Survivable networks,linear programming relaxations and the parsimonious property

We consider the survivable network design problem — the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and thek-edge-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worst-case analyses of two heuristics for the survivable network design problem.The research of both authors was partially supported by the National Science Foundation under grant ECS-8717970 and the Leaders for manufacturing program at MIT.     

A probabilistic analysis of the set covering problem

A probabilistic analysis of the minimum cardinality set covering problem (SCP) is developed, considering a stochastic model of the (SCP), withn variables andm constraints, in which the entries of the corresponding (m, n) incidence matrix are independent Bernoulli distributed random variables, each with constant probabilityp of success. The behaviour of the optimal solution of the (SCP) is then investigated as bothm andn grow asymptotically large, assuming either an incremental model for the evolution of the matrix (for each size, the matrixA is obtained bordering a matrix of smaller size by new columns and rows) or an independent one (for each size, an entirely new set of entries forA are considered). Two functions ofm are identified, which represent a lower and an upper bound onn in order the (SCP) to be a.e. feasible and not trivial. Then, forn lying within these bounds, an asymptotic formula for the optimum value of the (SCP) is derived and shown to hold a.e.The performance of two simple randomized algorithms is then analyzed. It is shown that one of them produces a solution value whose ratio to the optimum value asymptotically approaches 1 a.e. in the incremental model, but not in the independent one, in which case the ratio is proved to be tightly bounded by 2 a.e. Thus, in order to improve the above result, a second randomized algorithm is proposed, for which it is proved that the ratio between the approximate solution value and the optimum approaches 1 a.e. also in the independent model.     

Colouring random graphs

We discuss some results concerned with the behaviour of colouring algorithms on large random graphs.     

An introduction to timetabling

A huge variety of timetabling models have been described in the OR literature; they range from the weekly timetable of a school to the scheduling of courses or exams in a university. Graphs and networks have proven to be useful in the formulation and solution of such problems. Various models will be described with an emphasis on graph theoretical models.     

Performance Analysis of Cyclical Simulated Annealing Algorithms

Generalized hill climbing (GHC) algorithms provide a framework for modeling local search algorithms for addressing intractable discrete optimization problems. Measures for assessing the finite-time performance of GHC algorithms have been developed using this framework, including the expected number of iterations to visit a predetermined objective function value level. This paper analyzes how the expected number of iterations to visit a predetermined objective function value level can be estimated for cyclical simulated annealing. Cyclical simulated annealing uses a cooling schedule that cycles through a set of temperature values. Computational results with traveling salesman problem instances taken from TSPLIB show how the expected number of iterations to visit solutions with predetermined objective function levels can be estimated for cyclical simulated annealing.AMS 2000 Subject Classification 90-08 Computational Methods: Local Search, 90C59 Heuristics: Simulated Annealing     

An effective solution for a real cutting stock problem in manufacturing plastic rolls

We confront a practical cutting stock problem from a production plant of plastic rolls. The problem is a variant of the well-known one dimensional cutting stock, with particular constraints and optimization criteria defined by the experts of the company. We start by giving a problem formulation in which optimization criteria have been considered in linear hierarchy according to expert preferences, and then propose a heuristic solution based on a GRASP algorithm. The generation phase of this algorithm solves a simplified version which is rather similar to the conventional one dimensional cutting stock. To do that, we propose a Sequential Heuristic Randomized Procedure (SHRP). Then in the repairing phase, the solution of the simplified problem is transformed into a solution to the real problem. For experimental study we have chosen a set of problem instances of com-mon use to compare SHRP with another recent approach. Also, we show by means of examples, how our approach works over instances taken from the real production process. All authors are supported by MEC-FEDER Grant TIN2007-67466-C02-01 and by contract CN-05-127 of the University of Oviedo and the company ERVISA, and by FICYT under grant BP04-021.     

Interactive decision software and computer graphics for architectural and space planning

We describe the development and successful implementation of a decision support system now being used by several leading firms in the architecture and space planning industries. The system, which we call SPDS (spatial programming design system) has the following characteristics: (i) user-friendly convenience features permitting architects and space planners to operate the system without being experienced programmers; (ii) interactive capabilities allowing the user to control and to manipulate relevant parameters, orchestrating conditions to which his or her intuition provides valuable input; (iii) informative and understandable graphics, providing visual displays of interconnections that the computer itself treats in a more abstract methematical form; (iv) convenient ways to change configurations, and to carry out ‘what if’ analyses calling on the system’s decision support capabilities; (v) a collection of new methods, invisible to the user, capable of generating good solutions to the mathematical programming problems that underlie each major design component. These new methods succeed in generating high quality solutions to a collection of complex discrete, highly nonlinear problems. While these problems could only be solved in hours, or not at all, with previously existing software, the new methods obtain answers in seconds to minutes on a minicomputer. Major users, including Dalton, Dalton, Newport, and Marshal Erdwin, report numerous advantages of the system over traditional architectural design methods.