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Title: the multi-level bottleneck assignment problem: complexity and solution methods.

Abstract: We study the multi-level bottleneck assignment problem (MBA), which has important applications in scheduling and quantitative finance. Given a weight matrix, the task is to rearrange entries in each column such that the maximum sum of values in each row is as small as possible. We analyze the complexity of this problem in a generalized setting, where there are restrictions in how values in columns can be permuted. We present a lower bound on its approximability by giving a non-trivial gap reduction from three-dimensional matching to MBA. To solve MBA, a greedy method has been used in the literature. We present new solution methods based on an extension of the greedy method, an integer programming formulation, and a column generation heuristic. In computational experiments we show that it is possible to outperform the standard greedy approach by around 10% on random instances.

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## The Multi-level Bottleneck Assignment Problem: Complexity and Solution Methods

We study the multi-level bottleneck assignment problem (MBA), which has important applications in scheduling and quantitative finance. Given a weight matrix, the task is to rearrange entries in each column such that the maximum sum of values in each row is as small as possible. We analyze the complexity of this problem in a generalized setting, where there are restrictions in how values in columns can be permuted. We present a lower bound on its approximability by giving a non-trivial gap reduction from three-dimensional matching to MBA. To solve MBA, a greedy method has been used in the literature. We present new solution methods based on an extension of the greedy method, an integer programming formulation, and a column generation heuristic . In computational experiments we show that it is possible to outperform the standard greedy approach by around 10

Trivikram Dokka

Marc Goerigk

## Related Research

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## On the bottleneck assignment problem

- Contributed Papers
- Published: April 1977
- Volume 21 , pages 451–458, ( 1977 )

## Cite this article

- A. Ravindran 1 &
- V. Ramaswami 2

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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.

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Nicholson, T. A. J. , A Method for Optimizing Permutation Problems and its Industrial Applications , Combinatorial Mathematics and its Applications, Edited by D. J. A. Welsh, Academic Press, New York, New York, 1971.

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Charnes, A. , and Cooper, W. W. , The Stepping Stone Method of Explaining Linear Programming Calculations in Transportation Problems , Management Science, Vol. 1, pp. 49–69, 1954–55.

Ford, L. R. , and Fulkerson, D. R. , Flows in Networks , Princeton University Press, Princeton, New Jersey, 1962.

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## Author information

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Department of Industrial Engineering, Purdue University, West Lafayette, Indiana

A. Ravindran ( Associate Professor )

Department of Statistics, Purdue University, West Lafayette, Indiana

V. Ramaswami ( Graduate Student )

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## Additional information

Communicated by C. T. Leondes

This work was carried out by the junior author while holding a Purdue University Fellowship.

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## About this article

Ravindran, A., Ramaswami, V. On the bottleneck assignment problem. J Optim Theory Appl 21 , 451–458 (1977). https://doi.org/10.1007/BF00933089

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Issue Date : April 1977

DOI : https://doi.org/10.1007/BF00933089

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## COMMENTS

In this sensitivity analysis, we provide a sufficient but not necessary condition for the invariance of the optimal assignment, in the form of an interval test. n×m Let Λ ⊆ R be an n × m array of intervals over the extended reals. For each edge e ∈ E, let [−λe, λe] be the interval corresponding to edge e. Remark 1.

The bottleneck assignment problem (BAP) has the objective of minimising the most costly allocation of a task to an agent. Under certain conditions the structure of ... neck clusters and provide conditions when the solution to Problem 1 is found by merging matchings M 1 and M 2. 4.1 A Bound on the Optimal BAP Solution Theorem 1. Under ...

We consider two versions of bottleneck (or min-max) generalized assignment problem (BGAP) under capacity uncertainty: Task-BGAP and Agent-BGAP. A robust optimization approach is employed to study this issue. The decision maker's degree of risk aversion and the penalty weighting parameter are incorporated into the objective function. A state-of-the-art linearization method is introduced ...

The problem of assessing the sensitivity of a bottleneck edge is directly comparable to the sensitivity of the bottleneck assignment for the lexicographic assignment, see Burkard & Rendl (1991), and the solution of Problem 2 is a conservative estimate of the sensitivity from Problem 1.

The linear bottleneck assignment problem (LBAP), which is a variation of the classical assignment problem (CAP), seeks to minimize the longest completion time rather than the sum of the completion times when a number of jobs are to be assigned to the same number of workers. Several procedures have been proposed in the current literature to convert the LBAP into an equivalent CAP and then apply ...

