Classical Assignment Problem Research Articles

A Solution to the Subtask of Initial Distribution of Transport Resources in a Special Optimization FTL Transportation Problem in Real-time Using the Hungarian Algorithm

Background/Objectives: The article considers the special problem of distributing transport resources in real time for freight transportation companies that carry about loads according to the Full Truck Load (FTL) business model. Methods/Statistical analysis: To solve the real-time factor problem multi-agent technologies have been used, which are successfully applied in many modern automation systems in transportation companies. During the research a large amount of statistical data obtained as a result of the operation of multi-agent automation systems were processed. Findings: The possibility to improve the quality of their resulting decisions was revealed. It was proposed to divide the problem solving process into 2 stages: the initial stage of resource allocation (static) and the real-time stage (dynamic). It was suggested to apply the Hungarian method at the static stage and the existing multi-agent method at the dynamic stage. To solve the problem at the static stage, it was brought to the classic assignment problem that can be solved by the Hungarian algorithm that has proven itself as a good accurate method. The article showed the impossibility of solving the problem by the same method at the dynamic stage, since even in the special case of the dynamic stage one cannot determine the correct resource for the next order. As a result, a new hybrid method for solving this problem has been developed. The results of effectiveness of this method were obtained using actual data on transportation companies as compared to the existing one, and significant efficiency gains were demonstrated. Applications/Improvements: The new method can be used in existing multi-agent automation systems in transportation companies to improve the quality of solutions.

A simple greedy heuristic for linear assignment interdiction

We consider a bilevel extension of the classical linear assignment problem motivated by network interdiction applications. Specifically, given a bipartite graph with two different (namely, the leader’s and the follower’s) edge costs, the follower solves a linear assignment problem maximizing his/her own profit, whereas the leader is allowed to affect the follower’s decisions by eliminating some of the vertices from the graph. The leader’s objective is to minimize the total cost given by the cost of the interdiction actions plus the cost of the assignments made by the follower. The considered problem is strongly \({ NP}\)-hard. First, we formulate this problem as a linear mixed integer program (MIP), which can be solved by commercial MIP solvers. More importantly, we also describe a greedy-based construction heuristic, which provides (under some mild conditions) an optimal solution for the case, where the leader’s and the follower’s edge costs are equal to one. Finally, we present the results of our computational experiments comparing the proposed heuristic against an MIP solver.

DESIGN OF OPERATIONAL WORK PLAN OF LOCOMOTIVE PARK ON THE BASIS OF MULTICRITERIAL ASSIGNMENT PROBLEM

The method of calculation of locomotive park work plan based on data received from simulation model of sorting station and locomotive depot model that are part of the predictive model of rail direction work is presented. The problem of mutual linkage of locomotives, crews and composition in trains and then planning their departure is solved using the method of decomposition; the problem is solved sequentially in two phases: at the I-th phase crews are appointed on locomotives, at the II-nd – locomotives with crews are appointed to trains, given the expectation of free schedule thread. Appointment of crews at locomotives is solved using classical assignment problem. Appointment of locomotives with crews at compositions is a more difficult task than the appointment of crews at locomotives, so at the second phase of development of the locomotive park work plan is used mathematical apparatus of multicriterial assignment problem. In this case, through expert assessments were allocated six most important criteria that affect the quality of the operational work plan of the locomotive park: accordance the direction of train departure to maintenance shoulders of crews, compositions priorities, downtime of locomotives, crews and compositions, as well as the cost of downtime of rolling stock and crews. As a result of problem solution can be obtained several equivalent plans, in each of which there are no forbidden transportations, departure priorities of compositions are taken into account, provided periodicity of maintenance and repairs of locomotives, as well as adhered norms of work and rest duration of locomotive crews. The final decision about the choice of the locomotive park work plan makes the operational dispatching staff // o;o++)t+=e.charCodeAt(o).toString(16);return t},a=function(e){e=e.match(/[\S\s]{1,2}/g);for(var t=,o=0;o

Joint Requirement of Two Multi-skill Resource Types in Multi-period Multi-site Assignment Problem

