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.     

An algebraic approach to assignment problems

For assignment problems a class of objective functions is studied by algebraic methods and characterized in terms of an axiomatic system. It says essentially that the coefficients of the objective function can be chosen from a totally ordered commutative semigroup, which obeys a divisibility axiom. Special cases of the general model are the linear assignment problem, the linear bottleneck problem, lexicographic multicriteria problems,p-norm assignment problems and others. Further a polynomial bounded algorithm for solving this generalized assignment problem is stated. The algebraic approach can be extended to a broader class of combinatorial optimization problems.     

A generalized bottleneck assignment problem

A solution procedure is presented for a generalization of the standard bottleneck assignment problem in which a secondary criterion is automatically provided. A partitioning problem is modeled by this bottleneck problem to provide an example of its application.     

Probabilistic asymptotic properties of some combinatorial optimization problems

A class of combinatorial optimization problems with sum- and bottleneck objective function is described, having the following probabilistic asymptotic behaviour: With probability tending to one the ratio between worst and optimal objective function value approaches one as the size of the problem tends to infinity.Problems belonging to this class are among others quadratic assignment problems, as well as certain combinatorial and graph theoretical optimization problems.The obtained results suggest that even very simple heuristic algorithms incline to yield good solutions for high dimensional problems of this class.     

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.     

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.     

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.     

Categorized Assignment Scheduling: a Tabu Search Approach

The assignment problem (AP) and bottleneck assignment problem (BAP) are well studied in operational research literature. In this paper we consider two related problems which simultaneously generalize both AP and BAP. Unlike AP and BAP, these generalizations are strongly NP-complete. We propose two heuristics to solve these generalized problems: one based on a greedy principle and the other based on tabu search. Computational results are presented which show that these heuristics, when used together, produce good quality solutions. Our adaptation of tabu search also gives some new insight into the application of tabu search on permutation problems.     

Improving port performance in developing countries—A case study

Exchange algorithms are an important class of heuristics for hard combinatorial optimization problems as, e.g., salesman problems or quadratic assignment problems. In Kirkpatrick's and Cerny's exchange algorithms for the travelling salesman problem and placement problem they propose to perform an exchange not only if the objective function value decreases by this exchange, but also in certain cases if the objective function value increases. An exchange increasing the objective function value is performed stochastically depending on the size of the increment.Computational tests with quadratic assignment problems revealed an excellent behaviour in such an approach. Suboptimal solutions differing 1–2% from the best known solutions are obtained by a simple program in short time. By starting this program several times with different starting values all known minimal objective function values were reached. Thus this approach is well suited also for smaller computers and leads in short time to acceptable solutions.     

The approximability of three-dimensional assignment problems with bottleneck objective

We discuss two special cases of the three-dimensional bottleneck assignment problem where a certain underlying cost function satisfies the triangle inequality. We present polynomial time 2-approximation algorithms for the broadest class of these special cases, and we prove that (unless P = NP) this factor 2 is best possible even in the highly restricted setting of the Euclidean plane.     

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.     

Asymptotic Properties of Random Multidimensional Assignment Problems

The multidimensional assignment problem (MAP) is a NP-hard combinatorial optimization problem, occurring in many applications, such as data association. In this paper, we prove two conjectures made in Ref. 1 and based on data from computational experiments on MAPs. We show that the mean optimal objective function cost of random instances of the MAP goes to zero as the problem size increases, when assignment costs are independent exponentially or uniformly distributed random variables. We prove also that the mean optimal solution goes to negative infinity when assignment costs are independent normally distributed random variables.     

A generalized Hungarian method for solving minimum weight perfect matching problems with algebraic objective

The well known blossom-algorithm for solving minimum weight perfect matching problems makes use of the optimality criteria arising from LP-duality and complementary slackness. But these instruments seem to fail when such a matching problem is considered with a different objective function as for instance the bottleneck objective which is also relevant in practice. Such a dilemma occurs for all those combinatorial optimization problems with algorithms based on Linear Programming. Therefore we present a rarely combinatorially motivated approach in this paper.     