The bottleneck assignment problem (BAP) has the objective of minimising the most costly allocation of a task to an agent. ... Then, we introduce bottle- neck clusters and provide conditions when the solution to Problem 1 is found by merging matchings M1 and M2. 4.1 A Bound on the Optimal BAP Solution Theorem 1. Under Assumptions 1 and 2, it ...

This problem is known as the multi-level bottleneck assignment problem (MBAP). The MBAP with m = 2 is a special case of the classical bottleneck assignment problem and belongs to a class of high dimensional (axial) generalization of the well-known assignment problem, see Burkard et al. (2009). Therefore, MBAP is a hypergraph version of ...

The Linear Bottleneck Assignment Problem. In: Assignment and Matching Problems: Solution Methods with FORTRAN-Programs. Lecture Notes in Economics and Mathematical Systems, vol 184.

and we want to minimize the time at which the last task is completed, then we have a bottleneck assignment problem. Formally, the bottleneck assignment problem is defined as follows, where c;. is the cost of assigning resource i to task j: For a summary of major results on and algorithms for solving the bottleneck assignment problem, see §6.2 ...

Two versions of bottleneck (or min-max) generalized assignment problem (BGAP) under capacity uncertainty are considered: Task-BGAP and Agent- BGAP and a robust optimization approach is employed. We consider two versions of bottleneck (or min-max) generalized assignment problem (BGAP) under capacity uncertainty: Task-BGAP and Agent-BGAP. A robust optimization approach is employed to ...

Reviewer: Vladik Ya. Kreinovich A linear bottleneck assignment problem (LBAP) is defined as follows: we have n objects of the first type, and n objects of the second type, and we know the "damage" c ij caused by matching the i th object of the first class to the j th object of the second class.

These. problems can be formulated as follows. Let A = {ajj} fJ=1 be a n x n matrix of real numbers. which we further call weights. In the bottleneck (capacity) assignment problem, we ask to. minimize (maximize) the largest (smallest) element over all possible sets of n entries in A, one. from each row and column.

2 Spivey: Bottleneck Assignment Problem Mathematics of Operations Research (), pp. , c 20 INFORMS of maxft ijgover all possible matchings.But the former is, by de nition, R, and the latter is ˝(match;B~). (Lemma 1.1 is similar to the ideas behind the class of threshold algorithms used to solve the bottleneck

The Multi-level Bottleneck Assignment Problem: Complexity and Solution Methods. Trivikram Dokka, Marc Goerigk. We study the multi-level bottleneck assignment problem (MBA), which has important applications in scheduling and quantitative finance. Given a weight matrix, the task is to rearrange entries in each column such that the maximum sum of ...

A Generalized Bottleneck Assignment Problem. Abstract. A solution procedure is presented fora generalization of the standard bottleneck assignment problem in which a secondary criterion is automatically provided. problem is modeled A partitioning by this bottleneck problem to provide an of its example application. KeyWords.

Semantic Scholar extracted view of "Bottleneck generalized assignment problems" by J. Mazzola et al. ... It is shown that an optimal solution to this problem is a basic feasible solution to a slightly modified generalized transportation problem, and a branch-and-bound solution procedure prevents possible splitting of a job among computers. ...

In combinatorial optimization, a field within mathematics, the linear bottleneck assignment problem (LBAP) is similar to the linear assignment problem.. In plain words the problem is stated as follows: There are a number of agents and a number of tasks.Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.

Abstract. The min-max version of the generalized assignment problem is considered. We introduce relaxations and show that they produce, as sub-problems, min-max versions of the multiple-choice knapsack problem and of the 0-1 knapsack problem. It is proved that such problems can be solved exactly in polynomial time.

The classic bottleneck assignment problem is a deterministic one in which the completion times are constants. The deterministic bottleneck assignment problem is first introduced by Fulkerson et al. . Gross proves the min-max theorem for the bottleneck assignment problem (Gross 1959). Researchers propose many solution methods for the ...

We study the multi-level bottleneck assignment problem (MBA), which has important applications in scheduling and quantitative finance. Given a weight matrix, the task is to rearrange entries in each column such that the maximum sum of values in each row is as small as possible.

A simple algorithm for solving either of two different bottleneck assignment problems, which requires finding an assignment of men to machines in a serial production line to maximize the rate of flow through the line. Abstract : A simple algorithm for solving either of two different bottleneck assignment problems is described in this paper. The one problem requires finding an assignment of men ...

critical solution is 0 with associated neighborhood B(O, 1) = C*. This amounts to a relabeling, if necessary. Let o- ~ G. Clearly, if o'(1) = 1 or or e C*, ... bottleneck assignment problem is concerned with maximizing the minimum efficiency resulting from an assignment. Such an objective function arises in