time if no batch contains more than two orders. This result is not really surprising but it justifies the development of approximation and/or enumerative optimization algorithms for the general case. Our primary goal is to develop a branch-and-price optimization algorithm for the problem. To this end, we model the problem as a generalized set partitioning problem and present a column generation algorithm to solve its linear programming relaxation. Furthermore, we develop a new approximation algorithm for the problem. Finally, we test the performance of the branch-and-price algorithm and the approximation algorithm on a comprehensive set of instances. The computational experiments show the compelling performance of both algorithms.
A set of n jobs has to be scheduled on a single machine which can handle only one job at a time. Each job requires a given positive uninterrupted processing time and has a positive weight. The problem is to find a schedule that minimizes the sum of weighted deviations of the job completion times from a given common due date d, which is smaller than the sum of the processing times. We prove that this problem is NP-hard even if all job weights are equal. In addition, we present a pseudopolynomial algorithm that requires O(n2d) time and O(nd) space.
The just-in-time concept in manufacturing has aroused interest in machine scheduling problems with earliness-tardiness penalties. In particular, common due date problems, which are structurally less complicated than problems with general due dates, have emerged as an interesting and proli c eld of research. We prove that so-called almost common due date problems, in which the due date d j and processing time p j of each j o b J j (j = 1 : : : n ) are such that d j 2 D D+ p j ] for some constant D, are structurally less complicated also. Our main contribution is an O(n 2 ) t i m e dynamic programming algorithm for the almost common due date problem with large D. The dynamic programming algorithm is interesting in its own right, since the optimality principle behind it applies to other common due date and almost common due date problems as well. 1980 Mathematics Subject Classi cation (1991): 9 0 B 3 5 .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.