Equivalent weights for lexicographic multi-objective programs: Characterizations and computations

A recent paper [19] demonstrated the existence of a set of equivalent weights for which the optimal solution set of a preemptive priority multi-objective program is precisely equal to the set of optimal solutions to the resulting nonpreemptive program with the objective function given as a linear weighting of the multiple objectives. This paper addresses two further issues. Firstly, for some important special cases or applications, it is demonstrated that not only is the computation of a set of equivalent weights feasible, but it is also highly desirable. Two algorithms are presented to compute a set of equivalent weights. One method is a direct specialization of the approach adopted in [19], whereas the second approach is an alternative technique. The latter method is shown to yield weights of uniformly smaller values than the former method, while being of the same computational complexity, and is hence preferable. Secondly, as opposed to constructing one vector of equivalent weights, a characterization is provided for the entire set of equivalent weights.     

Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems

We consider multi-objective convex optimal control problems. First we state a relationship between the (weakly or properly) efficient set of the multi-objective problem and the solution of the problem scalarized via a convex combination of objectives through a vector of parameters (or weights). Then we establish that (i) the solution of the scalarized (parametric) problem for any given parameter vector is unique and (weakly or properly) efficient and (ii) for each solution in the (weakly or properly) efficient set, there exists at least one corresponding parameter vector for the scalarized problem yielding the same solution. Therefore the set of all parametric solutions (obtained by solving the scalarized problem) is equal to the efficient set. Next we consider an additional objective over the efficient set. Based on the main result, the new objective can instead be considered over the (parametric) solution set of the scalarized problem. For the purpose of constructing numerical methods, we point to existing solution differentiability results for parametric optimal control problems. We propose numerical methods and give an example application to illustrate our approach.     

Pruned Pareto-optimal sets for the system redundancy allocation problem based on multiple prioritized objectives

In this paper, a new methodology is presented to solve different versions of multi-objective system redundancy allocation problems with prioritized objectives. Multi-objective problems are often solved by modifying them into equivalent single objective problems using pre-defined weights or utility functions. Then, a multi-objective problem is solved similar to a single objective problem returning a single solution. These methods can be problematic because assigning appropriate numerical values (i.e., weights) to an objective function can be challenging for many practitioners. On the other hand, methods such as genetic algorithms and tabu search often yield numerous non-dominated Pareto optimal solutions, which makes the selection of one single best solution very difficult. In this research, a tabu search meta-heuristic approach is used to initially find the entire Pareto-optimal front, and then, Monte-Carlo simulation provides a decision maker with a pruned and prioritized set of Pareto-optimal solutions based on user-defined objective function preferences. The purpose of this study is to create a bridge between Pareto optimality and single solution approaches.     

Parallel machine scheduling with a convex resource consumption function

We consider some problems of scheduling jobs on identical parallel machines where job-processing times are controllable through the allocation of a nonrenewable common limited resource. The objective is to assign the jobs to the machines, to sequence the jobs on each machine and to allocate the resource so that the makespan or the sum of completion times is minimized. The optimization is done for both preemptive and nonpreemptive jobs. For the makespan problem with nonpreemptive jobs we apply the equivalent load method in order to allocate the resources, and thereby reduce the problem to a combinatorial one. The reduced problem is shown to be NP-hard. If preemptive jobs are allowed, the makespan problem is shown to be solvable in O(n²) time. Some special cases of this problem with precedence constraints are presented and the problem of minimizing the sum of completion times is shown to be solvable in O(n log n) time.     

A linear optimization problem to derive relative weights using an interval judgement matrix

In this paper, we propose a new Decision Making model, enabling to assess a finite number of alternatives according to a set of bounds on the preference ratios for the pairwise comparisons between alternatives, that is, an “interval judgement matrix”. In the case in which these bounds cannot be achieved by any assessment vector, we analyze the problem of determining of an efficient or Pareto-optimal solution from a multi-objective optimization problem. This multi-objective formulation seeks for assessment vectors that are near to simultaneously fulfil all the bound requirements imposed by the interval judgement matrix. Our new model introduces a linear optimization problem in order to define a consistency index for the interval matrix. By solving this optimization problem it can be associated a weakly efficient assessment vector to the consistency index in those cases in which the bound requirements are infeasible. Otherwise, this assessment vector fulfils all the bound requirements and has geometrical properties that make it appropriate as a representative assessment vector of the set of feasible weights.     

