Generalized Assignment Problem Research Articles

Supplier selection for global service providers: a decision support system

In this paper, we develop a decision support system (DSS) aimed at solving a real-world supplier selection problem (SSP) for a global service provider (GSP) operating in the facility management (FM) industry. The GSP provides its customers with FM services, which are subcontracted to external suppliers selected on the basis of multiple criteria, like economic soundness, quality of service, capacity, and closeness. The SSP is formulated as a multi-objective generalized assignment problem, where the quality and the closeness of the selected suppliers are maximized, whereas a penalty produced by overcapacity assignments is minimized. The quality of each supplier is computed by applying a weighted sum method, resulting from a multi-criteria decision analysis in which the criteria weights are determined through an Analytic Hierarchy Process. The DSS is developed using a modular architecture with a relational database, a supplier evaluator, and a simulator, as well as an additional user-friendly interface. The simulator relies on a rolling horizon algorithm and three alternative configurations to assign contracts to suppliers. The effectiveness of the DSS is assessed by means of extensive computational experiments on historical data. The results show a significant average improvement of 25% compared to the solution adopted by the company.

Integrating Statistical Simulation and Optimization for Redundancy Allocation in Smart Grid Infrastructure

It is a critical issue to allocate redundancy to critical smart grid infrastructure for disaster recovery planning. In this study, a framework to combine statistical prediction methods and optimization models for the optimal redundancy allocation problem is presented. First, statistical simulation methods to identify critical nodes of very large-scale smart grid infrastructure based on the topological features of embedding networks are developed, and then a linear integer programming model based on generalized assignment problem (GAP) for the redundancy allocation of critical nodes in smart grid infrastructure is presented. This paper aims to contribute to the field by employing a general redundancy allocation problem (GRAP) model from high-order nonlinear to linear model transformation. The model is specifically implemented in the context of smart grid infrastructure. The innovative linear integer programming model proposed in this paper capitalizes on the logarithmic multiplication property to reframe the inherently nonlinear resource allocation problem (RAP) into a linearly separable function. This reformulation markedly streamlines the problem, enhancing its suitability for efficient and effective solutions. The findings demonstrate that the combined approach of statistical simulation and optimization effectively addresses the size limitations inherent in a sole optimization approach. Notably, the optimal solutions for redundancy allocation in large grid systems highlight that the cost of redundancy is only a fraction of the economic losses incurred due to weather-related outages.

Optimization of Harvesting Priority of Sugarcane Farms by the Generalized Assignment Problem

Optimization of Harvesting Priority of Sugarcane Farms by the Generalized Assignment Problem

Approximation algorithms for scheduling parallel machines with an energy constraint in green manufacturing

Motivated by current green manufacturing standards, in this paper we study a parallel-machine scheduling model in which the energy cost incurred on each machine is machine-dependent and proportional to the load of the machine. The objective is to determine a production schedule with the minimum makespan subject to the energy constraint that the total energy cost does not exceed a given bound. We provide a technical lemma that enables us to design a very efficient approximation algorithm with a worst-case bound that can arbitrarily approach 43, improving on the existing performance ratio of 33+14≈1.686 in the literature. By introducing the concept of a monotonic schedule, we are able to develop the first polynomial time approximation scheme for this scheduling problem. The scheduling problem studied in this paper is an important special case of the generalized assignment problem (GAP). Our techniques and results bring new insights into research on the GAP.

A three-phase matheuristic algorithm for the multi-day task assignment problem

This paper considers a multi-day task assignment model that introduces several features of practical relevance into the widely-studied generalized assignment problem. This model includes a significantly increased number of variables and constraints compared to the task assignment models investigated in the literature and thus is computationally challenging. For solving this problem, we propose an innovative three-phase matheuristic algorithm that first employs a construction phase to quickly produce a reasonable quality solution and then alternates between an intensification phase to reach local optima and a diversification phase to drive the search into new regions. The construction phase decomposes the original problem into a sequence of smaller subproblems, solves each subproblem with the Gurobi optimizer, and aggregates the solutions from the subproblems to produce a feasible solution. The intensification phase executes an iterative variable fixing heuristic that divides the solution space into different neighborhoods and iteratively explores each neighborhood by solving the reduced model. The diversification phase solves a modified model that adds a distance component into the original objective function. Computational experiments demonstrate that our proposed algorithm outperforms Gurobi, LocalSolver and Tabu Search in terms of both solution quality and computational time. The best solutions found by our algorithm have percentage gaps to the upper bounds (attained by Gurobi and LocalSolver) ranging from 0.82% to 2.79%, indicating that they are very close to the optimal solutions. In addition, experimental analysis has been carried out to identify the impact of some of the key components of the proposed algorithm which are contributing to its superior performance. The benchmark instances generated for our study are made available to the public for future research works on this problem.

