Multi-criteria assignment problem with incompatibility and capacity constraints

The considered assignment problem generalizes its classical counterpart by the existence of some incompatibility constraints limiting the assignment of tasks to processing units within groups of mutually exclusive tasks. The groups are defined for each processing unit and the constraints allow at most one task from each group to be assigned to the corresponding processing unit. The processing units can normally process a certain number of tasks without any cost; this capacity can be extended, however, at some extra marginal cost that is non-decreasing with the number of additional tasks. Each task has to be assigned to exactly one processing unit and has some preference for the assignment; it is expressed for each pair ‘task-processing unit’ by a dissatisfaction degree. The quality of feasible assignments is evaluated by three criteria: g ₁-the maximum dissatisfaction of tasks, g ₂-the total dissatisfaction of tasks, g ₃-the total cost of processing units. If there is no feasible assignment, tasks and processing units creating a blocking configuration are identified and all actions of unblocking are proposed. Formal properties of blocking configurations and unblocking actions are proven, and an interactive procedure for exploring the set of non-dominated assignments is described together with illustrative examples processed by special software.     

Complexity analysis of an assignment problem with controllable assignment costs and its applications in scheduling

We extend the classical linear assignment problem to the case where the cost of assigning agent j to task i is a multiplication of task i’s cost parameter by a cost function of agent j. The cost function of agent j is a linear function of the amount of resource allocated to the agent. A solution for our assignment problem is defined by the assignment of agents to tasks and by a resource allocation to each agent. The quality of a solution is measured by two criteria. The first criterion is the total assignment cost and the second is the total weighted resource consumption. We address these criteria via four different problem variations. We prove that our assignment problem is NP-hard for three of the four variations, even if all the resource consumption weights are equal. However, and somewhat surprisingly, we find that the fourth variation is solvable in polynomial time. In addition, we find that our assignment problem is equivalent to a large set of important scheduling problems whose complexity has been an open question until now, for three of the four variations.     

Local search heuristics for multi-index assignment problems with decomposable costs

The multi-index assignment problem (MIAP) with decomposable costs is a natural generalization of the well-known assignment problem. Applications of the MIAP arise, for instance, in the field of multi-target multi-sensor tracking. We describe an (exponentially sized) neighbourhood for a solution of the MIAP with decomposable costs, and show that one can find a best solution in this neighbourhood in polynomial time. Based on this neighbourhood, we propose a local search algorithm. We empirically test the performance of published constructive heuristics and the local search algorithm on random instances; a straightforward iterated local search algorithm is also tested. Finally, we compute lower bounds to our problem, which enable us to assess the quality of the solutions found.     

Fast approximation algorithms for bi-criteria scheduling with machine assignment costs

We consider parallel machine scheduling problems where the processing of the jobs on the machines involves two types of objectives. The first type is one of two classical objective functions in scheduling theory: either the total completion time or the makespan. The second type involves an actual cost associated with the processing of a specific job on a given machine; each job-machine combination may have a different cost. Two bi-criteria scheduling problems are considered: (1) minimize the maximum machine cost subject to the total completion time being at its minimum, and (2) minimize the total machine cost subject to the makespan being at its minimum. Since both problems are strongly NP-hard, we propose fast heuristics and establish their worst-case performance bounds.     

Integer programming models for the multidimensional assignment problem with star costs

We consider a variant of the multidimensional assignment problem (MAP) with decomposable costs in which the resulting optimal assignment is described as a set of disjoint stars. This problem arises in the context of multi-sensor multi-target tracking problems, where a set of measurements, obtained from a collection of sensors, must be associated to a set of different targets. To solve this problem we study two different formulations. First, we introduce a continuous nonlinear program and its linearization, along with additional valid inequalities that improve the lower bounds. Second, we state the standard MAP formulation as a set partitioning problem, and solve it via branch and price. These approaches were put to test by solving instances ranging from tripartite to 20-partite graphs of 4 to 30 nodes per partition. Computational results show that our approaches are a viable option to solve this problem. A comparative study is presented.     

A self-adaptive projection and contraction algorithm for the traffic assignment problem with path-specific costs

This paper considers solving a special case of the nonadditive traffic equilibrium problem presented by Gabriel and Bernstein [Transportation Science 31 (4) (1997) 337–348] in which the cost incurred on each path is made up of the sum of the arc travel times plus a path-specific cost for traveling on that path. A self-adaptive projection and contraction method is suggested to solve the path-specific cost traffic equilibrium problem, which is formulated as a nonlinear complementarity problem (NCP). The computational effort required per iteration is very modest. It consists of only two function evaluations and a simple projection on the nonnegative orthant. A self-adaptive technique is embedded in the projection and contraction method to find suitable scaling factor without the need to do a line search. The method is simple and has the ability to handle a general monotone mapping F. Numerical results are provided to demonstrate the features of the projection and contraction method.     

