A Tabu search heuristic for the generalized assignment problem

This paper considers the generalized assignment problem (GAP). It is a well-known NP-hard combinatorial optimization problem that is interesting in itself and also appears as a subproblem in other problems of practical importance. A Tabu search heuristic for the GAP is proposed. The algorithm uses recent and medium-term memory to dynamically adjust the weight of the penalty incurred for violating feasibility. The most distinctive features of the proposed algorithm are its simplicity and its flexibility. These two characteristics result in an algorithm that, compared to other well-known heuristic procedures, provides good quality solutions in competitive computational times. Computational experiments have been performed in order to evaluate the behavior of the proposed algorithm.     

Bees algorithm for generalized assignment problem

Bees algorithm (BA) is a new member of meta-heuristics. BA tries to model natural behavior of honey bees in food foraging. Honey bees use several mechanisms like waggle dance to optimally locate food sources and to search new ones. This makes them a good candidate for developing new algorithms for solving optimization problems. In this paper a brief review of BA is first given, afterwards development of a BA for solving generalized assignment problems (GAP) with an ejection chain neighborhood mechanism is presented. GAP is a NP-hard problem. Many meta-heuristic algorithms were proposed for its solution. So far BA is generally applied to continuous optimization. In order to investigate the performance of BA on a complex integer optimization problem, an attempt is made in this paper. An extensive computational study is carried out and the results are compared with several algorithms from the literature.     

An approximation algorithm for the generalized assignment problem

The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing jobj on machinei requires timep _ij and incurs a cost ofc _ij ; each machinei is available forT _i time units, and the objective is to minimize the total cost incurred. Our main result is as follows. There is a polynomial-time algorithm that, given a valueC, either proves that no feasible schedule of costC exists, or else finds a schedule of cost at mostC where each machinei is used for at most 2T _i time units.We also extend this result to a variant of the problem where, instead of a fixed processing timep _ij , there is a range of possible processing times for each machine—job pair, and the cost linearly increases as the processing time decreases. We show that these results imply a polynomial-time 2-approximation algorithm to minimize a weighted sum of the cost and the makespan, i.e., the maximum job completion time. We also consider the objective of minimizing the mean job completion time. We show that there is a polynomial-time algorithm that, given valuesM andT, either proves that no schedule of mean job completion timeM and makespanT exists, or else finds a schedule of mean job completion time at mostM and makespan at most 2T. Research partially supported by an NSF PYI award CCR-89-96272 with matching support from UPS, and Sun Microsystems, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.Research supported in part by a Packard Fellowship, a Sloan Fellowship, an NSF PYI award, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

An algorithm for the generalized quadratic assignment problem

This paper reports on a new algorithm for the Generalized Quadratic Assignment problem (GQAP). The GQAP describes a broad class of quadratic integer programming problems, wherein M pair-wise related entities are assigned to N destinations constrained by the destinations’ ability to accommodate them. This new algorithm is based on a Reformulation Linearization Technique (RLT) dual ascent procedure. Experimental results show that the runtime of this algorithm is as good or better than other known exact solution methods for problems as large as M=20 and N=15. Current address of P.M. Hahn: 2127 Tryon Street, Philadelphia, PA 19146-1228, USA.     

A branch and bound algorithm for the generalized assignment problem

This paper describes what is termed the generalized assignment problem. It is a generalization of the ordinary assignment problem of linear programming in which multiple assignments of tasks to agents are limited by some resource available to the agents. A branch and bound algorithm is developed that solves the generalized assignment problem by solving a series of binary knapsack problems to determine the bounds. Computational results are cited for problems with up to 4 000 0–1 variables, and comparisons are made with other algorithms.This research was partly supported by ONR Contracts N00014-67-A-0126-0008 and N00014-67-A-0126-0009 with the Center for Cybernetic Studies, The University of Texas.     

The equilibrium generalized assignment problem and genetic algorithm

The well-known generalized assignment problem (GAP) is to minimize the costs of assigning n jobs to m capacity constrained agents (or machines) such that each job is assigned to exactly one agent. This problem is known to be NP-hard and it is hard from a computational point of view as well. In this paper, follows from practical point of view in real systems, the GAP is extended to the equilibrium generalized assignment problem (EGAP) and the equilibrium constrained generalized assignment problem (ECGAP). A heuristic equilibrium strategy based genetic algorithm (GA) is designed for solving the proposed EGAP. Finally, to verify the computational efficiency of the designed GA, some numerical experiments are performed on some known benchmarks. The test results show that the designed GA is very valid for solving EGAP.     

Design of a heuristic algorithm for the generalized multi-objective set covering problem

Computational Optimization and Applications - Set covering optimization problems (SCPs) are important and of broad interest because of their extensive applications in the real world. This study...     

Reducing the elastic generalized assignment problem to the standard generalized assignment problem

The elastic generalized assignment problem (eGAP) is a natural extension of the generalized assignment problem (GAP) where the capacities are not fixed but can be adjusted; this adjustment can be expressed by continuous variables. These variables might be unbounded or restricted by a lower or upper bound, respectively. This paper concerns techniques aiming at reducing several variants of eGAP to GAP, which enables us to employ standard approaches for the GAP. This results in a heuristic, which can be customized in order to provide solutions having an objective value arbitrarily close to the optimal.     