Learning Individualized Hyperparameter Settings

The performance of optimization algorithms, and consequently of AI/machine learning solutions, is strongly influenced by the setting of their hyperparameters. Over the last decades, a rich literature has developed proposing methods to automatically determine the parameter setting for a problem of interest, aiming at either robust or instance-specific settings. Robust setting optimization is already a mature area of research, while instance-level setting is still in its infancy, with contributions mainly dealing with algorithm selection. The work reported in this paper belongs to the latter category, exploiting the learning and generalization capabilities of artificial neural networks to adapt a general setting generated by state-of-the-art automatic configurators. Our approach differs significantly from analogous ones in the literature, both because we rely on neural systems to suggest the settings, and because we propose a novel learning scheme in which different outputs are proposed for each input, in order to support generalization from examples. The approach was validated on two different algorithms that optimized instances of two different problems. We used an algorithm that is very sensitive to parameter settings, applied to generalized assignment problem instances, and a robust tabu search that is purportedly little sensitive to its settings, applied to quadratic assignment problem instances. The computational results in both cases attest to the effectiveness of the approach, especially when applied to instances that are structurally very different from those previously encountered.

Dynamic Reliability Management of Multigateway IoT Edge Computing Systems

The emerging paradigm of edge computing envisions to overcome the shortcomings of cloud-centric Internet of Things (IoT) by providing data processing and storage capabilities closer to the source of data. Accordingly, IoT edge devices, with the increasing demand of computation workloads on them, are prone to failures more than ever. Hard failures in hardware due to aging and reliability degradation are particularly important since they are irrecoverable, requiring maintenance for the replacement of defective parts, at high costs. In this article, we propose a novel dynamic reliability management (DRM) technique for multigateway IoT edge computing systems to mitigate degradation and defer early hard failures. Taking advantage of the edge computing architecture, we utilize gateways for computation offloading with the primary goal of maximizing the battery lifetime of edge devices, while satisfying the Quality of Service (QoS) and reliability requirements. We present a two-level management scheme, which work together to 1) choose the offloading rates of edge devices; 2) assign edge devices to gateways; and 3) decide multihop data flow routes and rates in the network. The offloading rates are selected by a hierarchical multitimescale distributed controller. We assign edge devices by solving a bottleneck generalized assignment problem (BGAP) and compute optimal flows in a fully distributed fashion, leveraging the subgradient method. Our results, based on real measurements and trace-driven simulation, demonstrate that the proposed scheme can achieve a similar battery lifetime and better QoS compared to the state-of-the-art approaches while satisfying reliability requirements, where other approaches fail by a large margin.

Sensitivity analysis of combinatorial optimization problems using evolutionary bilevel optimization and data mining

Sensitivity analysis in general deals with the question of how changes in input data of a model affect its output data. In the context of optimization problems, such an analysis could, for instance, address how changes in capacity constraints affect the optimal solution value. Although well established in the domain of linear programming, sensitivity analysis approaches for combinatorial optimization problems are model-specific, limited in scope and not applicable to practical optimization problems. To overcome these limitations, Schulte et al. developed the concept of bilevel innovization. By using evolutionary bilevel optimization in combination with data mining and visualization techniques, bilevel innovization provides decision-makers with deeper insights into the behavior of the optimization model and supports decision-making related to model building and configuration. Originally introduced in the field of evolutionary computation, most recently bilevel innovization has been proposed as an approach to sensitivity analysis for combinatorial problems in general. Based on previous work on bilevel innovization, our paper illustrates this concept as a tool for sensitivity analysis by providing a comprehensive analysis of the generalized assignment problem. Furthermore, it is investigated how different algorithms for solving the combinatorial problem affect the insights gained by the sensitivity analysis, thus evaluating the robustness and reliability of the sensitivity analysis results.