A model for allocated versus actual costs in assignment and transportation problems

We present a simple mathematical model which will relate the actual cost spent in accomplishing a task to the dollars budgeted for that task. In the specific instances of assignment and transportation problems we show how to minimize total dollars spent given total dollars allocated. We show furthermore how to quantitatively measure the work done along each arc in such problems. The total work, which will measure how fixed costs are realized across various arcs for a given prescribed effort, can then be minimized. It is shown that this, in general, leads to a third type of optimal solution which is different from those optimal solutions obtained by minimizing either total cost or total dollars allocated.     

On solving a large-scale problem on facility location and customer assignment with interaction costs along a time horizon

A 0-1 quadratic programming model is presented for solving the strategic problem of timing the location of facilities and the assignment of customers to facilities in a multi-period setting. It is assumed that all parameters are known and, on the other hand, the quadratic character of the objective function is due to considering the interaction cost incurred by the joint assignment of customers belonging to different categories to a facility at a period. The plain use of a state-of-the-art MILP engine with capabilities for dealing with quadratic terms does not give any advantage over the matheuristic algorithm proposed in this work. In fact, the MILP engine was frequently running out of memory before reaching optimality for the equivalent mixed 0-1 linear formulation, being its best lower bound at that time instant too far from the incumbent solution for the large-sized instances which we have worked with. As an alternative, a fix-and-relax algorithm is presented. A deep computational comparison between MILP alternatives is performed, such that fix-and-relax provides a solution value very close to (and, frequently, a better than) the one provided by the MILP engine. The time required by fix-and-relax is very affordable, being frequently two times smaller than the time required by the MILP engine.     

Characterization of facility assignment costs for a location-inventory model under truckload distribution

We consider a two-stage distribution system, where the first stage consists of potential distribution centres (DCs) and the second stage consists of geographically dispersed existing retailers. Our goal is to determine the set of open DCs and assignment of open DCs to retailers simultaneously with inventory decisions of retailers. In addition to the DC-specific fixed facility location costs, we explicitly model the inventory replenishment and holding costs at the retailers and truckload transportation costs between the DCs and the retailers. The transportation costs are subject to truck/cargo capacity, leading to an integrated location-inventory problem with explicit cargo costs. We develop a mixed-integer nonlinear model and analyse its structural properties leading to exact expressions for the so-called implied facility assignment costs and imputed per-unit per-mile transportation costs. These expressions analytically demonstrate the interplay between strategic location and tactical inventory/transportation decisions in terms of resulting operational costs. Although both the theory and practice of integrated logistics have recognized the fact that strategic and tactical decisions are interrelated, to the best of our knowledge, our paper is the first to offer closed-form results demonstrating the relationship explicitly. We propose an efficient solution approach utilizing the implied facility assignment costs and we demonstrate that significant savings are realizable when the inventory decisions and cargo costs are modelled explicitly for facility location purposes.     

Using fuzzy integral for evaluating subjectively perceived travel costs in a traffic assignment model

A good traffic assignment model can be a powerful tool to describe the characteristics of traffic behavior in a road network. The traffic assignment results often play an important role in transportation planning, e.g., an optimal and economical network design. Many traditional traffic assignment models rely heavily on the travel cost function established by Wardrop’s principles; however, the Wardrop’s travel cost function has been proven to be weak for explaining the uncertainty and interactivity of traffic among links. This study tries to construct a traffic assignment model that is different from Wardrop’s in many aspects. First, it considers the cross-effect among the links. Second, a fuzzy travel cost function is established based on the possibility concept instead of precise calculation of traffic volumes. Third, the techniques of fuzzy measure and fuzzy integral are applied to calculate the subjectively perceived travel costs during traffic assignment. Furthermore, in order to validate our model, a detailed network with 22 nodes and 36 links is used to illustrate it. Study results show that our model explains more interactivity and uncertainty of traffic among links when compared with the traditional model of Wardrop’s.     

Task assignment with controlled and autonomous agents

We analyse assignment problems in which not every agent is controlled by the central planner. The autonomous agents search for vacant tasks guided by their own preference orders over available tasks. The goal of the central planner is to maximise the total value of the assignment, taking into account the behaviour of the uncontrolled agents. Such optimisation problems arise in numerous real-world situations, ranging from organisational economics to “crowdsourcing” and disaster response. We show that the problem faced by the central planner can be transformed into a mixed integer bilevel optimisation problem. Then we demonstrate how this program can be reduced to a disjoint bilinear program, which is much more manageable computationally.     

Resource assignment with preference conditions

This paper deals with a modification of the standard assignment problem, where subsets of resources express preferences in being, or not being, assigned together to the same activity. The problem arises in several real settings, among which the job assignment of the crew personnel of an airline company. We provide an integer programming formulation for both the Split Preference Problem, where couples of assignees do not want to work together, and for the Join Preference Problem, where, oppositely, couples of assignees want to work together. The mathematical nature of the two problems is indeed different, as for the first one it is possible to determine a minimum cost flow formulation on a suitable graph, and thus a polynomial time algorithm, while for the second one we face a NP-hard problem and device some heuristic solution approaches. Experimental tests conducted on instances of variable size confirm the effectiveness of the models and of the algorithms proposed.     