A heuristic procedure for solving the quadratic assignment problem

An heuristic approach to the solution of the quadratic assignment problem is presented. A simple procedure is used to get a good feasible starting point, then the problem is solved as a nonlinear program (ignoring the integrality conditions) using MINOS, and lastly the near integer solution is converted into an integer feasible solution using an heuristic procedure. The results compare favourably with other procedures in the literature. A superior solution to the 19 × 19 hospital layout problem is found.     

Relaxation heuristics for a generalized assignment problem

We propose relaxation heuristics for the problem of maximum profit assignment of n tasks to m agents (n > m), such that each task is assigned to only one agent subject to capacity constraints on the agents. Using Lagrangian or surrogate relaxation, the heuristics perform a subgradient search obtaining feasible solutions. Relaxation considers a vector of multipliers for the capacity constraints. The resolution of the Lagrangian is then immediate. For the surrogate, the resulting problem is a multiple choice knapsack that is again relaxed for continuous values of the variables, and solved in polynomial time. Relaxation multipliers are used with an improved heuristic of Martello and Toth or a new constructive heuristic to find good feasible solutions. Six heuristics are tested with problems of the literature and random generated problems. Best results are less than 0.5% from the optimal, with reasonable computational times for an AT/386 computer. It seems promising even for problems with correlated coefficients.     

Model-hierarchical column generation and heuristic for the routing and wavelength assignment problem

The routing and wavelength assignment (RWA) problem typically occurs in wavelength division multiplexing optical networks. Given a number of available wavelengths, we consider here the problem of maximising the number of accepted connections with respect to the clash and continuity constraints. We first propose a new strategy which combines two existing models. This leads to an improved column generation scheme. We also present two heuristics to compute feasible solutions: a hybrid heuristic and the integer solution at the root node of the column generation. Our approaches are compared with the best existing results on a set of classic RWA instances.     

A heuristic algorithm for the truckload and less-than-truckload problem

The delivery of goods from a warehouse to local customers is an important and practical problem of a logistics manager. In reality, we are facing the fluctuation of demand. When the total demand is greater than the whole capacity of owned trucks, the logistics managers may consider using an outsider carrier.Logistics managers can make a selection between a truckload (a private truck) and a less-than-truckload carrier (an outsider carrier). Selecting the right mode to transport a shipment may bring significant cost savings to the company.In this paper, we address the problem of routing a fixed number of trucks with limited capacity from a central warehouse to customers with known demand. The objective of this paper is developing a heuristic algorithm to route the private trucks and to make a selection of less-than-truckload carriers by minimizing a total cost function. Both the mathematical model and the heuristic algorithm are developed. Finally, some computational results and suggestions for future research are presented.     

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.     

The elastic generalized assignment problem

The generalized assignment problem (GAP) has been studied by numerous researchers over the past 30 years or so. Simply stated, one must find a minimum-cost assignment of tasks to agents such that each task is assigned to exactly one agent and such that each agent's resource capacity is honoured. The problem is known to be NP-hard. In this paper, we study the elastic generalized assignment problem (EGAP). The elastic version of GAP allows agent resource capacity to be violated at additional cost. Another version allows undertime costs to be assessed as well if an agent's resource capacity is not used to its full extent. The EGAP is also NP-hard. We describe a special-purpose branch-and-bound algorithm that utilizes linear programming cuts, feasible solution generators, Lagrangean relaxation and subgradient optimization. We present computational results on a large collection of randomly generated ‘hard’ problems with up to 4000 binary variables.     

A new algorithm for the assignment problem

We propose a new algorithm for the classical assignment problem. The algorithm resembles in some ways the Hungarian method but differs substantially in other respects. The average computational complexity of an efficient implementation of the algorithm seems to be considerably better than the one of the Hungarian method. In a large number of randomly generated problems the algorithm has consistently outperformed an efficiently coded version of the Hungarian method by a broad margin. The factor of improvement increases with the problem dimensionN and reaches an order of magnitude forN equal to several hundreds.Work supported by Grant NSF ENG-7906332.     

A heuristic algorithm for the strip packing problem

The two-dimensional strip packing problem is to pack a given set of rectangles into a strip with a given width and infinite height so as to minimize the required height of the packing. From the computational point of view, the strip packing problem is an NP-hard problem. With the B*-tree representation, this paper first presents a heuristic packing strategy which evaluates the positions used by the rectangles. Then an effective local search method is introduced to improve the results and a heuristic algorithm (HA) is further developed to find a desirable solution. Computational results on randomly generated instances and popular test instances show that the proposed method is efficient for the strip packing problem.     

A survey of algorithms for the generalized assignment problem

This paper surveys algorithms for the well-known problem of finding the minimum cost assignment of jobs to agents so that each job is assigned exactly once and agents are not overloaded. All approaches seem to be based on branch-and-bound with bound supplied through heuristics and through relaxations of the primal problem formulation. From the survey one can select building blocks for the design of one's own tailor-made algorithm. The survey also reveals that although just about every mathematical programming technique was tried on this problem, there is still a lack of a representative set of test problems on which competing enumeration algorithms can be compared, as well as a shortage of effective heuristics.     

Generalized cover facet inequalities for the generalized assignment problem

The generalized assignment problem is that of finding an optimal assignment of agents to tasks, where each agent may be assigned multiple tasks and each task is performed exactly once. This is an NP-complete problem. Algorithms that employ information about the polyhedral structure of the associated polytope are typically more effective for large instances than those that ignore the structure. A class of generalized cover facet-defining inequalities for the generalized assignment problem is derived. These inequalities are based upon multiple knapsack constraints and are derived from generalized cover inequalities.