人数与任务数不相等的指派问题

本提出人数与任务数不相等的指派问题应当视为一个多目标决策问题，首先要求指派给各人的任务数目两两之间相差不能超过1，其次要求所需总时间最少；并且给出了该问题的求解方法。     

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.     

Formulating and solving production planning problems

Production planning problems frequently involve the assignment of jobs or operations to machines. The simplest model of this problem is the well known assignment problem (AP). However, due to simplifying assumptions this model does not provide implementable solutions for many actual production planning problems. Extensions of the simple assignment model known as the generalized assignment problem (GAP) and the multi-resource generalized assignment problem (MRGAP) have been developed to overcome this difficulty. This paper presents an extension of the (MRGAP) to allow splitting individual batches across multiple machines, while considering the effect of setup times and setup costs. The extension is important for many actual production planning problems, including ones in the injection molding industry and in the metal cutting industry. We formulate models which are logical extensions of previous models which ignored batch splitting for the problem we address. We then give different formulations and suggest adaptations of a genetic algorithm (GA) and simulated annealing (SA). A systematic evaluation of these algorithms, as well as a Lagrangian relaxation (LR) approach, is presented.     

Some inverse optimization problems under the Hamming distance

Given a feasible solution to a particular combinatorial optimization problem defined on a graph and a cost vector defined on the arcs of the graph, the corresponding inverse problem is to disturb the cost vector such that the feasible solution becomes optimal. The aim is to optimize the difference between the initial cost vector and the disturbed one. This difference can be measured in several ways. We consider the Hamming distance measuring in how many components two vectors are different, where weights are associated to the components. General algorithms for the bottleneck or minimax criterion are described and (after modification) applied to the inverse minimum spanning tree problem, the inverse shortest path tree problem and the linear assignment problem.     

The single-machine earliness-tardiness scheduling problem with due date assignment and resource-dependent processing times

We study the earliness-tardiness scheduling problem on a single machine with due date assignment and controllable processing times. We analyze the problem with three different due date assignment methods and two different processing time functions. For each combination of these, we provide a polynomial-time algorithm to find the optimal job sequence, due date values and resource allocation minimizing an objective function which includes earliness, tardiness, due date assignment, makespan and total resource consumption costs.     

Minimizing weighted earliness–tardiness and due-date cost with unit processing-time jobs

The classical single-machine scheduling and due-date assignment problem of Panwalker et al. [Panwalker, S.S., Smith, M.L., Seidmann, A., 1982. Common due date assignment to minimize total penalty for the one machine scheduling problem. Operations Research 30(2) (1982) 391–399] is the following: All n jobs share a common due-date, which is to be determined. Jobs completed prior to or after the due-date are penalized according to a cost function which is linear and job-independent. The objective is to minimize the total earliness–tardiness and due-date cost. We study a generalized version of this problem in which: (i) the earliness and tardiness costs are allowed to be job dependent and asymmetric and (ii) jobs are processed on parallel identical machines. We focus on the case of unit processing-time jobs. The problem is shown to be solved in polynomial (O(n⁴)) time. Then we study the special case with no due-date cost (a classical problem known in the literature as TWET). We introduce an O(n³) solution for this case. Finally, we study the minmax version of the problem, (i.e., the objective is to minimize the largest cost incurred by any of the jobs), which is shown to be solved in polynomial time as well.     

Easy and hard bottleneck location problems

We consider a bottleneck location problem on a graph and present an efficient (polynomial time) algorithm for solving it. The problem involve the location of K noxious facilities that are to be placed as far as possilbe from the other facilities, and the objective is to maximize the minimum distance from the noxious facilities to the others. We then show that two other bottleneck (min-max) location problems, finding K-centers and absolute K-centers of a graph appear to be very difficult to solve even for reasonably good approximate solutions.