Dimension reduction for a multi-resource general assignment problem by decomposable costs for a vehicle compound

In the finished vehicle logistics on vehicle compounds, the handling requires a high quote of personnel requirement, i.e. workers, who transfer the vehicles from or to traffic carriers. For an efficient assignment of orders to workers and shuttles, which transfer the workers across the terminal, this problem can be formulated as a (3D-)multi-resource general assignment problem to obtain a time-minimal solution of workers to orders and an optional shuttle usage. However, as this optimisation problem faces the curse of dimensionality with increasing numbers of workers, shuttles and orders, the assignment problem is decomposed into several 2D-sub-problems to ensure a faster execution and hence, allow for larger problem sizes. We describe the approach to split up the problem, develop the mathematical formulation, and present the simulations for both approaches and compare their results.

Surrogate “Level-Based” Lagrangian Relaxation for mixed-integer linear programming

Mixed-Integer Linear Programming (MILP) plays an important role across a range of scientific disciplines and within areas of strategic importance to society. The MILP problems, however, suffer from combinatorial complexity. Because of integer decision variables, as the problem size increases, the number of possible solutions increases super-linearly thereby leading to a drastic increase in the computational effort. To efficiently solve MILP problems, a “price-based” decomposition and coordination approach is developed to exploit 1. the super-linear reduction of complexity upon the decomposition and 2. the geometric convergence potential inherent to Polyak’s stepsizing formula for the fastest coordination possible to obtain near-optimal solutions in a computationally efficient manner. Unlike all previous methods to set stepsizes heuristically by adjusting hyperparameters, the key novel way to obtain stepsizes is purely decision-based: a novel “auxiliary” constraint satisfaction problem is solved, from which the appropriate stepsizes are inferred. Testing results for large-scale Generalized Assignment Problems demonstrate that for the majority of instances, certifiably optimal solutions are obtained. For stochastic job-shop scheduling as well as for pharmaceutical scheduling, computational results demonstrate the two orders of magnitude speedup as compared to Branch-and-Cut. The new method has a major impact on the efficient resolution of complex Mixed-Integer Programming problems arising within a variety of scientific fields.

The parallel AGV scheduling problem with battery constraints: A new formulation and a matheuristic approach

Nowadays, automated guided vehicles (AGVs) are frequently used in larger systems, known as AGV-based transportation systems, for the movement of goods and materials from one location to another. The design of an efficient and effective AGV-based transportation system requires to address many tactical and operational issues. Among the others, the scheduling of transfer jobs on the AGVs represents one of the main operational issues that has to be solved to overcome delays in production and material handling processes. In this context, a significant research activity on AGV systems and on related scheduling problems has been conducted in the last twenty years. However, most of the contributions neglected the issues related to the AGV battery depletion and recharge. Thus, in this work, we study the AGV scheduling problem with battery constraints (ASP-BC). It consists in determining the scheduling of transfer jobs and charging operations of a fleet of homogeneous AGVs such that the makespan of the handling process is minimized. The methodological contribution of our work is twofold. On one side, we propose an original mixed integer linear programming formulation based on the bottleneck generalized assignment problem. On the other side, we propose a three step matheuristic based on the sequential solution of the two subproblems arising from the natural decomposition of the ASP-BC and a local search heuristic. The proposed approaches have been tested and validated on simulated and real instances provided by a manufacturing company. The results show the effectiveness and the scalability of the proposed solution methods.

Online generalized assignment problem with historical information

The rapid development of online platforms has inspired a wide range of applications for timely resources allocations, such as the hotel booking, the cargo logistics, the cloud servers and so on. Motivated by such needs, we study the online versions of the famous generalized assignment problem (GAP) and the packing problem (also known as d-GAP) in the classic random order model, where the online items arrive over time randomly and uniformly and request specific offline resources. Along a recent line of research that uses historical information to improve the performance of online algorithms, we design effective competitive algorithms for both online GAP and d-GAP (d⩾2) with augmentation of historical information. Our algorithms are inspired by Albers et al.’s sequential approach (Albers et al., 2021). If no historical information can be accessed, our algorithm for online GAP reduces to Albers et al.’s algorithm, and our algorithm for online d-GAP (d⩾2) outperforms the current best algorithm (Kesselheim et al., 2018). The practical performance of the proposed algorithms is explored via experiments on both synthetic and real-life datasets. In particular, the positive effect of historical information can be verified by the experiment results.

