Generalized Assignment Problem Research Articles

Improved Lagrangian bounds and heuristics for the generalized assignment problem

Modified Lagrangian bounds are proposed for the generalized assignment problem. The approach is based on a double decomposable structure of the formulation. A family of greedy heuristics is considered to get Lagrangian based feasible solutions. Numerical results for problem instances with number of agents close to number of tasks are provided.

Genelleştirilmiş Atama Probleminin Hypergraf Çözümü ve Uzaysal Veri Kümelerine Uygulaması

In the simple task assignment problem, at most one task should be assigned to each agent; this constraint is relaxed in the multiple task assignment problems. The goal of the well-known Generalized Assignment Problem is to assign tasks to agents such that the capacity of the agent does not exceed its limits as it minimizes the total cost. In this study we present a novel approach to solve the general assignment problem by using the hypergraphs. Hypergraphs can be considered as the generalization of the graphs in such way that an edge can connect any number of vertices, and can be seen as the set systems. In the hypergraph multi-assignment problem, we are looking for a cost minimizing solution to tasks assignment to the agents which are the individual hyperedges. For this purpose, we first determine the tasks as hyperedges and then obtain the vertex cover of the simple graph representation of a hypergraph. Amongst the all possible covers, we choose the cost minimizing one as the solution.

A Hybrid Heuristic in GPU-CPU Based on Scatter Search for the Generalized Assignment Problem

In the Generalized Assignment Problem (GAP), tasks must be allocated to machines with limited resources, in order to minimize processing costs. This problem has several industrial applications and often appears as a substructure in other combinatorial optimization problems. We propose a hybrid method inspired by Scatter Search metaheuristic, that efficiently generates a pool of solutions using a Tabu list criteria and an Ejection Chain mechanism. Common characteristics are extracted from the pool and solutions are combined by exploring a restricted search space, as a Binary Programming (BP) model. This method was implemented as a parallel approach to run in a Graphics Processing Unit (GPU). Experimental results show that the proposed method is very competitive to the algorithms found in the literature. On average, a gap of 0.09% is obtained over a set of 45 instances, when compared to lower bounds. Due to the integration of the method with an exact BP solver, it was capable of proving the optimality of small size instances, also finding new best known solutions for 21 instances. In terms of computational times, the proposed method performs on average 8 times faster than literature, also indicating that the proposed approach is scalable and robust for practical applications.

A modified genetic algorithm for a special case of the generalized assignment problem

A modified genetic algorithm for a special case of the generalized assignment problem

Generalized assignment problem: Truthful mechanism design without money

We propose truthful approximation mechanisms for strategic variants of the generalized assignment problem (GAP) in a payment-free environment. In GAP, a set of items has to be optimally assigned to a set of bins without exceeding the capacity of any singular bin. In our strategic variant, bins are held by strategic agents and each agent may hide its willingness to receive some items in order to obtain items of higher values. The model has applications in auctions with budgeted bidders.

Social network ad allocation and optimization: a geometric mapping-based approach

With the increasing popularity of online social networks (SNS), many advertisers choose to post their advertisements (ads) within SNS. Advertising activity on SNS has grown rapidly and is now a billion-dollar business. For example, Facebook has reached more than 1 million active advertisers who contribute 90% of its revenue. In the SNS advertising model, advertisers participating in a SNS ad campaign benefit from the effects of viral marketing and network diffusion. Modern SNS serve as advertising agents and take advantage of the network diffusion to attract advertisers and charge for the cascading impressions. The optimal ad allocation task is the problem of choosing the ad allocation plan that maximizes revenue for the SNS. Considering that users have diffusion abilities and limited daily impressions, and advertisers have various bidding prices and budget concerns, a feasible plan that obeys the constraints is difficult to find. The solution to this problem lies in the space of \({\mathbb {N}}_0^{|Ads|\times |Users|}\), which makes direct optimization unattractive. In this work, we study SNS advertising business models and formulate the SNS ad allocation problem. We show the problem is NP-hard and propose two dimension reduction schemes together with novel relaxation techniques. Our dimension reduction technique is formulated based on SNS user profile-based bidding scenario as well as social influence-based billing policies. We show the core ideas for dimension reduction are applicable to generalized assignment problems in bipartite graphs. We further draw connections between geometric mapping for complex network and the SNS ad allocation problem and map the SNS onto 2-D geometric space in order to relax the problem to geometric region allocation problems. We develop an optimization framework and solve the relaxed problem as a series of linear programs. Our proposed method is able to reduce the dimensionality of the original problem significantly, run two to four orders of magnitude faster, and reach 95% of the optimal solution. In addition, we discuss several extensions of our approach, including shape design in the geometric space for incorporating domain constraints in allocation strategies, more comprehensive real-world social influence models, as well as an alternative relaxation approach and its application in generalized assignment problems.