Multistate components assignment problem with optimal network reliability subject to assignment budget

The typical assignment problem for finding the optimal assignment of a set of components to a set of locations in a system has been widely studied in practical applications. However, this problem mainly focuses on maximizing the total profit or minimizing the total cost without considering component’s failure. In practice, each component should be multistate due to failure, partially failure, or maintenance. That is, each component has several capacities with a probability distribution and may fail. When a set of multistate components is assigned to a system, the system can be treated as a stochastic-flow network. The network reliability is the probability that d units of homogenous commodity can be transmitted through the network successfully. The multistate components assignment problem to maximize the network reliability is never discussed. Therefore, this paper focuses on solving this problem under an assignment budget constraint, in which each component has an assignment cost. The network reliability under a components assignment can be evaluated in terms of minimal paths and state-space decomposition. Subsequently an optimization method based on genetic algorithm is proposed. The experimental results show that the proposed algorithm can be executed in a reasonable time.     

The incompatibility of individualism and ordinalism

A formal proof is offered of the fact that Individualism and Ordinalism imply the controversial condition A3 used by Kemp and Ng (1976) to show the non-existence of an individualistic social welfare function based only on ordinal preferences but objected to by Samuelson (1977) as unreasonable. Mayston's argument against our definition of ordinality is refuted; his notion of ‘true ordinality’ involves elements of cardinality and/or goes against individualism.     

Due-date assignment and parallel-machine scheduling with deteriorating jobs

In this paper we study the problem of scheduling n deteriorating jobs on m identical parallel machines. Each job's processing time is a nondecreasing function of its start time. The problem is to determine an optimal combination of the due-date and schedule so as to minimize the sum of the due-date, earliness and tardiness penalties. We show that this problem is NP-hard, and we present a heuristic algorithm to find near-optimal solutions for the problem. When the due-date penalty is 0, we present a polynomial time algorithm to solve it.     

Fleet assignment and routing with schedule synchronization constraints

This paper introduces a new type of constraints, related to schedule synchronization, in the problem formulation of aircraft fleet assignment and routing problems and it proposes an optimal solution approach. This approach is based on Dantzig–Wolfe decomposition/column generation. The resulting master problem consists of flight covering constraints, as in usual applications, and of schedule synchronization constraints. The corresponding subproblem is a shortest path problem with time windows and linear costs on the time variables and it is solved by an optimal dynamic programming algorithm. This column generation procedure is embedded into a branch and bound scheme to obtain integer solutions. A dedicated branching scheme was devised in this paper where the branching decisions are imposed on the time variables. Computational experiments were conducted using weekly fleet routing and scheduling problem data coming from an European airline. The test problems are solved to optimality. A detailed result analysis highlights the advantages of this approach: an extremely short subproblem solution time and, after several improvements, a very efficient master problem solution time.     

Capacitated transit assignment with loading priorities

In a transit network involving vehicles with rigid capacities, we advocate the use of strategies for describing consumer behavior. At each boarding node, a user sorts the transit lines in decreasing order of preference, and boards the first vehicle in this list whose residual capacity is nonzero. Since a users position in the queue varies from day to day, the delay experienced is stochastic. This leads to an equilibrium problem where, at a solution, users are assigned to strategies that minimize their expected delay. This situation is formulated as a variational inequality, whose cost mapping is discontinuous and strongly asymmetric, due to the priority of current passengers over incoming users. We prove that the solution set is nonempty and provide numerical results obtained by an efficient solution algorithm.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Fonds pour la formation de chercheurs et laide à la recherche (FCAR).Mathematics Subject Classification (2000):20E28, 20G40, 20C20Accepted: December 20, 2003     

Due dates assignment and JIT scheduling with equal-size jobs

This paper deals with due date assignment and just-in-time scheduling for single machine and parallel machine problems with equal-size jobs where the objective is to minimize the total weighted earliness–tardiness and due date cost. These two problems, but with a common due date to be calculated, were shown to be polynomially solvable in O(n⁴)$O(n^4)$ time. We first show that this complexity can be reduced to O(n³)$O(n^3)$ by modeling the single machine scheduling problem as an assignment problem without necessary due date enumeration. We next prove that the general case with identical parallel machines and a given set of assignable due dates where the cardinality of this set is bounded by a constant number is still polynomially solvable.     

Due-date assignment and single machine scheduling with deteriorating jobs

We study a scheduling problem with deteriorating jobs, that is, jobs whose processing times are an increasing function of their start times. We consider the case of a single machine and linear job-independent deterioration. The problem is to determine an optimal combination of the due-date and schedule so as to minimize the sum of due-date, earliness and tardiness penalties. We give an O(n log n) time algorithm to solve this problem.     

Controlled diffusions with boundary-crossing costs

This paper considers control of nondegenerate diffusions in a bounded domain with a cost associated with the boundary-crossings of a subdomain. Existence of optimal Markov controls and a verification theorem are established.Research supported by ARO Contract No. DAAG29-84-K-0005 and AFOSR 85-0227.