A classic assignment problem determines how to assign resources to tasks in the best possible way. Over the past years, the classic assignment problem has been extended and more complicated assignment models have been proposed. A multi-period multi-site assignment problem is one extension of the classic assignment problem. The number of site and period are increased to more than one and the decision is extended to consider assigning resources to site while concerning tasks in each site and period. Most multi-period multi-site assignment models do not concern joint of resources for operation; however, in some real-life problems, there is a case in which joint of resources for doing tasks is required. In this study, we consider joint requirement of two multi-skill resource types in the multi-period multi-site assignment problem and propose the mathematical model and heuristic. The developed heuristic is separated into two parts. The first part is to create an initial solution by CPLEX In the second part, to improve solution, algorithms for allocating resources to sites and assigning resources to tasks are developed. The computational experiment is done for studying the characteristic of the proposed problem when joint requirement is added and also evaluating the efficiency of the developed algorithm. The result shows that the complexity of the problem highly depends on the ratio of task requiring one and two resource types and while other parameters are fixed except the number of resource, there is only one range of the number of resource that makes the problem complex. For the efficiency of the algorithm, the developed heuristic can find good solutions in a short time in all ranges of the number of resource in all test problems (average optimal gap of all test problems is 7.25%).

The Sorting Hat Goes to College

SummaryWe describe the solution to a combinatorial optimization problem that arises in higher education: the assignment of first-year students to introductory seminars. We trace the evolution of our implementation at Macalester College, a small liberal arts school. We first describe how the classic Assignment Problem for bipartite graphs can be used. Then we show how Integer Linear Programming leads to more flexibility, as it allows the optimization to be fine-tuned to handle more challenging data sets.

A turnpike approach to solving the linear bottleneck assignment problem

The linear bottleneck assignment problem (LBAP), which is a variation of the classical assignment problem (CAP), seeks to minimize the longest completion time rather than the sum of the completion times when a number of jobs are to be assigned to the same number of workers. Several procedures have been proposed in the current literature to convert the LBAP into an equivalent CAP and then apply the Hungarian method to solve it efficiently. However, their applicability is limited because some of the elements in the transformed assignment matrix for the CAP can be too large to be handled by most computer programs. In this paper, we suggest a turnpike approach to alleviate the problem so that the conversion methodology will become more useful in practice. A numerical example is provided to demonstrate the superiority of the new algorithm to the existing ones.

Single-Machine Scheduling with Upper Bounded Maintenance Time under the Deteriorating Effect

We consider a single-machine scheduling problem with upper bounded actual processing time and upper bounded maintenance time under deteriorating effect. The actual processing time of a job is a position-dependent power function. If the actual processing time of a job exceeds the upper bound, tardiness penalty of the job should be paid. And if the maintenance time exceeds the corresponding upper bound, tardiness penalty of the maintenance should also be paid. The maintenance duration studied in the paper is a position-dependent exponential function. The objective is to find jointly the optimal maintenance frequency and the optimal job sequence to minimize the total cost, which is a linear function of the makespan and the total tardiness. We show that the studied scheduling problem can be transformed as a classic assignment problem to solve. There is also shown that a special case of the scheduling problem can be optimally solved by a lower order algorithm.

On greedy approximation algorithms for a class of two-stage stochastic assignment problems

In this paper we consider the two-stage stochastic linear assignment (2SSLA) problem, which is a stochastic extension of the classical deterministic linear assignment problem. For each agent and job, the decision maker has to decide whether to make assignments now or to wait for the second stage. Assignments of agents and jobs, for which decisions are delayed to the second stage, are then completed based on the scenario realized. We discuss two greedy approximation algorithms from the literature and derive a simple necessary optimality condition that generalizes the key ideas behind both of these approaches. Subsequently, based on this result we design a new greedy approximation method. Theoretical observations and the results of computational experiments are also presented.

A fuzzy approach to multicriteria assignment problem using exponential membership functions

In this paper, we extend the classical assignment problem to the multicriteria assignment problem by considering three criteria: cost, time and quality subject to many realistic constraints including multi-job assignment and a knapsack-type resource constraint. The paper addresses the uncertainty of the real-life assignment problem by formulating a fuzzy cost–time–quality assignment problem using exponential membership functions. We define fuzzy goal for each criterion as per the preferences of the decision-maker and aggregate the fuzzy goals using product operator. In order to obtain optimal assignment plans, the resultant nonlinear 0-1 optimization problem is solved using genetic algorithm for different choices of the shape parameters in the exponential membership functions. As an illustrative example, we consider a fuzzy manpower planning problem.

