Bottleneck Objective Function Research Articles

Mixed Graph Colouring as Scheduling a Partially Ordered Set of Interruptible Multi-Processor Tasks with Integer Due Dates

We investigate relationships between scheduling problems with the bottleneck objective functions (minimising makespan or maximal lateness) and problems of optimal colourings of the mixed graphs. The investigated scheduling problems have integer durations of the multi-processor tasks (operations), integer release dates and integer due dates of the given jobs. In the studied scheduling problems, it is required to find an optimal schedule for processing the partially ordered operations, given that operation interruptions are allowed and indicated subsets of the unit-time operations must be processed simultaneously. First, we show that the input data for any considered scheduling problem can be completely determined by the corresponding mixed graph. Second, we prove that solvable scheduling problems can be reduced to problems of finding optimal colourings of corresponding mixed graphs. Third, finding an optimal colouring of the mixed graph is equivalent to the considered scheduling problem determined by the same mixed graph. Finally, due to the proven equivalence of the considered optimisation problems, most of the results that were proven for the optimal colourings of mixed graphs generate similar results for considered scheduling problems, and vice versa.

Max-type reliability in uncertain post-disaster networks through the lens of sensitivity and stability analysis

The functionality of infrastructures, particularly in densely populated areas, is greatly impacted by natural disasters, resulting in uncertain networks. Thus, it is important for crisis management professionals and computer-based systems for transportation networks (such as expert systems) to utilize trustworthy data and robust computational methodologies when addressing convoluted decision-making predicaments concerning the design of transportation networks and optimal routes. This study aims to evaluate the vulnerability of paths in post-disaster transportation networks, with the aim of facilitating rescue operations and ensuring the safe delivery of supplies to affected regions. To investigate the problem of links' tolerances in uncertain networks and the resiliency and reliability of paths, an uncertainty theory-based model that employs minmax optimization with a bottleneck objective function is used. The model addresses the uncertain maximum reliable paths problem, which takes into account uncertain risk variables associated with links. Rather than using conventional methods for calculating the deterministic tolerances of a single element in combinatorial optimization, this study introduces a generalization of stability analysis based on tolerances while the perturbations in a group of links are involved. The analysis defines set tolerances that specify the minimum and maximum values that a designated group of links could simultaneously fluctuate while maintaining the optimality of the max-type reliable paths. The study shows that set tolerances can be considered as well-defined and proposes computational methods to calculate or bound such quantities - which were previously unresearched and difficult to measure. The model and methods are demonstrated to be both theoretically and numerically efficient by applying them to four subnetworks from our case study. In conclusion, this study provides a comprehensive approach to addressing uncertainty in reliability problems in networks, with potential applications in various fields.

Information-Theoretic Odometry Learning

In this paper, we propose a unified information theoretic framework for learning-motivated methods aimed at odometry estimation, a crucial component of many robotics and vision tasks such as navigation and virtual reality where relative camera poses are required in real time. We formulate this problem as optimizing a variational information bottleneck objective function, which eliminates pose-irrelevant information from the latent representation. The proposed framework provides an elegant tool for performance evaluation and understanding in information-theoretic language. Specifically, we bound the generalization errors of the deep information bottleneck framework and the predictability of the latent representation. These provide not only a performance guarantee but also practical guidance for model design, sample collection, and sensor selection. Furthermore, the stochastic latent representation provides a natural uncertainty measure without the needs for extra structures or computations. Experiments on two well-known odometry datasets demonstrate the effectiveness of our method.

On the complexity of the bilevel minimum spanning tree problem

We consider the bilevel minimum spanning tree (BMST) problem where the leader and the follower choose a spanning tree together, according to different objective functions. We show that this problem is NP-hard, even in the special case where the follower only controls a matching. Moreover, we give some evidence that BMST might even remain hard in case the follower controls only few edges. On the positive side, we present a ( | V | − 1 ) $$ \left(|V|-1\right) $$ -approximation algorithm for BMST, where | V | $$ \mid V\mid $$ is the number of vertices. Moreover, we show that 2-approximating BMST is fixed-parameter tractable and that, in case of uniform costs on leader's edges, even solving BMST exactly is fixed-parameter tractable. We finally consider bottleneck variants of BMST and settle the complexity landscape of all combinations of sum or bottleneck objective functions for the leader and follower, for the optimistic as well as the pessimistic setting.

A note on the complexity of the bilevel bottleneck assignment problem

We establish the NP-completeness of the variant of the bilevel assignment problem, where the leader and the follower both have bottleneck objective functions and were the follower behaves according to the optimistic rule. This result settles a problem that has been left open by Klinz & Gassner [4OR 7:379–394, 2009].

