Combinatorial optimization tolerances calculated in linear time

For a given optimal solution to a combinatorial optimization problem, we show, under very natural conditions, the equality of the minimal values of upper and lower tolerances, where the upper tolerances are calculated for the given optimal solution and the lower tolerances outside the optimal solution. As a consequence, the calculation of such tolerances can now be done in linear time, while all current methods use quadratic time.

Thecomposition of semi finished inventories at a solid board plant

A solid board factory produces rectangular sheets of cardboard in two different formats, namely large formats and small formats. The production process consists of two stages separated by an inventory point. In the first stage a cardboard machine produces the large formats. In the second stage a part of the large formats is cut into small formats by a separate rotary cut machine. Due to very large setup times, technical restrictions, and trim losses, the cardboard machine is not able to produce these small formats. The company follows two policies to satisfy customer demands for rotary cut format orders. When the company applies the first policy, then for each customer order an ‘optimal’ large format (with respect to trim loss) is determined and produced on the cardboard machine. In case of the second policy, a stock of a restricted number of large formats is determined in such a way that the expected trim loss is minimal. The rotary cut format order then uses the most suitable standard large format from the stock. Currently, the dimensions of the standard large formats in the semi finished inventory are based on intuitive motives, with an accent on minimizing trim losses. From the trim loss perspective it is most efficient to produce each rotary cut format from a specific large format. On the other hand, if there is only one large format in each caliper, the variety is minimal, but the trim loss might be inacceptably high. On average, the first policy results in a lower trim loss. In order to make efficiently use of the two machines and to meet customer’s due times the company applies both policies. In this paper we concentrate on the second policy, taking into account the various objectives and restrictions of the company. The purpose of the company is to have not too many different types of large formats and an acceptable amount of trim loss. The problem is formulated as a minimum clique covering problem with alternatives (MCCA), which is presumed to be NP-hard. We solve the problem by using an appropriate heuristic, which is built into a decision support system. Based on a set of real data, the actual composition of semi finished inventories is determined. The paper concludes with computational experiments.

Balinski-Tucker simplex tableaus : dimensions, degeneracy degrees, and interior points of optimal faces

This paper introduces a general, formal treatment of dynamic constraints, i.e., constraints on the state changes that are allowed in a given state space. Such dynamic constraints can be seen as representations of "real world" constraints in a managerial context. The notions of transition, reversible and irreversible transition, and transition relation will be introduced. The link with Kripke models (for modal logics) is also made explicit. Several (subtle) examples of dynamic constraints will be given. Some important classes of dynamic constraints in a database context will be identified, e.g. various forms of cumulativity, non-decreasing values, constraints on initial and final values, life cycles, changing life cycles, and transition and constant dependencies. Several properties of these dependencies will be treated. For instance, it turns out that functional dependencies can be considered as "degenerated" transition dependencies. Also, the distinction between primary keys and alternate keys is reexamined, from a dynamic point of view.

Degeneracy degrees of constraint collections

New product development is one of the most powerful but difficult activities in business. It is also a very important factor affecting final product quality. There are many techniques available for new product development. Experimental design is now regarded as one of the most significant techniques. In this article, we will discuss how to use the technique of experimental design in developing a new product - an extrusion press. In order to provide a better understanding of this specific process, a brief description of the extrusion press is presented. To ensure the successful development of the extrusion press, customer requirements and expectations were obtained by detailed market research. The critical and non-critical factors affecting the performance of the extrusion press were identified in preliminary experiments. Through conducting single factorial experiments, the critical factorial levels were determined. The relationships between the performance indexes of the extrusion press and the four critical factors were determined on the basis of multi-factorial experiments. The mathematical models for the performance of the extrusion press were established according to a central composite rotatable design. The best combination of the four critical factors and the optimum performance indexes were determined by optimum design. The results were verified by conducting a confirmatory experiment. Finally, a number of conclusions became evident.

Branch and peg algorithms for the simple plant location problem

The simple plant location problem is a well-studied problem in combinatorial optimization. It is one of deciding where to locate a set of plants so that a set of clients can be supplied by them at the minimum cost. This problem of ten appears as a subproblem in other combinatorial problems. Several branch and bound techniques have been developed to solve these problems. In this paper we present a few techniques that enhance the performance of branch and bound algorithms. The new algorithms thus obtained are called branch and peg algorithms, where pegging refers to assigning values to variables outside the branching process. We present exhaustive computational experiments which show that the new algorithms generate less than 60% of the number of subproblems generated by branch and bound algorithms, and in certain cases require less than 10% of the execution times required by branch and bound algorithms.

Data Correcting Algorithms in Combinatorial Optimization

This paper describes data correcting algorithms. It provides the theory behind the algorithms and presents the implementation details and computational experience with these algorithms on the asymmetric traveling salesperson problem, the problem of maximizing submodular functions, and the simple plant location problem.

The data-correcting algorithm for supermodular functions, with applications to quadratic cost partition and simple plant location problems

New product development is one of the most powerful but difficult activities in business. It is also a very important factor affecting final product quality. There are many techniques available for new product development. Experimental design is now regarded as one of the most significant techniques. In this article, we will discuss how to use the technique of experimental design in developing a new product - an extrusion press. In order to provide a better understanding of this specific process, a brief description of the extrusion press is presented. To ensure the successful development of the extrusion press, customer requirements and expectations were obtained by detailed market research. The critical and non-critical factors affecting the performance of the extrusion press were identified in preliminary experiments. Through conducting single factorial experiments, the critical factorial levels were determined. The relationships between the performance indexes of the extrusion press and the four critical factors were determined on the basis of multi-factorial experiments. The mathematical models for the performance of the extrusion press were established according to a central composite rotatable design. The best combination of the four critical factors and the optimum performance indexes were determined by optimum design. The results were verified by conducting a confirmatory experiment. Finally, a number of conclusions became evident.

Iterative Patching and the Asymmetric Traveling Salesman Problem

Although Branch and Bound (BnB) methods are among the most widely used techniques for solving hard problems, it is still a challenge to make these methods smarter. In this paper, we investigate iterative patching, a technique in which a fixed patching procedure is applied at each node of the BnB search tree for the Asymmetric Traveling Salesman Problem. Computational experiments show that iterative patching results in general in search trees that are smaller than the usual classical BnB trees, and that solution times are lower for usual random and sparse instances. Furthermore, it turns out that, on average, iterative patching with the Contract-or-Patch procedure of Glover, Gutin, Yeo and Zverovich (2001) and the Karp-Steele procedure are the fastest, and that ?iterative? Modified Karp-Steele patching generates the smallest search trees.