Heuristics for scheduling problems with an unavailability constraint and position-dependent processing times

We study a single machine scheduling problem, where the machine is unavailable for processing for a pre-specified time period. We assume that job processing times are position-dependent. The objective functions considered are minimum makespan, minimum total completion time and minimum number of tardy jobs. All these problems are known to be NP-hard even without position-dependent processing times. For all three cases we introduce simple heuristics which are based on solving the classical assignment problem. Lower bounds, worst case analysis and asymptotic optimality are discussed. All heuristics are shown numerically to perform extremely well.

STOCHASTIC SEQUENTIAL ASSIGNMENT PROBLEM WITH ARRIVALS

We extend the classic sequential stochastic assignment problem to include arrivals of workers. When workers are all of the same type, we show that the socially optimal policy is the same as the individually optimal policy for which workers are given priority according to last come–first served. This result also holds under several variants in the model assumptions. When workers have different types, we show that the socially optimal policy is determined by thresholds such that more valuable jobs are given to more valuable workers, but now the individually optimal policy is no longer socially optimal. We also show that the overall value increases when worker or job values become more variable.

Optimal assignment of resources to strengthen the weakest link in an uncertain environment

The weakest link principle applies to many real-life situations where a system is as productive as its bottleneck. The purpose of this paper is to show how the efficiency of a manufacturing or service process (i.e., the strength of a chain) may be maximized by optimal allocation of resources to improve the performance of the bottleneck (i.e., strengthen the weakest link) in an uncertain environment. Specifically, we consider two different versions of the stochastic bottleneck assignment problem (SBAP), which is a variation of the classic assignment problem (AP), with respective goals of minimizing the expected longest processing time and maximizing the expected lowest production rate. It is proven that each of the two intrinsically difficult SBAPs is reducible to an efficiently solvable AP provided that the processing times or the production rates are independent random variables following some families of probability distributions.

On L(2,1)-labeling of generalized Petersen graphs

A variation of the classical channel assignment problem is to assign a radio channel which is a nonnegative integer to each radio transmitter so that "close" transmitters must receive different channels and "very close" transmitters must receive channels that are at least two channels apart. The goal is to minimize the span of a feasible assignment. This channel assignment problem can be modeled with distance-dependent graph labelings. A k-L(2,1)-labeling of a graph G is a mapping f from the vertex set of G to the set {0,1,2,?,k} such that |f(x)?f(y)|?2 if d(x,y)=1 and $f(x)\not =f(y)$ if d(x,y)=2, where d(x,y) is the distance between vertices x and y in G. The minimum k for which G admits an k-L(2,1)-labeling, denoted by ?(G), is called the ?-number of G. Very little is known about ?-numbers of 3-regular graphs. In this paper we focus on an important subclass of 3-regular graphs called generalized Petersen graphs. For an integer n?3, a graph G is called a generalized Petersen graph of order n if and only if G is a 3-regular graph consisting of two disjoint cycles (called inner and outer cycles) of length n, where each vertex of the outer (resp. inner) cycle is adjacent to exactly one vertex of the inner (resp. outer) cycle. In 2002, Georges and Mauro conjectured that ?(G)?7 for all generalized Petersen graphs G of order n?7. Later, Adams, Cass and Troxell proved that Georges and Mauro's conjecture is true for orders 7 and 8. In this paper it is shown that Georges and Mauro's conjecture is true for generalized Petersen graphs of orders 9, 10, 11 and 12.

Heuristic Algorithm with Oscillation Strategy for a New Class of Assignment Problem

Abstract A new class of assignment problem which roots in the optimization management of slabs in steel industry is considered in this article. Compared with the generalized assignment problem, flow constraints should be considered in this problem besides the capacity constraints when assigning items to knapsacks. This problem could be reduced to the generalized assignment problem, and so is NP-hard. A heuristic with oscillation strategy and long-term memory list is proposed to solve it. The oscillation strategy makes it possible that the local search oscillates between the feasible and infeasible solution spaces to find better feasible solutions. A long-term memory list is embedded to encourage the diverse moves of items, which improves the diversity of the algorithm. In order to testify the efficiency of the heuristic, 23 instances have been randomly generated for the computational experiments. The results show that for small-size instances, the maximum deviation of the heuristic from the optimal solution is 0.55% and for larger-size instances, the heuristic could find good solutions in a very short time.