Lexicographic bottleneck mixed-model assembly line balancing problem: Artificial bee colony and tabu search approaches with optimised parameters

The lexicographic bottleneck assembly line balancing problem is a recently introduced problem which aims at obtaining a smooth workload distribution among workstations. This is achieved hierarchically. The workload of the most heavily loaded workstation is minimised, followed by the workload of the second most heavily loaded workstation and so on. This study contributes to knowledge by examining the application of the lexicographic bottleneck objective on mixed-model lines, where more than one product model is produced in an inter-mixed sequence. The main characteristics of the lexicographic bottleneck mixed-model assembly line balancing problem are described with numerical examples. Another contribution of the study is the methodology used to deal with the complex structure of the problem. Two effective meta-heuristic approaches, namely artificial bee colony and tabu search, are proposed. The parameters of the proposed meta-heuristics are optimised using response surface methodology, which is a well-known design of experiments technique, as a unique contribution to the expert and intelligent systems literature. Different from the common tendency in the literature (which aims to optimise one parameter at a time), all parameters are optimised simultaneously. Therefore, it is shown how a complex production planning problem can be solved using sophisticated artificial intelligence techniques with optimised parameters. The methodology used for parameter setting can be applied to other metaheuristics for solving complex problems in practice. The performances of both algorithms are assessed using well-known test problems and it is observed that both algorithms find promising solutions. Artificial bee colony algorithm outperforms tabu search in minimising the number of workstations while tabu search shows a better performance in minimising the value of lexicographic bottleneck objective function.

Exact solution approaches for bilevel assignment problems

We consider the bilevel assignment problem in which each decision maker (i.e., the leader and the follower) has its own objective function and controls a distinct set of edges in a given bipartite graph. The leader acts first by choosing some of its edges. Subsequently, the follower completes the assignment process. The edges selected by the leader and the follower are required to constitute a perfect matching. In this paper we propose an exact solution approach, which is based on a branch-and-bound framework and exploits structural properties of the assignment problem. Extensive computational experiments with linear sum and linear bottleneck objective functions are conducted to demonstrate the performance of the developed methods. While the considered problem is known to be NP-hard in general, we also describe some restricted cases that can be solved in polynomial time.

Time minimizing transportation problem with fractional bottleneck objective function

This paper is aimed at studying the Time Minimizing Transportation Problem with Fractional Bottleneck Objective Function (TMTP-FBOF). TMTP-FBOF is related to a Lexicographic Fractional Time Minimizing Transportation Problem (LFTMTP), which will be solved by a lexicographic primal code. An algorithm is also developed to determine an initial efficient basic solution to this TMTP-FBOF. The developed TMTP-FBOF Algorithm is supported by a real life example of Military Transportation Problem of Indian Army.

Multiobjective combinatorial optimization problems with a cost and several bottleneck objective functions: An algorithm with reoptimization

This paper addresses multicriteria combinatorial optimization problems involving one cost and several bottleneck objective functions. An algorithm is developed which generates the minimal complete set of Pareto-optimal solutions. This algorithm runs in polynomial time as long as the single objective problem considering only the cost function can be solved polynomially. A reoptimization procedure is used to accelerate the convergence of the algorithm. Applications are given. Computational results on randomly generated instances and planar grid graphs concerning the minimum cost spanning tree and the shortest path problem are presented.

An efficient algorithm for the Steiner Tree Problem with revenue, bottleneck and hop objective functions

This paper considers a tricriteria Steiner Tree Problem arising in the design of telecommunication networks. The objective functions consist of maximizing the revenue and of minimizing the maximal distance between each pair of interconnected nodes, as well as the maximal number of arcs between the root and each node. A polynomial algorithm is developed for the generation of a minimal complete set of Pareto-optimal Steiner trees. Optimality proofs are given and computational experience on a set of randomly generated problems is reported.

On algorithms for the tricriteria shortest path problem with two bottleneck objective functions

This paper addresses a tricriteria path problem involving two bottleneck objective functions and a cost. It presents an enhanced method that computes shortest paths in subnetworks, obtained by restricting the set of arcs according to the bottleneck values in order to find the minimal complete set of Pareto-optimal solutions, and taking into account the objective values of the determined shortest paths to reduce the number of considered subnetworks, and thus the number of solved shortest path problems. A labeling procedure for the problem is also developed. The algorithms are compared with the previous literature. Moreover a variant of the first method is presented. Its aim is to choose the solutions with the best bottleneck value when the cost is the same. Results for random instances reveal that the enhanced method is the fastest, and that, in average, it runs in less than 20 s for networks with 30 000 nodes, an average degree of 20 and 1000 distinct bottleneck values. Its variant that avoids ties improved the former version up to 15% for costs in [ 1 , 10 ] .

