Frits C. R. Spieksma

Minimizing the number of tool switches on a flexible machine

This article analyzes a tool switching problem arising in certain flexible manufacturing environments. A batch of jobs have to be successively processed on a single flexible machine. Each job requires a subset of tools, which have to be placed in the tool magazine of the machine before the job can be processed. The tool magazine has a limited capacity, and, in general, the number of tools needed to produce all the jobs exceeds this capacity. Hence, it is sometimes necessary to change tools between two jobs in a sequence. The problem is then to determine a job sequence and an associated sequence of loadings for the tool magazine, such that the total number of tool switches is minimized. This problem has been previously considered by several authors; it is here revisited, both from a theoretical and from a computational viewpoint. Basic results concerning the computational complexity of the problem are established. Several heuristics are proposed for its solution, and their performance is computationally assessed.

On the approximability of an interval scheduling problem

In this paper we consider a general interval scheduling problem. The problem is a natural generalization of finding a maximum independent set in an interval graph. We show that, unless =, this maximization problem cannot be approximated in polynomial time within arbitrarily good precision. On the other hand, we present a simple greedy algorithm that delivers a solution with a value of at least 1/2 times the value of an optimal solution. Finally, we investigate the quality of an LP-relaxation of a formulation for the problem, by establishing an upper bound on the ratio between the value of the LP-relaxation and the value of an optimal solution. Copyright

The assembly of printed circuit boards: A case with multiple machines and multiple board types

In this paper a typical situation arising in the assembly of printed circuit boards is investigated. The planning problem we face is how to assemble boards of different types using a single line of placement machines. From a practical viewpoint, the multiplicity of board types adds significantly to the complexity of the problem, which is already very hard to solve in the case of a single board type. In addition, relatively few studies deal with the multiple board type case. We propose a solution procedure based on a hierarchical decomposition of the planning problem. An important subproblem in this decomposition is the so-called feeder rack assignment problem. By taking into account as much as possible the individual board type characteristics (as well as the machine characteristics) we heuristically solve this problem. The remaining subproblems are solved using constructive heuristics and local search methods. The solution procedure is tested on real-life instances. It turns out that, in terms of the makespan, we can substantially improve the current solutions. Keywords: heuristics, PCB-assembly, feeder rack assignment problem.

Approximation algorithms for three-dimensional assignment problems with triangle inequalities

Consider the following classical formulation of the (axial) three-dimensional assignment problem (3DA) (see e.g. Balas and Saltzman (1989)). Given is a complete tripartite graph Kn,n, n = (I∪J∪K,(I × J) ∪ (I × K) ∪ (J × K)), where I, J, K are disjoint sets of size n, and a cost c ijk for each triangle (i, j, k) ∈ I × J × K. The problem 3DA is to find a subset A of n triangles, A⊆ I × J × K, such that every element of I ∪ J ∪ K occurs in exactly one triangle of A, and the total cost c(A) = ∑(i,j,k)∈Ac ijk is minimized. Some recent references to this problem are Balas and Saltzman (1989), Frieze (1974), Frieze and Yadegar (1981), Hansen and Kaufman (1973).

Exact algorithms for procurement problems under a total quantity discount structure

In this paper, we study the procurement problem faced by a buyer who needs to purchase a variety of goods from suppliers applying a so-called total quantity discount policy. This policy implies that every supplier announces a number of volume intervals and that the volume interval in which the total amount ordered lies determines the discount. Moreover, the discounted prices apply to all goods bought from the supplier, not only to those goods exceeding the volume threshold. We refer to this cost-minimization problem as the total quantity discount (TQD) problem. We give a mathematical formulation for this problem and argue that not only it is NP-hard, but also that there exists no polynomial-time approximation algorithm with a constant ratio (unless P = NP). Apart from the basic form of the TQD problem, we describe four variants. In a first variant, the market share that one or more suppliers can obtain is constrained. Another variant allows the buyer to procure more goods than strictly needed, in order to reach a lower total cost. We also consider a setting where the buyer needs to pay a disposal cost for the extra goods bought. In a third variant, the number of winning suppliers is limited, both in general and per product. Finally, we investigate a multi-period variant, where the buyer not only needs to decide what goods to buy from what supplier, but also when to do this, while considering the inventory costs. We show that the TQD problem and its variants can be solved by solving a series of min-cost flow problems. Finally, we investigate the performance of three exact algorithms (min-cost flow based branch-and-bound, linear programming based branch-and-bound, and branch-and-cut) on randomly generated instances involving 50 suppliers and 100 goods. It turns out that even the large instances of the basic problem are solved to optimality within a limited amount of time. However, we find that different algorithms perform best in terms of computation time for different variants.