Heuristic routing algorithm toward scalable distributed generalized assignment problem

Distributed generalized assignment problem (D-GAP) is very popular in scalable multi-agent systems. However, existing algorithms are either not effective or efficient in large-scale or highly dynamic domains owing to limited communications and computational resources. In this paper, we present a novel approach named intelligent routing algorithm (IRA) to address this issue. In IRA, in order to reduce communication load, a decentralized model for agents is proposed to jointly search for optimized solutions. Moreover, due to the complexity of distributed generalized assignment problem (D-GAP) in a massive multi-agent system where agents cannot perform optimal search based on their local views, we propose a heuristic algorithm that can find an approximate optimized solution. By inferring knowledge from their previous communicated searches, agents are able to predict how to deploy future similar searches more efficiently. If an agent can solve some parts of D-GAP well, similar searches will be sent to it. By taking advantage of the accumulation effect to agents’ local knowledge, agents can independently make simple decisions with highly reliable performance and limited communication overheads. The simulation and the experimental results demonstrate the feasibility and efficiency of our algorithm.

Two stage decision making approach for Sensor Mission Assignment Problem

Sensor Mission Assignment (SMA) is the process of assigning sensors to missions in the best way, which may depend on the cost of using individual sensors and the requirements of individual missions. SMA is Np-complete and is a special case of Generalized Assignment Problem. The significant bottlenecks in SMA are energy conservation and uncertainty in the demand of the missions. In order to conserve energy, some sensors are considered to be in sleeping state while others remain active for sensing. In this paper, Sensor Mission Assignment problem is studied in the context of Generalized Assignment Problem combined with decision making approach. Two stage decision making approach is formulated to determine the minimum number of sleeping sensors to be activated so as to assign exactly one sensor to each mission optimally subject to some of the energy resource constraints and environmental constraints imposed on it. The method draws upon the existing generalized assignment problem and the decision making approaches by analyzing trade-offs among desirable value of objective function and the constraints that include all the parameters. The proposed method is applied to the simulation on a small sized wireless sensor network and it is shown that the method is energy efficient. The proposed method provides more holistic point of view on the factors impacting sensor mission assignment.

A Decision Support Approach for Postal Delivery and Waste Collection Services

This paper presents an urban-decision support system (U-DSS) devoted to manage, in a unified framework, the logistic services of the smart cities, such as postal delivery (PD) and waste collection (WC) services. The U-DSS architecture is proposed by describing its main components. In particular, this paper focuses on the core of the U-DSS, i.e., the model component that provides the solutions of a general vehicle assignment and routing optimization problem with the aim of minimizing the length of the routes and satisfying time and capacity constraints. In order to solve the vehicle routing problems in reasonable time, a two-phase heuristic algorithm is proposed based on a clustering strategy and a farthest insertion heuristic for the solution of a traveling salesman problem. The applicability of the proposed U-DSS is enlightened by comparing the proposed heuristic algorithm solutions with the mixed integer linear programming problem solutions of the PD and WC services. Moreover, the discussion of the real case studies of the city of Bari (Italy) assesses the proposed approach.

Optimization-Based Very Large-Scale Neighborhood Search for Generalized Assignment Problems with Location/Allocation Considerations

This paper introduces a novel class of generalized assignment problems with location/allocation considerations that arises in several applications including retail shelf space allocation. We consid...

Exact Solution Methods for a Generalized Assignment Problem with Location/Allocation Considerations

We investigate modeling approaches and exact solution methods for a generalized assignment problem with location/allocation (GAPLA) considerations. In contrast with classical generalized assignment problems, each knapsack in GAPLA is discretized into consecutive segments having different levels of attractiveness. To maximize a total reward function, the decision maker decides not only about item knapsack assignments, but also the specific location of items within their assigned knapsacks and their total space allocation within prespecified lower and upper bounds. Mathematical programming formulations are developed for single and multiple knapsack variants of this problem along with valid inequalities, preprocessing routines, and model enhancements. Further, a branch-and-price algorithm is devised for a set partitioning reformulation of GAPLA, and is demonstrated to yield substantial computational savings over solving the original formulation using branch-and-bound/cut solvers such as CPLEX over challenging problem instances.