Tools for Multicoloring with Applications to Planar Graphs and Partial k-Trees

We study graph multicoloring problems, motivated by the scheduling of dependent jobs on multiple machines. In multicoloring problems, vertices have lengths which determine the number of colors they must receive, and the desired coloring can be either contiguous (nonpreemptive schedule) or arbitrary (preemptive schedule). We consider both the sum-of-completion times measure, or the sum of the last color assigned to each vertex, as well as the more common makespan measure, or the number of colors used. In this paper, we study two fundamental classes of graphs: planar graphs and partial k-trees. For both classes, we give a polynomial time approximation scheme (PTAS) for the multicoloring sum, for both the preemptive and nonpreemptive cases. On the other hand, we show the problem to be strongly NP-hard on planar graphs, even in the unweighted case, known as the sum coloring problem. For a nonpreemptive multicoloring sum of partial k-trees, we obtain a fully polynomial time approximation scheme. This is based on a pseudo-polynomial time algorithm that holds for a general class of cost functions. Finally, we give a PTAS for the makespan of a preemptive multicoloring of partial k-trees that uses only O(log n) preemptions. These results are based on several properties of multicolorings and tools for manipulating them, which may be of more general applicability.     

Optimal discrete-valued control computation

In this paper, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transformation, we transform this optimal control problem into an equivalent problem subject to additional linear and quadratic constraints. The feasible region defined by these additional constraints is disconnected, and thus standard optimization methods struggle to handle these constraints. We introduce a novel exact penalty function to penalize constraint violations, and then append this penalty function to the objective. This leads to an approximate optimal control problem that can be solved using standard software packages such as MISER. Convergence results show that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude the paper with some numerical results for two difficult train control problems.     

Dominance,potential optimality and alternative ranking in imprecise multi-attribute decision making

In this paper, we introduce a methodology based on an additive multiattribute utility function that does not call for precise estimations of the inputs, such as utilities, attribute weights and performances of decision alternatives. The information about such inputs is assumed to be in the form of ranges, which constitute model constraints and give rise to nonlinear programming problems. This has significant drawbacks for outputting the sets of non-dominated and potentially optimal alternatives for such problems, and we, therefore, propose their transformation into equivalent linear programming problems. The set of non-dominated and potentially optimal alternatives is a non-ranked set and can be very large, which makes the choice of the most preferred alternative very difficult. The above problem is solved by proposing several methods for alternative ranking. An application to the disposal of surplus weapons-grade plutonium is considered, showing the advantages of this approach.     

A Process for Determining Priorities and Weights for Large-Scale Linear Goal Programmes

Many planning models can be formulated as large-scale linear goal-programming problems in which the analyst and user must establish thousands of objective-function weights that reflect the priorities of the many goals. How to select such weights so as to have the resulting optimal solution be a suitable compromise solution is the main focus of this paper. We first describe the problem setting that gave rise to the need, here military personnel planning, and then a process by which a set of goal priorities and objective-function weights can be developed using Saaty's analytic hierarchy process.     

Inverse multi-objective combinatorial optimization

Inverse multi-objective combinatorial optimization consists of finding a minimal adjustment of the objective functions coefficients such that a given set of feasible solutions becomes efficient. An algorithm is proposed for rendering a given feasible solution into an efficient one. This is a simplified version of the inverse problem when the cardinality of the set is equal to one. The adjustment is measured by the Chebyshev distance. It is shown how to build an optimal adjustment in linear time based on this distance, and why it is right to perform a binary search for determining the optimal distance. These results led us to develop an approach based on the resolution of mixed-integer linear programs. A second approach based on a branch-and-bound is proposed to handle any distance function that can be linearized. Finally, the initial inverse problem is solved by a cutting plane algorithm.     

Enhanced Efficiency in Multi-objective Optimization

For a given multi-objective optimization problem, we introduce and study the notion of α-proper efficiency. We give two characterizations of such proper efficiency: one is in terms of exact penalization and the other is in terms of stability of associated parametric problems. Applying the aforementioned characterizations and recent results on global error bounds for inequality systems, we obtain verifiable conditions for α-proper efficiency. For a large class of polynomial multi-objective optimization problems, we show that any efficient solution is α-properly efficient under some mild conditions. For a convex quadratically constrained multi-objective optimization problem with convex quadratic objective functions, we show that any efficient solution is α-properly efficient with a known estimate on α whenever its constraint set is bounded. Finally, we illustrate our achieved results with examples, and give an example to show that such an enhanced efficiency property may not hold for multi-objective optimization problems involving C ^∞-functions as objective functions.     