A Study of Team Recommended Generalized Assignment Methods

This study considers the team recommendation problem as a generalized assignment problem. Firstly, a formal description of the team recommendation problem is given; secondly, a team-recommended generalized assignment model (TRGAM) is established based on the work ability value of alternative members, the comprehensive work ability value of the team as the core concern index, the importance weight of team tasks and the energy allocation weight of team members as the fusion strategy of the data; thirdly, a solution method for the standard case of TRGAM is designed using the enumeration method and Hungarian algorithms (BEM⊕HM–TRGAMs) as local computational tools; fourthly, the alternative member set refinement methods and standardization measures for TRGAM are given; finally, BEM⊕HM–TRGAMs are analyzed using specific arithmetic examples. The theoretical analysis and experimental results show that TRGAM has good structural features and interpretability and BEM⊕HM–TRGAMs can effectively solve the TRGAM solving problem.

A time-triggered dimension reduction algorithm for the task assignment problem

The task assignment problem is fundamental in combinatorial optimisation, aiming at allocating one or more tasks to a number of agents while minimizing the total cost or maximizing the overall assignment benefit. This problem is known to be computationally hard since it is usually formulated as a mixed-integer programming problem. In this paper, we consider a novel Time-Triggered Dimension Reduction Algorithm (TTDRA). We propose convexification approaches to convexify both the constraints and the cost function for the general non-convex assignment problem. The computational speed is accelerated via our time-triggered dimension reduction scheme, where the triggering condition is designed based on the optimality tolerance and the convexity of the cost function. Optimality and computational efficiency are verified via numerical simulations on benchmark examples.

Generalized Assignment for Multi-Robot Systems via Distributed Branch-And-Price

In this article, we consider a network of agents that has to self-assign a set of tasks while respecting resource constraints. One possible formulation is the generalized assignment problem, where the goal is to find a maximum payoff while satisfying capability constraints. We propose a purely distributed branch-and-price algorithm to solve this problem in a cooperative fashion. Inspired by classical (centralized) branch-and-price schemes, in the proposed algorithm, each agent locally solves small linear programs, generates columns by solving simple knapsack problems, and communicates to its neighbors a fixed number of basic columns. We prove finite-time convergence of the algorithm to an optimal solution of the problem. Then, we apply the proposed scheme to a generalized assignment scenario, in which a team of robots has to serve a set of tasks. We implement the proposed algorithm in a Robot Operating System testbed and provide experiments for a team of heterogeneous robots solving the assignment problem.

Solving the reliability-oriented generalized assignment problem by Lagrangian relaxation and Alternating Direction Method of Multipliers

The well-known generalized assignment problem has many real-world applications. The assignment costs between agents and tasks affected by several factors could be unstable and uncertain. In this paper, we assume that the means and variances of assignment costs are known in advance. The idea is that the decision-maker aims to minimize the total assignment costs not only on average, but also to keep their variability as small as possible. Then, a reliability-oriented model with a nonlinear and concave objective function is formulated, and two decomposition-based methods are systematically developed for this challenging problem. The task allocation constraints are dualized and the Lagrangian relaxed problem is broken into many reliability-oriented knapsack subproblems. By the Lagrangian substitution technique, each subproblem is further decomposed into a standard knapsack problem and a simple univariate concave minimization problem. A lower bound is constructed and multipliers are optimized in dual problems by the subgradient method. Feasible solutions can be generated from the results of these reliability-oriented subproblems by Lagrangian heuristic. To further improve the convergence and solution quality, an Alternating Direction Method of Multipliers (ADMM) based decomposition approach is proposed. The augmented Lagrangian relaxed problem is split into a sequence of subproblems by the block coordinate descent method. To cope with the quadratic terms and the concave terms in these subproblems, linearization and Lagrangian substitution techniques are applied. Feasible solutions are produced from these subproblems, and solution quality can be evaluated by the lower bound provided by the Lagrangian relaxed problem. Numerical experiments are conducted on the test cases transformed from the standard benchmark instances, and the ADMM-based method has superior convergence and solution quality.