Optimization-Based Very Large-Scale Neighborhood Search for Generalized Assignment Problems with Location/Allocation Considerations

This paper introduces a novel class of generalized assignment problems with location/allocation considerations that arises in several applications including retail shelf space allocation. We consider a set of items where each item may represent a family of products and a set of variable-sized knapsacks that may represent shelves, which comprise contiguous segments having distinct attractiveness. The decision-maker seeks to assign the set of items to these knapsacks, specify their segment assignments within knapsacks, and determine their total allocated space within predetermined lower/upper bounds in a fashion that maximizes a reward-based objective function. We develop an effective optimization-based very large-scale neighborhood search (VLNS), which greatly outperforms the best solution identified by CPLEX within one CPU hour, whereas general-purpose solver heuristics failed to provide feasible solutions to most of the larger instances within a time limit comparable to the VLNS algorithm run times. Our computational study was carried out on randomly generated computationally challenging instances with up to 210 items and 42 knapsacks and on a case study motivated by a shelf space allocation problem. Our results demonstrate that the proposed approach consistently produces high-quality solutions.

Dynamic Job Assignment: A Column Generation Approach with an Application to Surgery Allocation

We consider the assignment of jobs to agents in a stochastic and dynamic setting. Focus is on a dynamic scenario with due dates and service levels reflecting the completion of jobs within certain deadlines. Due dates and other relevant characteristics for currently uncompleted jobs generated in the past are known, but the consumption of resources needed for their completion is stochastic. Distributions for the generation of future jobs as well as their characteristics are known. Capacity is limited, and an arriving job that cannot be assigned to an agent within its due date must be outsourced. Outsourcing is accompanied by a cost. We develop an optimization model based on column generation for the assignment of known and future jobs to agents such that the expected cost of outsourcing is minimum. The model is an extension of a generalized assignment problem and provides an allocation of known as well as tentative future jobs to agents. The model is embedded in a rolling horizon framework and subjected to a series of computational tests. The results indicate that taking stochastic information about future job arrivals into account in the assignment of jobs to agents implies an improved performance. The model is highly relevant in the context of patient scheduling in an operating theater. For this reason patient scheduling constitutes the story line in the development of the model.

Population extremal optimisation for discrete multi-objective optimisation problems

The power to solve intractable optimisation problems is often found through population based evolutionary methods. These include, but are not limited to, genetic algorithms, particle swarm optimisation, differential evolution and ant colony optimisation. While showing much promise as an effective optimiser, extremal optimisation uses only a single solution in its canonical form – and there are no standard population mechanics. In this paper, two population models for extremal optimisation are proposed and applied to a multi-objective version of the generalised assignment problem. These models use novel intervention/interaction strategies as well as collective memory in order to allow individual population members to work together. Additionally, a general non-dominated local search algorithm is developed and tested. Overall, the results show that improved attainment surfaces can be produced using population based interactions over not using them. The new EO approach is also shown to be highly competitive with an implementation of NSGA-II.

Fuzzy Multiobjective Modeling and Optimization for One-Shot Multiattribute Exchanges With Indivisible Demand

The modern economy involves a variety of marketplaces, and the Internet has led to the development of a new efficient marketplace—the one-shot multiattribute exchange. It is an important decision problem for a matchmaker (or broker) to achieve the optimal trade matching in one-shot multiattribute exchanges; however, to the best of our knowledge, there has been little work on this issue under fuzzy environments. This paper proposes an optimal matching approach for one-shot multiattribute exchanges with simultaneous fuzzy information and indivisible demand considerations. First, we employ fuzzy set theory to represent the traders’ orders with fuzzy information and then put forward a calculation method of the matching degree based on the improved fuzzy information axiom. Second, on the basis of the matching degree, we construct a fuzzy multiobjective programming model for one-shot multiattribute exchanges with indivisible demand. Afterward, the credibility measure is introduced to convert the model into a crisp one. The crisp model belongs to a class of multiobjective nonlinear general assignment problems and has NP-hard complexity. In order to solve the crisp model effectively, we develop a problem-specified metaheuristic algorithm, i.e., multiobjective discrete differential evolution. Finally, we conduct comprehensive computational experiments on numerical examples to illustrate the application and performance of the proposed model and algorithm.

