Discrete OptimizationLagrangian relaxation guided problem space search heuristics for generalized assignment problems
Introduction
The generalized assignment problem (GAP) has many applications and can be usually found as a subproblem within more complex problems. A non-exhaustive list of such problems includes computer network problems, vehicle routing and scheduling problems, and resource allocation and scheduling problems in manufacturing and service. We use machine scheduling terminology to introduce and present the development of the algorithm tested in the paper; this terminology and the associated approach are directly applicable to other domains with little change. In this machine scheduling environment, the GAP attempts to find the minimum-cost assignment of a set of jobs with certain resource requirements to a given set of capacitated machines, without violating the machines’ capacity limitations.
There are many exact algorithms and heuristics developed to solve the GAP. Exact algorithms include branch and bound-based algorithms, branch-and-cut and branch-and-price algorithms (e.g., Ross and Soland, 1975, Nauss, 2003, Savelsbergh, 1997). Some heuristics use the linear programming relaxation (Trick, 1992). Others use search techniques such as genetic algorithms (e.g., Chu and Beasley, 1997), and tabu search algorithms (e.g., Higgins, 2001, Díaz and Fernández, 2001, Yagiura et al., 2004). Lagrangian relaxation (LR) is arguably one of the heavily used methods to solve general integer programming problems (Fisher, 1981). Most papers on GAP also have applied Lagrangian relaxation to the GAP and relax one of the two constraint sets. Some of these papers have proposed combining LR with surrogate relaxation (e.g., Narciso and Lorena, 1999). The Lagrangian decomposition (LD) method has been also applied to GAP (e.g., Haddadi, 1999). More recently, LR and subgradient methods have been integrated in branch and bound schemes (e.g., Haddadi and Ouzia, 2001, Haddadi and Ouzia, 2004).
In this paper, we present another algorithm that combines LR and its associated subgradient optimization procedure with a search technique based on problem space search (PSS). PSS uses the problem input data perturbations (as opposed to traditional solution space search, where the search is directly over solutions). The idea of perturbation is not new and has been used in the literature before. Storer et al. (1992) present the first example of PSS within the context of scheduling. Charon and Hudry (1993) use a very similar noising method applied to clique partitioning problem. Their noising methods are based on elementary (or local) transformations applied to solution(s). Codenotti et al. (1996) use the perturbations applied to intermediate solutions of large-scale Euclidean TSP benchmark instances. Here the main difference is that PSS aims the “noising” or the “perturbations” applied to the problem data rather than intermediate solution(s). The main idea is to combine the strengths of LR and PSS to find improved solutions to the GAP.
Our goal is to present the general framework of the LR-PSS-based algorithm, especially in the context of the subgradient optimization used to solve the Lagrangian relaxation problem, and to show the effectiveness of the combined approach by applying it to an extensive set of challenging benchmark instances from the literature.
To give an overall idea, we first construct a Lagrangian relaxation of the GAP by relaxing the machine capacity constraints in the integer programming formulation of the GAP. The relaxed problem’s solution gives a lower bound on the optimal solution of the GAP, which can be improved using the subgradient optimization procedure. As the lower bound solutions are rarely feasible, the solution of the relaxed problem in each iteration of the procedure is also used to find heuristic solutions to the original GAP and produce upper bounds. As this “feasibility restoration” in fact does not guarantee capacity-feasible solutions due to its structure, a PSS-based enhancement is introduced: The problem data in each iteration is temporarily perturbed a given number of times to increase the chances of obtaining capacity-feasible solutions with potentially improved objective values. At algorithm termination, the best feasible solution and the best lower bound are reported. The two primary factors affecting the PSS enhancement are (1) the number of perturbations in each iteration, and (2) the amount of perturbation of the problem data. We test the effectiveness of the algorithm using the problem instances from the literature.
Section snippets
Literature review
Ross and Soland (1975) define the GAP and Fisher and Jaikumar (1981) proves that it is NP-hard. However, it is important to solve this problem efficiently as its applications are found in many real life problems addressing process assignments to computer networks (Balachandran, 1976), facility location (Ross and Soland, 1977), vehicle routing and its extensions such as constraints on total travel and delivery times for the vehicles (Fisher and Jaikumar, 1981, Baker and Sheasby, 1999).
The GAP is
Generalized assignment problem
We can define the GAP as assigning n jobs to a set of m capacitated resources/machines with minimum cost. Each machine i has a limited amount bi of capacity and each job j must be assigned to one and only one machine. Assigning job j to machine i (subject to the availability of machine i) requires aij units of machine i’s capacity, which costs cij when assigned. Let xij (defined for all i and j) be a binary decision variable indicating whether (xij = 1) or not (xij = 0) we assign job j to machine i
Feasibility restoration heuristic
The two ideas (denoted by LBR and RGT) for finding feasible assignments and calculating upper bounds are based on two different strategies. LBR is based on finding a good ‘machine’ for a randomly selected ‘job’ where the ‘goodness’ is derived from the lower bound solution, and RGT is based on finding a good ‘ordering’ of ‘job-machine’ pairs where the ordering is derived from a regret (or desirability) calculation. The hybrid version (HBR) combines the two ideas into a single scheme.