An Iterated Dual Substitution Approach for Binary Integer Programming Problems Under the Min-Max Regret Criterion

We consider binary integer programming problems with the min-max regret objective function under interval objective coefficients. We propose a heuristic framework, the iterated dual substitution (iDS) algorithm, which iteratively invokes a dual substitution heuristic and excludes from the search space any solution already checked in previous iterations. In iDS, we use a best scenario–based lemma to improve performance. We apply iDS to four typical combinatorial optimization problems: the knapsack problem, the multidimensional knapsack problem, the generalized assignment problem, and the set covering problem. For the multidimensional knapsack problem, we compare the iDS approach with two algorithms widely used for problems with the min-max regret criterion: a fixed-scenario approach, and a branch-and-cut approach. The results of computational experiments on a broad set of benchmark instances show that the proposed iDS approach performs best on most tested instances. For the knapsack problem, the generalized assignment problem, and the set covering problem, we compare iDS with state-of-the-art results. The iDS algorithm successfully updates best-known records for a number of benchmark instances. Summary of Contribution: This paper proposes a heuristic framework for binary integer programming (BIP) problems with the min-max regret objective function under interval objective coefficients. We selected four representative NP-hard combinatorial optimization problems: the knapsack problem, the multidimensional knapsack problem, the set covering problem, and the generalized assignment problem. We show the effectiveness and efficiency of the approach by comparing with state-of-the-art results.

Combining data reduction, MIP solver and iterated local search for generalized assignment

ABSTRACT This work is motivated by a memory allocation problem (MAP) in an embedded system which is modeled as a generalized assignment problem (GAP) with side constraints. Thus, this paper aims to design a fast and sufficiently accurate method for GAP that will be used in future research to obtain a solution for MAP. Since this solution barely satisfies the conflict constraints, it will be repaired and then improved using local search . In the current research, the proposed method for GAP combines data reduction, a MIP solver, and iterated local search (ILS). The generic data reduction procedure is based on useful information provided by applying subgradient optimization to the Lagrangian relaxation of the knapsack constraints. In practice, it can eliminate up to of GAP variables. The reduced GAP left by this procedure is small and sparse. Furthermore, its optimal solution can be easily extended to an optimal or near-optimal solution of GAP. An ILS metaheuristic is designed for solving the reduced GAP and the entire method is tested on a widely used benchmark of large and difficult instances. Its comparison with recently published methods shows that it is competitive in terms of solution quality; it is also by far the fastest.

Verification of multi-layered assignment problems

The class of assignment problems is a fundamental and well-studied class in the intersection of Social Choice, Computational Economics and Discrete Allocation. In a general assignment problem, a group of agents expresses preferences over a set of items, and the task is to allocate items to agents in an “optimal” way. A verification variant of this problem includes an allocation as part of the input, and the question becomes whether this allocation is “optimal”. In this paper, we generalize the verification variant to the setting where each agent is equipped with multiple incomplete preference lists: Each list (called a layer) is a ranking of items in a possibly different way according to a different criterion. In particular, we introduce three multi-layer verification problems, each corresponds to an optimality notion that weakens the notion of global optimality (that is, pareto optimality in multiple layers) in a different way. Informally, the first notion requires that, for each group of agents whose size is exactly some input parameter k, the agents in the group will not be able to trade their assigned items among themselves and benefit in at least \(\alpha \) layers; the second notion is similar, but it concerns all groups of size at most k rather than exactly k; the third notion strengthens these notions by requiring that groups of k agents will not be part of possibly larger groups that benefit in at least \(\alpha \) layers. We study the three problems from the perspective of parameterized complexity under several natural parameterizations such as the number of layers, the number of agents, the number of items, the number of allocated items, the maximum length of a preference list, and more. We present an almost comprehensive picture of the parameterized complexity of the problems with respect to these parameters.

Approximation algorithms for the generalized incremental knapsack problem

We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of n items, each associated with a non-negative weight, and T time periods with non-decreasing capacities \(W_1 \le \dots \le W_T\). When item i is inserted at time t, we gain a profit of \(p_{it}\); however, this item remains in the knapsack for all subsequent periods. The goal is to decide if and when to insert each item, subject to the time-dependent capacity constraints, with the objective of maximizing our total profit. Interestingly, this setting subsumes as special cases a number of recently-studied incremental knapsack problems, all known to be strongly NP-hard. Our first contribution comes in the form of a polynomial-time \((\frac{1}{2}-\epsilon )\)-approximation for the generalized incremental knapsack problem. This result is based on a reformulation as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys–Tardos algorithm for the generalized assignment problem. Combined with further enumeration-based self-reinforcing ideas and new structural properties of nearly-optimal solutions, we turn our algorithm into a quasi-polynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.