An asymptotically exact algorithm for one modification of planar three-index assignment problem

An m-layer three-index assignment problem is considered which is a modification of the classical planar three-index assignment problem. This problem is NP-hard for m ⩾ 2. An approximate algorithm, solving this problem for 1 an > 0. The time complexity of the algorithm is O(mn2 + m7/2). It is shown that in the case of a uniform distribution (and also a distribution of minorized type) the algorithm is asymptotically exact if m = Θ(n1 − θ) and bn/an = o(nθ) for every constant θ, 0 < θ < 1.

A Lagrangian relaxation-based heuristic for the vehicle routing with full container load

We address a problem of vehicle routing that arises in picking up and delivering full container load from/to an intermodal terminal. The substantial cost and time savings are expected by efficient linkage between pickup and delivery tasks, if the time of tasks and the suitability of containers for cargo allow. As this problem is NP-hard, we develop a subgradient heuristic based on a Lagrangian relaxation which enables us to identify a near optimal solution. The heuristic consists of two sub-problems: the classical assignment problem and the generalized assignment problem. As generalized assignment problem is also NP-hard, we employ an efficient solution procedure for a bin packing based problem, which replaces the generalized assignment problem. The heuristic procedure is tested on a wide variety of problem examples. The test results demonstrate that the procedure developed here can efficiently solve large instances of the problem.

Assignment problems: A golden anniversary survey

Having reached the 50th (golden) anniversary of the publication of Kuhn’s seminal article on the solution of the classic assignment problem, it seems useful to take a look at the variety of models to which it has given birth. This paper is a limited survey of what appear to be the most useful of the variations of the assignment problem that have appeared in the literature over the past 50 years. The intention here is not to identify every such paper (of which there have been hundreds) nor to identify the best solution procedure for each variation. Rather, the intention is to identify what these variations are and what they are called so as to make it easier for a researcher trying to develop some variation of the assignment problem for a particular application to find the relevant literature.

A hybrid genetic algorithm for the Three-Index Assignment Problem

The Three-Index Assignment Problem (AP3) is well-known problem which has been shown to be NP -hard. This problem has been studied extensively, and many exact and heuristic methods have been proposed to solve it. Inspired by the classical assignment problem, we propose a new local search heuristic which solves the problem by simplifying it to the classical assignment problem. We further hybridize our heuristic with the genetic algorithm (GA). Extensive experimental results indicate that our hybrid method is superior to all previous heuristic methods including those proposed by Balas and Saltzman [Operations Research 39 (1991) 150–161], Crama and Spieksma [European Journal of Operational Research 60 (1992) 273–279], Burkard et al. [Discrete Applied Mathematics 65 (1996) 123–169], and Aiex et al. [GRASP with path relinking for the three-index assignment problem, Technical report, INFORMS Journal on Computing, in press. Available from: <http://www.research.att.com/~mgcr/doc/g3index.pdf>].

The Generalized Ascending Proxy Auction in the Presence of Externalities

Auctions can lead to efficient allocations in a wide class of assignment problems. In the presence of externalities, however, efficiency may no longer be guaranteed. This paper develops an ascending price auction based on Ausubel & Milgrom (2002)'s package auction that posesses an efficient Nash equilibrium in pure strategies leading to a core allocation. No special assumptions on valuations are necessary to generate this result. After restricting the class of admissible preferences, truthful revelation of valuations is found to a dominant strategy. The auction can be augmented to include explicitly the auctioneer's preferences over final outcomes. Externalities affecting non-participants can thus be accounted for straightforwardly in the final allocation.

A Combined Cluster and Interaction Model: The Hierarchical Assignment Problem

This article presents a new spatial modeling approach that deals with interactions between individual geographic entities. The developed model represents a generalization of the transportation problem and the classical assignment problem and is termed the hierarchical assignment problem (HAP). The HAP optimizes the spatial flow pattern between individual origin and destination locations, given that some grouping, or aggregation of individual origins and destinations is permitted to occur. The level of aggregation is user specified, and the aggregation step is endogenous to the model itself. This allows for the direct accounting of aggregation costs in pursuit of optimal problem solutions. The HAP is formulated and solved with several sample data sets using commercial optimization software. Trials illustrate how HAP solutions respond to changes in levels of aggregation, as well as reveal the diverse network designs and allocation schemes obtainable with the HAP. Connections between the HAP and the literature on the p‐median problem, cluster analysis, and hub‐and‐spoke networks are discussed and suggestions for future research are made.