The tricriterion shortest path problem with at least two bottleneck objective functions

Abstract The focus of this paper is on the tricriterion shortest path problem where two objective functions are of the bottleneck type, for example MinMax or MaxMin. The third objective function may be of the same kind or we may consider, for example, MinSum or MaxProd. Let p ( n ) be the complexity of a classical single objective algorithm responsible for this third function, where n is the number of nodes and m be the number of arcs of the graph. An O( m 2 p ( n )) algorithm is presented that can generate the minimal complete set of Pareto-optimal solutions. Finding the maximal complete set is also possible. Optimality proofs are given and extensions for several special cases are presented. Computational experience for a set of randomly generated problems is reported.

The computational complexity of bilevel assignment problems

In bilevel optimization problems there are two decision makers, the leader and the follower, who act in a hierarchy. Each decision maker has his own objective function, but there are common constraints. This paper deals with bilevel assignment problems where each decision maker controls a subset of edges and each edge has a leader’s and a follower’s weight. The edges selected by the leader and by the follower need to form a perfect matching. The task is to determine which edges the leader should choose such that his objective value which depends on the follower’s optimal reaction is maximized. We consider sum- and bottleneck objective functions for the leader and follower. Moreover, if not all optimal reactions of the follower lead to the same leader’s objective value, then the follower either chooses an optimal reaction which is best (optimistic rule) or worst (pessimistic rule) for the leader. We show that all the variants arising if the leader’s and follower’s objective functions are sum or bottleneck functions are NP-hard if the pessimistic rule is applied. In case of the optimistic rule the problem is shown to be NP-hard if at least one of the decision makers has a sum objective function.

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 multiple objective time transportation problem with additional restrictions

This paper deals with Multiple Objective Time Transportation Problem with Additional Restrictions (MOTTP). This MOTTP with non-linear bottleneck objective functions is related to Lexicographic Multiple Time Transportation Problem with Additional Restrictions, which is solved by a lexicographic primal code. An algorithm and its underlying theory is developed to determine an initial efficient basic solution of this MOTTP. The developed algorithm is supported by a real life example of minimizing the shipping time, loading/unloading time and congestion time for transporting ore from source i to destination j for a Steel Industry.

A minimax resource allocation problem with variable resources

The problem of optimally allocating the resources to competing activities where the amounts of the resources are not only previosly given but are also to be determined subject to certain linear constraints is considered. The objective is to find such activity levels such that their weighted deviation from a prespecified target is as small as possible. A primal decomposition approach for the problem and derive an explicit formula for the piecewise affine-linear convex objective function of the upper level problem are suggested. The lower level problem is a parametric optimization problem with a bottleneck objective function. If the constraints on the amounts of the resources are all of knapsack type or if there is only one such constraint, then the upper level problem is equivalent to the lower level problem with fixed right-hand sides. In the general case, the problem can efficiently be solved by means of the level method.

Minimum-cost strong network orientation problems: Classification, complexity, and algorithms

In the minimum-cost strong network orientation problem (MCSO), we are given an undirected graph G = (V, E) with nonnegative edge lengths 𝓁(e) and a transportation schedule T = {(s1, t1, w1), …, (sk, tk, wk)}, where wi units of weight have to be transported from the source vertex si to the target vertex ti for i = 1, …, k. Let Gσ be a strongly connected orientation of G and let L be the length of the shortest (directed) path from si to ti in Gσ. The goal in the MCSO is to find a strongly connected orientation Gσ such that the overall cost of the orientation given by Σ wiL (sum case) or maxi=1,…,k wiL (bottleneck case) is minimized. The strong network orientation problem is motivated by the practical problem of designing the optimal unidirectional flow path of automated guided vehicles. In this paper, we investigate the MCSO from the algorithmic and complexity points of view and propose a classification scheme. In the first part of the paper, we identify several efficiently solvable cases of the MCSO with sum and bottleneck objective functions which arise if additional restrictions are imposed on the structure of the graph G, the edge lengths 𝓁(e), and/or the transportation schedule T. In the second part, we identify special cases of the MCSO which are NP-hard. © 1999 John Wiley & Sons, Inc. Networks 33: 57–70, 1999

On ranking of feasible solutions of a bottleneck linear programming problem

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 fast and simple algorithm for the bottleneck biconnected spanning subgraph problem

We study the problem of identifying an optimal biconnected spanning subgraph of an undirected graph with respect to a bottleneck objective function. In a graph with m edges and n nodes, our logarithm has a worst-case complexity of O( m + n log n). This improves the best available O( m log n) bound for the problem. The min-sum version of this problems is known to be NP-hard.

Lexicographic bottleneck problems

We study combinatorial optimization problems with bottleneck objective function, where any feasible solution has the same number of elements. In addition to minimizing the largest element of a feasible solution we are interested in minimizing also the second largest element, the third largest element, and so on. For this version of the bottleneck problem two generic solution procedures are developed and analyzed. The first is based on scaling the cost elements. In the second approach an optimal solution is constructed iteratively. Both methods have polynomial running time, provided the underlying sum optimization problem is polynomially solvable.