Solving the Generalized Assignment Problem: An Optimizing and Heuristic Approach

Lagrangean heuristics for location problems

Lagrangian relaxation based approaches to capacitated hub-and-spoke network design problem

This paper considers the generalized assignment problem (GAP). It is well-known NP-hard combinatorial optimization problem that is interesting in itself and also appears as a sub problem in other problems of practical importance. Line-up competition Differential Evolution algorithm for the GAP is proposed. The algorithm uses integer-coding structure, and introduces the idea of line-up competition. The experimental results indicates that, compared with other algorithms, this algorithm has the characteristics of quick convergence speed, of falling into local optimization rarely and of powerful optimization-searching ability and this algorithm can solve GAP effectively.

: An algorithm is presented for the two duty period scheduling problem. This integer programming problem has a binary constraint matrix with two sets of consecutive ones in each column. At each subproblem of a branch and bound procedure, subgradient optimization is used to maximize the value of a Lagrangean relaxation, which is a network flow problem. The algorithm is implemented for the two duty period set partitioning problem, with shortest path relaxations. A second algorithm utilizing the unique properties of prime numbers is developed for solving small subproblems. Computational results are reported for several large problems. (Author)

We propose a two-phase heuristic for the generalized assignment problem (GAP). The first phase—a generic variable-fixing method—heuristically eliminates up to 98% of the variables without sacrificing the solution quality. The second phase takes as input the small reduced GAP obtained during the first phase and applies a very large scale neighborhood search. The definition of the successive exponential size neighborhoods is guided by the subgradient method applied to the Lagrangian relaxation of the knapsack constraints via the reduced costs. Searching the proposed neighborhood is NP-hard and amounts to solving a monotone binary program (BP) with m constraints and p variables, where m and p are respectively the number of agents and tasks of the reduced GAP (monotone BPs are BPs with two nonzero coefficients of opposite sign per column). To the best of our knowledge, this is the first time the above ideas are exposed. Extensive testing on large scale GAP instances is presented and previously unknown better values for eight instances are obtained. Comparison to well-established methods shows that this new approach is competitive and constitutes a substantial addition to the arsenal of tools for solving GAP.

Dual formulations and subgradient optimization strategies for linear programming relaxations of mixed-integer programs

The classroom assignment problem: Complexity, size reduction and heuristics

A Tabu search heuristic for the generalized assignment problem

We consider the two-dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut. We develop a Lagrangean relaxation of a zero-one integer programming formulation of the problem and use it as a bound in a tree search procedure. Subgradient optimization is used to optimize the bound derived from the Lagrangean relaxation. Problem reduction tests derived from both the original problem and the Lagrangean relaxation are given. Incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved.

Discrete search path planning is known to be a NP-hard problem, and problem-solving methods proposed so far rely on heuristics with no way to reasonably estimate solution quality for practical size problems. Departing from traditional nonlinear model formulations, a novel information theoretic based approach using integer linear programming (ILP) is proposed to optimally solve the discrete open-loop centralized search path planning problem with anticipated feedback, involving a team of homogeneous agents. The approach takes advantage of objective function separability and conditional probability independence of observations to efficiently minimize expected system entropy. A network representation is exploited to simplify modeling, reduce constraint specification and speed-up problem-solving. The novelty of the approach consists in capturing false-alarm observations explicitly while proposing an original and efficient way to linearize the underlying non-linear expected entropy function required to properly represent target location uncertainty, making for the first time practical problems tractable. The proposed ILP formulation rapidly yields near-optimal solutions for realistic problems while providing for the first time, a robust lower bound through Lagrangian relaxation. Long planning problem horizon may be dynamically adapted by periodically solving new problem instances incorporating actual observation outcomes from previous episodes over receding horizons. Computational results clearly show the value of the approach in comparison to a myopic heuristic over various problem instances.

Integrated operations scheduling with delivery deadlines

The generalized assignment problem (GAP) determines the minimum cost assignment of n jobs to m agents such that each job is assigned to exactly one agent, subject to an agent's capacity. Existing solution algorithms have not solved problems with more than 100 decision variables. This paper designs an optimization algorithm for the GAP that effectively solves problems with up to 500 variables. Compared with existing procedures, this algorithm requires fewer enumeration nodes and shorter running times. Improved performance stems from: an enhanced Lagrangian dual ascent procedure that solves a Lagrangian dual at each enumeration node; adding a surrogate constraint to the Lagrangian relaxed model: and an elaborate branch-and-bound scheme. An empirical investigation of various problem structures, not considered in existing literature, is also presented.

The location-routing problem (LRP) considers facility-location and vehicle-routing decisions. These decisions have to be tackled simultaneously since they are interdependent. In this work, a two-stage LRP is considered wherein customers are to be served from a number of plants through a number of depots. Since LRP is an NP-hard problem, heuristic approach becomes the only alternative to solve the problem as the problem size increases. An ant colony optimisation (ACO)-based heuristic is proposed to solve the problem. For the purpose of performance evaluation of the proposed heuristic, various problem sizes are generated and the computational results are analysed. The results of this study reveal that the proposed heuristic is efficient for solving location routing problem in a two stage supply chain.

Matheuristics have become widespread and effective methods for tackling the generalized assignment problem (GAP) and many other NP-hard problems. In fact, in this book we have many such methods, ranging from metaheuristics and mathematical programming techniques but mainly to real matheuristics. In these methods we will see no parameter settings, but it is true: all these methods in the end rely on a good parameter setting. Here, in this chapter, we will learn of what can be seen as a good parameter setting. After having seen the parameters and the parameter configuration problem, we turn to the automatic algorithm configuration and then learn about three methods: ParamILS, SMAC, and irace. Then we learn about what these methods are good for and go into some details of automated configuration of existing algorithms, the integration of an algorithm engineering process up to the design of, in part, completely new algorithms. The chapter ends with a little example of how irace can be used to automatically configure a simple iterated local search method for the GAP.

Hierarchical design of an integrated production and 2-echelon distribution system