van de Sl Steef Velde

Order batching to minimize total travel time in a parallel-aisle warehouse

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.

Complexity of scheduling multiprocessor tasks with prespecified processor allocations

We investigate the computational complexity of scheduling multiprocessor tasks with prespecified processor allocations. We consider two criteria: minimizing schedule length and minimizing the sum of the task completion times. In addition, we investigate the complexity of problems when precedence constraints or release dates are involved.

Parallel Machine Scheduling by Column Generation

Parallel machine scheduling problems concern the scheduling of n jobs on m machines to minimize some function of the job completion times. If preemption is not allowed, then most problems are not only NP-hard, but also very hard from a practical point of view. In this paper, we show that strong and fast linear programming lower bounds can be computed for an important class of machine scheduling problems with additive objective functions. Characteristic of these problems is that on each machine the order of the jobs in the relevant part of the schedule is obtained through some priority rule. To that end, we formulate these parallel machine scheduling problems as a set covering problem with an exponential number of binary variables, n covering constraints, and a single side constraint. We show that the linear programming relaxation can be solved efficiently by column generation because the pricing problem is solvable in pseudo-polynomial time. We display this approach on the problem of minimizing total weighted completion time on m identical machines. Our computational results show that the lower bound is singularly strong and that the outcome of the linear program is often integral. Moreover, they show that our branch-and-bound algorithm that uses the linear programming lower bound outperforms the previously best algorithm.

Scheduling around a small common due date

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.

Minimizing total completion time and maximum cost simultaneously is solvable in polynomial time

We prove that the bicriteria single-machine scheduling problem of minimizing total completion time and maximum cost simultaneously is solvable in polynomial time. Our result settles a long-standing open problem.

Stronger Lagrangian bounds by use of slack variables: applications to machine scheduling problems

Lagrangian relaxation is a powerful bounding technique that has been applied successfully to manyNP-hard combinatorial optimization problems. The basic idea is to see anNP-hard problem as an “easy-to-solve” problem complicated by a number of “nasty” side constraints. We show that reformulating nasty inequality constraints as equalities by using slack variables leads to stronger lower bounds. The trick is widely applicable, but we focus on a broad class of machine scheduling problems for which it is particularly useful. We provide promising computational results for three problems belonging to this class for which Lagrangian bounds have appeared in the literature: the single-machine problem of minimizing total weighted completion time subject to precedence constraints, the two-machine flow-shop problem of minimizing total completion time, and the single-machine problem of minimizing total weighted tardiness.

Positioning automated guided vehicles in a loop layout

We address the problem of determining the home positions for m automated guided vehicles (AGVs) in a loop layout where n pickup points are positioned along the circumference (m<n). A home position is the location where idle AGVs are held until they are assigned to the next transportation task. The home positions need to be selected so as to minimize an objective function of the response times, where the response time for a pickup point is defined as the travel time to the pickup point from the nearest home location. For the unidirectional flow system, where all AGVs can move in one direction only, we first point out that the problem of minimizing an arbitrary regular cost function can quite straightforwardly be solved in O(n2) time if m=1 and in O(mnm) time if m2, which is polynomial for a fixed number m of AGVs. For m3, we can do better, however: we derive a generic O(mn3) time and O(mn) space dynamic programming algorithm for minimizing any regular function of the response times. For minimizing maximum response time a further gain in efficiency is possible: this problem can be solved in O(n2) time if m=2 and O(n2 logn) time if m3. Our results improve on earlier published work, where it was suggested that problems with m2 are NP-hard. For the bidirectional flow system, where the AGVs can move in both directions, the problem of determining the home locations is inherently much more difficult. Important objective functions like average response time and maximum response time can nonetheless still be minimized by the same types of algorithms and in the same amount of time as their unidirectional counterparts, once restrictive conditions apply such that the case m=1 can be solved in polynomial time. One such restrictive condition is that each AGV travels at constant speed.