Modeling and solving the periodic maintenance problem

We study the problem of scheduling maintenance services. Given is a set of mmachines and integral cost-coefficients ai and bi for each machine i (1

Approximation algorithms for multi-dimensional assignment problems with decomposable costs

Abstract The k-dimensional assignment problem with decomposable costs is formulated as follows. Given is a complete k-partite graph G = (X0 ∪ ⋯ ∪ Xk − 1, E), with |Xi| = p for each i, and a nonnegative length function defined on the edges of G. A clique of G is a subset of vertices meeting each Xi in exactly one vertex. The cost of a clique is a function of the lengths of the edges induced by the clique. Four specific cost functions are considered in this paper; namely, the cost of a clique is either the sum of the lengths of the edges induced by the clique (sum costs), or the minimum length of a spanning star (star costs) or of a traveling salesman tour (tour costs) or of a spanning tree (tree costs) of the induced subgraph. The problem is to find a minimum-cost partition of the vertex set of G into cliques. We propose several simple heuristics for this problem, and we derive worst-case bounds on the ratio between the cost of the solutions produced by these heuristics and the cost of an optimal solution. The worst-case bounds are stated in terms of two parameters, viz. k and τ, where the parameter τ indicates how close the edge length function comes to satisfying the triangle inequality.

An LP-based algorithm for the data association problem in multitarget tracking

In this work we present a linear programming (LP) based approach for solving the data association problem (DAP) in multiple target tracking. It is well-known that the DAP can be formulated as an integer program. We present a compact formulation of the DAP. To solve practical instances of the DAP we propose an algorithm that uses an iterated K-scan sliding window technique. In each iteration we solve the LP relaxation of an integer program and next apply a greedy rounding procedure. Computational experiments indicate that the quality of the solutions found is quite satisfactory.

Multi Index Assignment Problems: Complexity, Approximation, Applications

This chapter deals with approximation algorithms for and applications of multi index assignment problems (MIAPs). MIAPs and relatives of it have a relatively long history both in applications as well as in theoretical results, starting at least in the fifties (see e.g. [Motzkin, 1952], [Schell, 1955] and [Koopmans and Beckmann, 1957]). Here we intend to give the reader i) an idea of the range and diversity of practical problems that have been formulated as an MIAP, and ii) an overview on what is known on theoretical aspects of solving instances of MIAPs. In particular, we will discuss complexity and approximability issues for special cases of MIAPs. We feel that investigating special cases of MIAPs is an important topic since real-world instances almost always posses a certain structure that can be exploited when it comes to solving them.

A branch-and-bound algorithm for the two-dimensional vector packing problem

Scope and Purpose-Bin-packing problems arise when a number of non-overlapping objects have to be packed into a number of so-tailed bins, where a bin usually represents a fixed amount of space or time. Applications of bin-packing problems are quite diverse, and can be found in, for example, scheduling theory, VLSI-design, computer network design and in the field of cutting-stock problems. In this paper, a bin-packing problem is studied where each object has two requirements, and the objective is to find the minimum number of unit-capacity bins. In geometric terms: given a set of rectangles (where the length and width of each rectangle represent the two requirements (or dimensions) of an object), pack them in as few unit squares (the bins) as possible, such that the rectangles are placed corner to corner, in a diagonal fashion. The main cont~butions of this paper are the description of a new heuristic for this problem, and the presentation of two types of lower bounds. These bounds are used by a branch-and-bound algorithm, whose performance is tested on randomly generated instances.