Lexicographic balanced optimization problems

We introduce the lexicographic balanced optimization problem (LBaOP) and show that it can be solved efficiently if an associated lexicographic bottleneck problem can be solved efficiently. For special cases of cuts in a graph and base system of a matroid, improved algorithms are proposed. A generalization of LBaOP is also discussed.     

One-parametric bottleneck transportation problems

We discuss the bottleneck transportation problem with one nonlinear parameter in the bottleneck objective function. A finite sequence of feasible basic solutions which are optimal in subsequent closed parameter-intervals is generated using a primal method for the nonparametric subproblems. The best among three primal codes for solving these subproblems is selected on extensive computational comparisons. We discuss computational experience with the sequential method for the case of linear and quadratic dependence on one parameter. Observed computational behaviour is O((n ·m)), with 2.     

On weighting two criteria with a parameter in combinatorial optimization problems

Two criteria in a combinatorial problem are often combined in a weighted sum objective using a weighting parameter between 0 and 1. For special problem types, e.g., when one of the criteria is a bottleneck value, efficient algorithms are known that solve for a given value of the weighting parameter.     

Optimal Solution of a Bottleneck Problem with State Delay

Two computational procedures for solving a class of bottleneck problems with state delay are presented. The first one uses a direct computation scheme for the solution of the continuous-time problem, while the second one is obtained by discretizing the continuous-time problem using piecewise constant controls on small intervals. Examples show that these two approaches agree.     

On the bottleneck assignment problem

In this paper, a new approach for solving the bottleneck assignment problem is presented. The problem is treated as a special class of permutation problems which we call max-min permutation problems. By defining a suitable neighborhood system in the space of permutations and designating certain permutations as critical solutions, it is shown that any critical solution yields a global optimum. This theorem is then used as a basis to develop a general method to solve max-min permutation problems.This work was carried out by the junior author while holding a Purdue University Fellowship.     

Possibilistic bottleneck combinatorial optimization problems with ill-known weights

In this paper a general bottleneck combinatorial optimization problem with uncertain element weights modeled by fuzzy intervals is considered. A possibilistic formalization of the problem and solution concepts in this setting, which lead to compute robust solutions under fuzzy weights, are given. Some algorithms for finding a solution according to the introduced concepts and evaluating optimality of solutions and elements are provided. These algorithms are polynomial for bottleneck combinatorial optimization problems with uncertain element weights, if their deterministic counterparts are polynomially solvable.     

双目标瓶颈指派问题的遗传算法

给出一种双目标瓶颈指派问题的新模型,本模型结合了决策者和工人两方面的因素,特别之处在于考虑到了工人对工作的排名偏好.进而,将双目标瓶颈指派问题转化为单目标规划,并设计了解此问题的遗传算法,算法的解均为双目标瓶颈指派问题的Pareto最优解.     

On Steiner Versions of (bi)Connectivity in Network Problems

The notions of connectivity and biconnectivity can be generalized in the Steiner sense, i.e., they are restricted to a given subset of the vertices of a graph. We illustrate this generalization on two problems. The first problem is the bottleneck biconnected subgraph problem, the second one is the so-called bipartition problem. The adapted algorithms to solve the Steiner versions of these problems exploit depth-first search to attain respectively a running time of O(|E|+|V|log|V|) and O(|E|+|V|) with E denoting the set of edges and V the set of vertices of the given graph.Final version received: October 15, 2003     

On ranking of feasible solutions of a bottleneck linear programming problem

Summary An algorithm for the ranking of the feasible solutions of a bottleneck linear programming problem in ascending order of values of a concave bottleneck objective function is developed in this paper. The “best” feasible solution for a given value of the bottleneck objective is obtained at each stage. It is established that only the extreme points of a convex polytope need to be examined for the proposed ranking. Another algorithm, involving partitioning of the nodes, to rank the feasible solutions of the bottleneck linear programming problem is proposed, and numerical illustrations for both are provided.     

A note on the asymptotic behaviour of bottleneck problems

We generalize and sharpen results of Burkard and Fincke concerning the asymptotic behaviour of a certain class of combinatorial optimization problems with bottleneck objective function. In this way several open questions are answered.     

The stochastic bottleneck linear programming problem

In this paper we consider some stochastic bottleneck linear programming problems. We overview the solution methods in the literature. In the case when the coefficients of the objective functions are simple randomized, the minimum-risk approach will be used for solving these problems. We prove that, under some positivity conditions, these stochastic problems are reduced to certain deterministic bottleneck linear problems. An application of these problems to bottleneck spanning tree problems is given. Two simple numerical examples are presented. This paper was written when I.M. Stancu-Minasian was visiting the Instituto Complutense de Análisis Económico, in the Universidad Complutensen de Madrid, from October 1, 1997 to November 15, 1997 and from October 24, 1998 to November, 9, 1998, as invited researcher. He is grateful to the Institution.     

Multiple bottleneck assignment problem

We consider the multiple bottleneck assignment problem which subsumes the well known min-sum and bottleneck assignment problems. The problem arises in the context of flexible manufacturing systems, where the objective is to maximize the throughput of a production system with several flow shops, running in parallel, to produce a product. The problem is known to be strongly NP-hard. We propose a new algorithm for obtaining sharp lower bounds to the optimal objective value. Computational experiments are conducted to show the improvement over existing methods.     