Time-Cost Optimization Algorithm for Generalized Assignment Problem

NP-난제인 GAP에 대해 Guéret et al.은 O(MSUP4/SUP) 복잡도의 선형계획법으로 해를 얻고자 하였다. 반면에, 본 논문에서는 이 문제에 대해 O(mn) 복잡도로 최적 해를 얻을 수 있는 휴리스틱 알고리즘을 제안하였다. 제안된 방법은 GAP의 m × n 시간과 비용행렬에 대해 시간행렬 최적화 결정(선택)과 비용행렬 최적화 결정(교환)의 2단계 최적화(TSO) 기법과 m × n의 시간x비용행렬에 대해 결정 (선택)하는 1단계 최적화(OSO) 기법으로 적용하였다. 제안된 알고리즘은 O(mn) 수행 복잡도의 단순한 결정기법을 적용하였음에도 불구하고, 비용 측면에서는 O(mSUP4/SUP) 수행 복잡도를 가진 LP의 최적화 기법과 동일한 결과를 얻을 수 있었다. 또한, 기계 가동시간 측면에서는 LP에 비해 보다 적은 가동시간의 결과를 얻었다.

Pre-tactical optimization of runway utilization under uncertainty

Efficient planning of runway utilization is one of the main challenges in Air Traffic Management (ATM). It is important because runway is the combining element between airside and groundside. Furthermore, it is a bottleneck in many cases. In this paper, we develop a specific optimization approach for the pre-tactical planning phase that reduces complexity by omitting unnecessary information. Instead of determining arrival/departure times to the minute in this phase yet, we assign several aircraft to the same time window of a given size. The exact orders within those time windows can be decided later in tactical planning. Mathematically, we solve a generalized assignment problem on a bipartite graph. To know how many aircraft can be assigned to one time window, we consider separation requirements for consecutive aircraft types. In reality, however, uncertainty and inaccuracy almost always lead to deviations from the actual plan or schedule. Thus, we present approaches to incorporate uncertainty directly in our model in order to achieve a stabilization with respect to changes in the data. Namely, we use techniques from robust optimization and stochastic optimization. Further, we analyze real-world data from a large German airport to obtain realistic delay distributions, which turn out to be two-parametric Γ-distributions. Finally, we describe a simulation environment to test our new solution methods.

Display Optimization for Vertically Differentiated Locations Under Multinomial Logit Choice Preferences

We introduce a new optimization model, dubbed the display optimization problem, that captures a common aspect of choice behavior, known as the framing bias. In this setting, the objective is to optimize how distinct items (corresponding to products, web links, ads, etc.) are being displayed to a heterogeneous audience, whose choice preferences are influenced by the relative locations of items. Once items are assigned to vertically differentiated locations, customers consider a subset of the items displayed in the most favorable locations, before picking an alternative through Multinomial Logit choice probabilities.The main contribution of this paper is to derive a constant-factor approximation for the display optimization problem. Our algorithm relies on a decomposition into polynomially-many instances of the maximum generalized assignment problem with additional side constraints, constructed through approximate dynamic programming and randomization methods. The theoretical guarantees we attain are rather surprising, in light of strong inapproximability bounds for closely related models, when optimizing in the face of a parametric mixture of Multinomial Logit preferences. In computational experiments, our algorithm dominates various natural heuristics -- greedy methods, local-search algorithms, and priority rules -- with significant improvements of the expected revenue, ranging from 5% to 11% on synthetic instances, as well as better robustness.

E-Vision Decision Support System Implementation to Match Donors and Recipients Based on Gale-Shapely Algorithm for Cornea Transplantation Surgery

In this paper, we propose to transform the traditional cornea transplantation methods into an electronic exchange between the cornea donors and recipients in the cornea transplantation elective surgery. Preferential evaluations of recipient and donors (individuals / eye-bank) satisfactions are mathematically modeled, then the preference matrix is used as input for Gale Shapely matching algorithm. The results of m*n match happens to be a very transparent approach in a bilateral e-cornea transplantation environment. These matched results are compared with the results obtained using Generalized Assignment problem which produces NP-hard approximated matches. It is found that the proposed method produces stable matching, which is preference based and strategy proof and it also reduces the need for number of iterations for matching.

Zero Assignment Problem in RLC Networks

The paper deals with the problem of zero assignment in RLC networks by selection of appropriate values for the non- dynamical elements, the resistors. For a certain family of network redesign problems by the additive perturbations may be described as diagonal perturbations and such modifications are considered here. This problem belongs to the family of DAP problems (Determinantal Assignment Problem) and has common features with the pole assignment problem by decentralized output feedback and the zero assignment problem via structured additive perturbations. We demonstrate that the sufficient condition for generic zero assignment by selecting the resistors holds true. This condition is related to the rank of the differential of the related map and holds true generically when the degrees of freedom of the matrix of resistors exceeds the number of frequencies to be assigned (n > p + q). Using this result, we show through a generic example that the sufficient condition for the general zero assignment problems in RLC networks is satisfied and thus, zero assignment can be achieved via resistor determination.