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.     

Lagrangian relaxation guided problem space search heuristics for generalized assignment problems

We develop and test a heuristic based on Lagrangian relaxation and problem space search to solve the generalized assignment problem (GAP). The heuristic combines the iterative search capability of subgradient optimization used to solve the Lagrangian relaxation of the GAP formulation and the perturbation scheme of problem space search to obtain high-quality solutions to the GAP. We test the heuristic using different upper bound generation routines developed within the overall mechanism. Using the existing problem data sets of various levels of difficulty and sizes, including the challenging largest instances, we observe that the heuristic with a specific version of the upper bound routine works well on most of the benchmark instances known and provides high-quality solutions quickly. An advantage of the approach is its generic nature, simplicity, and implementation flexibility.     

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.     

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.     

Due-date assignment and maintenance activity scheduling problem

In the scheduling problem addressed in this note we have to determine: (i) the job sequence, (ii) the (common) due-date, and (iii) the location of a rate modifying (maintenance) activity. Jobs scheduled before (after) the due-date are penalized according to their earliness (tardiness) value. The processing time of a job scheduled after the maintenance activity decreases by a job-dependent factor. The objective is minimum total earliness, tardiness and due-date cost. We introduce a polynomial (O(n⁴)) solution for the problem.     

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.     

The generalized assignment problem with flexible jobs

The Generalized Assignment Problem (GAP) seeks an allocation of jobs to capacitated resources at minimum total assignment cost, assuming a job cannot be split among multiple resources. We consider a generalization of this broadly applicable problem in which each job must not only be assigned to a resource, but its resource consumption must also be determined within job-specific limits. In this profit-maximizing version of the GAP, a higher degree of resource consumption increases the revenue associated with a job. Our model permits a job’s revenue per unit resource consumption to decrease as a function of total resource consumption, which allows modeling quantity discounts. The objective is then to determine job assignments and resource consumption levels that maximize total profit. We develop a class of heuristic solution methods, and demonstrate the asymptotic optimality of this class of heuristics in a probabilistic sense.     

GRASP with path-relinking for the generalized quadratic assignment problem

The generalized quadratic assignment problem (GQAP) is a generalization of the NP-hard quadratic assignment problem (QAP) that allows multiple facilities to be assigned to a single location as long as the capacity of the location allows. The GQAP has numerous applications, including facility design, scheduling, and network design. In this paper, we propose several GRASP with path-relinking heuristics for the GQAP using different construction, local search, and path-relinking procedures. We introduce a novel approximate local search scheme, as well as a new variant of path-relinking that deals with infeasibilities. Extensive experiments on a large set of test instances show that the best of the proposed variants is both effective and efficient.     

Effective algorithm and heuristic for the generalized assignment problem

We present new Branch-and-Bound algorithm for the generalized assignment problem. A standard subgradient method (SM), used at each node of the decision tree to solve the Lagrangian dual, provides an upper bound. Our key contribution in this paper is a new heuristic, applied at each iteration of the SM, which tries to exploit the solution of the relaxed problem, by solving a smaller generalized assignment problem. The feasible solution found is then subjected to a solution improvement heuristic. We consider processing the root node as a Lagrangian heuristic. Computational comparisons are made with new existing methods.     

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.     

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.     

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.     

Surgical case scheduling as a generalized job shop scheduling problem

Surgical case scheduling allocates hospital resources to individual surgical cases and decides on the time to perform the surgeries. This task plays a decisive role in utilizing hospital resources efficiently while ensuring quality of care for patients. This paper proposes a new surgical case scheduling approach which uses a novel extension of the Job Shop scheduling problem called multi-mode blocking job shop (MMBJS). It formulates the MMBJS as a mixed integer linear programming (MILP) problem and discusses the use of the MMBJS model for scheduling elective and add-on cases. The model is illustrated by a detailed example, and preliminary computational experiments with the CPLEX solver on practical-sized instances are reported.     

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.     

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.