Maria Grazia Speranza

Heuristic algorithms for the portfolio selection problem with minimum transaction lots

The problem of selecting a portfolio has been largely faced in terms of minimizing the risk, given the return. While the complexity of the quadratic programming model due to Markowitz has been overcome by the recent progress in algorithmic research, the introduction of linear risk functions has given rise to the interest in solving portfolio selection problems with real constraints. In this paper we deal with the portfolio problem with minimum transaction lots. We show that in this case the problem of finding a feasible solution is, independently of the risk function, NP-complete. Moreover, given the mixed integer linear model, new heuristics are proposed which starting from the solution of the relaxed problem allow to find a solution close to the optimal one. The algorithms are based on the construction of mixed integer subproblems (using only a part of the securities available) formulated using the information obtained from the solution of the relaxed problem. The heuristics have been tested with respect to two disjoint time periods, using real data from the Milan Stock Exchange.

Minimization of logistic costs with given frequencies

We study the problem of shipping products from one origin to several destinations, when a given set of possible shipping frequencies is available. The objective of the problem is the minimization of the transportation and inventory costs. We present different heuristic algorithms and test them on a set of randomly generated problem instances. The heuristics are based upon the idea of solving, in a first phase, single link problems, and of locally improving the solution in subsequent phases.

A 13/12 approximation algorithm for bin packing with extendable bins

Abstract A set of items has to be assigned to a set of bins with size one. If necessary, the size of the bins can be extended. The objective is to minimize the total size, i.e., the sum of the sizes of the bins. The Longest Processing Time heuristic is applied to this NP-hard problem. For this approximation algorithm we prove a worst-case bound of 13 12 which is shown to be tight when the number of bins is even.

Inventory control on sequences of links with given transportation frequencies

This paper deals with the problem of minimizing the sum of the inventory and transportation costs in the multi-products logistic network with one origin, some intermediate nodes and one destination when a set of possible shipping frequencies is given. The problem is to determine for each link a periodic shipping strategy in order to minimize the total cost. We first propose a mixed integer linear programming model; then we present two more compact formulations of the problem obtained by computing the inventory cost through the aggregation of the inventory over time or over nodes. Finally, we present heuristic algorithms based either on the decomposition of the sequence or on the solution of a simpler problem through dynamic programming techniques.

Efficiency and effectiveness of normal schedules on three dedicated processors

Abstract A set of tasks has to be scheduled on three processors and each task requires that a set of the processors be available for a given processing time. The objective of the problem is to determine a nonpreemptive schedule with minimum makespan. The problem is known to be NP-hard in the strong sense. A normal schedule is such that all tasks requiring the same set of processors are scheduled consecutively. We show that, under a certain (uniform) probability distribution on the problem instances, in more than 95% of the instances the best normal schedule is optimal when the number of tasks grows to infinity. For the hard cases it is shown that the relative error produced by the best normal schedule is bounded by 5 4 . This result improves the bound of 4 3 known in the literature and the improved bound is shown to be tight.

Minimizing logistic costs in multistage supply chains

We consider the problem in which a set of products has to be shipped from a common origin to a common destination through one or several intermediate nodes with the objective of minimizing the sum of inventory and transportation costs when a set of possible shipping frequencies is given on each link. From the theoretical point of view, the main issue is the computation of the inventory cost in the intermediate nodes. From the computational point of view, given that the simpler single link problem is known to be NP-hard, we present four classes of heuristic algorithms. The first two classes are based on the decomposition of the sequence in links, the third class on the adaptation of the EOQ-type solution known for the continuous case, and the fourth on the optimal solution of a simpler problem through dynamic programming techniques. Finally, we compare them on a set of randomly generated problem instances.

Minimizing makespan in a multimode multiprocessor shop scheduling problem

We study the problem of multimode scheduling tasks on dedicated processors, with the objective of minimizing the maximum completion time. Each task can be undertaken in one among a set of predefined alternative modes, where each mode specifies a required set of dedicated processors and a processing time. At any time each processor can be used by a single task at most. General precedence constraints exist among tasks, and task preemption is not allowed. The problem consists of assigning a mode and a starting time to each task, respecting processor and precedence constraints, to minimize the time required to complete all tasks. The problem is NP-hard in several particular cases. In previous works, we studied algorithms in which a solution was obtained by means of an iterative procedure that combines mode assignment and sequencing phases separately. In this paper, we present some new heuristics where the decision on the mode assignment is taken on the basis of a partial schedule. Then, for each task, the mode selection and the starting time are chosen simultaneously considering the current processor usage. Different lower bounds are derived from a mathematical formulation of the problem and from a graph representation of a particular relaxed version of the problem. Heuristic solutions and lower bounds are evaluated on randomly generated test problems.

An approximation result for a duo-processor task scheduling problem

Abstract We consider the problem of scheduling tasks on a set of dedicated processors, where each task requires a subset of two processors be simultaneously available for a given processing time. The objective is to determine a nonpreemptive schedule with minimum completion time. By means of a graph theoretical formulation, we show that instances with up to 4 processors can be solved in polynomial time. With m = 2s + 1 processors (for s = 2, 3, …) and a minimum of m task types, we prove that the problem is NP-hard. Moreover, for this class of NP-hard instances, a simple O(m) approximation algorithm, whose performance ratio is bounded by 3s (2s + 1) , is given. The bound is shown to be tight.