A General Linea-Quadratic Optimization Problem

A linear-quadratic optimization problem is formulated in a dynamic programming manner. An updating formula for obtaining the solutions to such a problem is provided and illustrated using a few simple examples. This updating formula is also compared to a well-known updating formula for obtaining the inverses of symmetric positive-definite matrices. Numerical results are given.     

Chance constrained bottleneck spanning tree problem

In this paper we consider a stochastic version of the bottleneck spanning tree in which edge costs are independent random variables. Our problem is to find an optimal spanning tree and optimal satisficing level of the chance constraint with respect to the bottleneck (maximum cost) edge of the spanning tree. The problem is first transformed into a deterministic equivalent problem. Then a subproblem in order to solve the problem is introduced and the close relation between these problems is clarified. Finally, based on the relation, polynomial time solution procedures to solve the problem are proposed under two special functions of satisficing level which are given as examples to be solved easily.     

Relaxation of Minimax Optimal Control Problems with Infinite Horizon

A minimax optimal control problem with infinite horizon is studied. We analyze a relaxation of the controls, which allows us to consider a generalization of the original problem that not only has existence of an optimal control but also enables us to approximate the infinite-horizon problem with a sequence of finite-horizon problems. We give a set of conditions that are sufficient to solve directly, without relaxation, the infinite-horizon problem as the limit of finite-horizon problems.     

Machine duplication and part subcontracting in the presence of alternative cell locations in manufacturing cell design

In this paper a model and a solution algorithm are reported to simultaneously deal with the processes of machine duplication and part subcontracting in the presence of two significant design issues in cellular manufacturing systems: (i) alternative cell locations; and (ii) the maximum number of machines assigned to a cell. As the problem, formulated as a polynomial programming model, is shown NP-hard in the strong sense, a higher-level heuristic algorithm based upon a concept known as ‘tabu search’ is presented. An example (small) problem is solved to demonstrate the functionality of the algorithm. Additionally, the small problem is solved for its optimal solution under different scenarios, both with linear single-row and linear double-row layout arrangements, and the solutions obtained are shown to match with those obtained with the heuristic algorithm. A comparison of six different versions of tabu search-based heuristics (TSH 1–TSH 6) is performed to investigate the impact of long-term memory and the use of fixed versus variable tabu-list sizes. A carefully constructed statistical experiment, based on randomised complete-block design, is used to test the performance on three different problem structures, classified as small, medium and large. The results show that TSH 6 with variable tabu-list size and long-term memory based on minimal frequencies is preferred for the single-row layout, while TSH 4 with variable tabu-list size and no long-term memory is preferred for the double-row layout. When subject to budgetary restrictions, the proposed approach can be used by parts manufacturing companies to determine which of the following three actions should be undertaken for each bottleneck part: bottleneck part left as in the initial solution, all the bottleneck machines connected to it are duplicated, or the part subcontracted.     

A graph theory approach to subcontracting, machine duplication and intercell moves in cellular manufacturing

We address in this paper the problem of finding an optimal strategy for dealing with bottleneck machines and bottleneck parts in the cell formation process in group technology. Three types of economic decisions are considered: subcontracting, machine duplication and intercell moves. The problem is formulated as a minimum weighted node covering problem in a hypergraph, and we show that it can be solved in polynomial time by finding a maximum weighted stable set in a bipartite graph. We extend this result to cellular manufacturing systems in which the sequence of operations of each part is known in advance.     

Best possible heuristics for the bottleneck wandering salesperson and bottleneck vehicle routing problem

The purpose of this paper is to present heuristics for the bottleneck wandering salesperson problem and the bottleneck vehicle routing problem with the triangle inequality that are guaranteed to produce solutions that have a cost no more than twice the optimum. These heuristics are all best possible in the sense that the existence of a heuristic that guarantees a better ratio would imply that PP = NP, and this is widely believed to be false.     

Conjugate maps and duality in multiobjective optimization

This paper considers duality in convex vector optimization. A vector optimization problem requires one to find all the efficient points of the attainable value set for given multiple objective functions. Embedding the primal problem into a family of perturbed problems enables one to define a dual problem in terms of the conjugate map of the perturbed objective function. Every solution of the stable primal problem is associated with a certain solution of the dual problem, which is characterized as a subgradient of the perturbed efficient value map. This pair of solutions also provides a saddle point of the Lagrangian map.     

Optimal assignment of resources to strengthen the weakest link in an uncertain environment

The weakest link principle applies to many real-life situations where a system is as productive as its bottleneck. The purpose of this paper is to show how the efficiency of a manufacturing or service process (i.e., the strength of a chain) may be maximized by optimal allocation of resources to improve the performance of the bottleneck (i.e., strengthen the weakest link) in an uncertain environment. Specifically, we consider two different versions of the stochastic bottleneck assignment problem (SBAP), which is a variation of the classic assignment problem (AP), with respective goals of minimizing the expected longest processing time and maximizing the expected lowest production rate. It is proven that each of the two intrinsically difficult SBAPs is reducible to an efficiently solvable AP provided that the processing times or the production rates are independent random variables following some families of probability distributions.