All versions
Overall algorithm
We have discussed how we compute lower bounds and upper bounds, and PSS-based neighborhood search for given λ. As the main input to all the aforementioned components of the algorithm, the quality of Lagrange multipliers has significant effect on the performance of the overall algorithm. A good set of multipliers should produce a good set of weights, which in turn should lead to good lower bound and upper bound solutions. Hence, the overall idea behind Lagrangian relaxation is to update Lagrange
Algorithmic settings
The development and successful implementation of new heuristics often require tuning the parameters of the algorithm, as their effects on the algorithmic performance are not exactly known. We too have several parameters in the overall generic feasibility restoration heuristic and in the subgradient optimization procedure. Their proper selection is crucial for the quality of the solutions, the lower bounds, and the overall computational burden and convergence of the algorithm. Once these are
Test problems
There are five types of benchmark GAP instances (Chu and Beasley, 1997, Laguna et al., 1995) called types A, B, C, D, and E, generally in the increasing order of difficulty. We have two sets of data, differentiated according to their source: (1) Well-known Beasley’s OR-library data set that consists of smaller, once-challenging instances, and (2) the more recent data set from Yagiura et al. (2004) that consists of more challenging and larger instances (downloaded from //www.al.cm.is.nagoya-u.ac.jp/~yagiura/gap/
Computational results
The LR-PSS-based algorithm is programmed in C++ and run under the Suse Linux (version 2.6.8–24.14-smp) operating system. We perform the computational runs on a PC with an AMD Athlon 1.8 GHz processor. We present the LR-PSS computational results in 3 tables and compare them with the results from Díaz and Fernández (2001) and even more recent results obtained by Yagiura et al. (2004). Their efficient Tabu-search-based heuristics outperform almost all previously known heuristics. Hence, we believe
Conclusions and future research directions
We develop and test a heuristic that combines ideas from Lagrangian relaxation, subgradient optimization, and problem space search for the generalized assignment problem. We instantiate versions of this generic heuristic using two specific methods to restore feasibility and find high-quality solutions over the iterations of subgradient optimization. Our extensive tests conducted using the most challenging GAP instances in the literature show that the PSS-enhanced Lagrangian heuristic within
Acknowledgements
We acknowledge the detailed and insightful comments of three anonymous referees on earlier versions of this paper.
References (43)
- et al.
Extensions to the generalized assignment heuristic for vehicle routing
European Journal of Operational Research
(1999) - et al.
A survey of algorithms for the generalized assignment problem
European Journal of Operational Research
(1992) - et al.
A set partitioning heuristic for the generalized assignment problem
European Journal of Operational Research
(1994) - et al.
The noising method: A new method for combinatorial optimization
Operations Research Letters
(1993) - et al.
A genetic algorithm for the generalized assignment problem
Computers and Operations Research
(1997) - et al.
A tabu search heuristic for the generalized assignment problem
European Journal of Operational Research
(2001) - et al.
Effective algorithm and heuristic for the generalized assignment problem
European Journal of Operational Research
(2004) A dynamic tabu search for large-scale generalized assignment problems
Computers and Operations Research
(2001)- et al.
Tabu search for the multilevel generalized assignment problem
European Journal of Operational Research
(1995) - et al.
Lagrangian/surrogate relaxation for generalized assignment problems
European Journal of Operational Research
(1999)
A class of greedy algorithms for the generalized assignment problem
Discrete Applied Mathematics
A path relinking approach with ejection chains for the generalized assignment problem
European Journal of Operational Research
A rigorous comparison of alternative solution methods for the generalized assignment problem
Management Science
An integer generalized transportation model for optimal job assignment in computer networks
Operations Research
On improving relaxation methods by modified gradient techniques
Mathematical Programming
An investigation of the relationship between problem characteristics and algorithm performance: A case study of the generalized assignment problems
IIE Transactions on Operations Engineering
Perturbation: An efficient technique for the solution of very large instances of the Euclidean TSP
INFORMS Journal on Computing
A guide to vehicle routing heuristics
Journal of the Operational Research Society
A family of inequalities for the generalized assignment polytope
Operations Research Letters
The Lagrangian relaxation method for solving integer programming problems
Management Science
A generalized assignment heuristic for vehicle routing
Networks
Cited by (37)
Solving the reliability-oriented generalized assignment problem by Lagrangian relaxation and Alternating Direction Method of Multipliers
2022, Expert Systems with ApplicationsCitation Excerpt :Poore and Rijavec (1993) and Poore and Robertson (1997) solved the multidimensional assignment problems by Lagrangian relaxation based algorithms. Jeet and Kutanoglu (2007) proposed a heuristic method based on Lagrangian relaxation and problem space search to generate high-quality feasible solutions. Posta, Ferland, and Michelon (2011) solved the GAP by a simple depth-first Lagrangian branch-and-bound approach, and variable-fixing rules are applied to prune the search tree.
An integrated algorithm for solving multi-customer joint replenishment problem with districting consideration
2020, Transportation Research Part E: Logistics and Transportation ReviewA comparative study of formulations for a cross-dock door assignment problem
2020, Omega (United Kingdom)Solving the Wire-Harness Design Problem at a European car manufacturer
2019, European Journal of Operational ResearchCitation Excerpt :Some greedy algorithms have been developed for the GAP, where weighted functions are designed to measure the desirability of assigning an object to a resource, and then the assignments are performed according to a decreasing order of desirability (Romeijn & Morales, 2000). Lagrangian heuristics were also proposed for the GAP (Jeet & Kutanoglu, 2007; Litvinchev, Mata, Rangel, & Saucedo, 2010). They consist in performing a Lagrangian relaxation of the problem before using subgradient optimization or a greedy heuristic to repair the relaxed solution.
Improved differential evolution algorithms for solving generalized assignment problem
2016, Expert Systems with Applications