Identifying preferred solutions to Multi-Objective Binary Optimisation problems,with an application to the Multi-Objective Knapsack Problem

In this paper we present a new framework for identifying preferred solutions to multi-objective binary optimisation problems. We develop the necessary theory which leads to new formulations that integrate the decision space with the space of criterion weights. The advantage of this is that it allows for incorporating preferences directly within a unique binary optimisation problem which identifies efficient solutions and associated weights simultaneously. We discuss how preferences can be incorporated within the formulations and also describe how to accommodate the selection of weights when the identification of a unique solution is required. Our results can be used for designing interactive procedures for the solution of multi-objective binary optimisation problems. We describe one such procedure for the multi-objective multi-dimensional binary knapsack formulation of the portfolio selection problem.     

Eigenvalue Multiplicity Estimate in Semidefinite Programming

A semidefinite programming problem is a mathematical program in which the objective function is linear in the unknowns and the constraint set is defined by a linear matrix inequality. This problem is nonlinear, nondifferentiable, but convex. It covers several standard problems (such as linear and quadratic programming) and has many applications in engineering. Typically, the optimal eigenvalue multiplicity associated with a linear matrix inequality is larger than one. Algorithms based on prior knowledge of the optimal eigenvalue multiplicity for solving the underlying problem have been shown to be efficient. In this paper, we propose a scheme to estimate the optimal eigenvalue multiplicity from points close to the solution. With some mild assumptions, it is shown that there exists an open neighborhood around the minimizer so that our scheme applied to any point in the neighborhood will always give the correct optimal eigenvalue multiplicity. We then show how to incorporate this result into a generalization of an existing local method for solving the semidefinite programming problem. Finally, a numerical example is included to illustrate the results.     

Enumeration of All Possibly Optimal Vertices with Possible Optimality Degrees in Linear Programming Problems with a Possibilistic Objective Function

In this paper, we treat linear programming problems with fuzzy objective function coefficients. To such a problem, the possibly optimal solution set is defined as a fuzzy set. It is shown that any possibly optimal solution can be represented by a convex combination of possibly optimal vertices. A method to enumerate all possibly optimal vertices with their membership degrees is developed. It is shown that, given a possibly optimal extreme point with a higher membership degree, the membership degree of an adjacent extreme point is calculated by solving a linear programming problem and that all possibly optimal vertices are enumerated sequentially by tracing adjacent possibly optimal extreme points from a possibly optimal extreme point with the highest membership degree.     

An improved solution methodology for the arsenal exchange model (AEM)

We develop an iterative approach for solving a linear programming problem with prioritized goals. We tailor our approach to preemptive goal programming problems and take advantage of the fact that at optimality, most constraints are not binding. To overcome the problems posed by redundant constraints, our procedure ensures redundant constraints are not present in the problems we solve. We apply our approach to the arsenal exchange model (AEM). AEM allocates weapons to targets using linear programs (LPs) formulated by the model. Our methodology solves a subproblem using a specific subset of the constraints generated by AEM. Violated constraints are added to the original subproblem and redundant constraints are not included in any of the subproblems. Our methodology was used to solve five test cases. In four of the five test cases, our methodology produced an optimal integer solution. In all five test cases, solution quality was maintained or improved.     

The single-server scheduling problem with convex costs

Being probably one of the oldest decision problems in queuing theory, the single-server scheduling problem continues to be a challenging one. The original formulations considered linear costs, and the resulting policy is puzzling in many ways. The main one is that, either for preemptive or nonpreemptive problems, it results in a priority ordering of the different classes of customers being served that is insensitive to the individual load each class imposes on the server and insensitive to the overall load the server experiences. This policy is known as the cμ-rule. We claim and show that for convex costs, the optimal policy depends on the individual loads. Therefore, there is a need for an alternative generalization of the cμ-rule. The main feature of our generalization consists on first-order differences of the single stage cost function, rather than on its derivatives. The resulting policy is able to reach near optimal performances and is a function of the individual loads.     

Two optimality test for differentiable concave value functions in linear multi-objective programming problems

This note outlinestwo test which can be applied, under certain conditions, to solutions to linear multi-objective programming problems. The first test can be used to determine whether a given basic solution to the problem is optimal. The second test can be used to determine whether a given non-basic solution to the problem is optimal in a certain face of the set of feasible solutions of the problem.     

Minimum node covers and 2-bicritical graphs

The problem of finding a minimum cardinality set of nodes in a graph which meet every edge is of considerable theoretical as well as practical interest. Because of the difficulty of this problem, a linear relaxation of an integer programming model is sometimes used as a heuristic. In fact Nemhauser and Trotter showed that any variables which receive integer values in an optimal solution to the relaxation can retain the same values in an optimal solution to the integer program. We define 2-bicritical graphs and give several characterizations of them. One characterization is that they are precisely the graphs for which an optimal solution to the linear relaxation will have no integer valued variables. Then we show that almost all graphs are 2-bicritical and hence the linear relaxation almost never helps for large random graphs.This research was supported in part by the National Research Council of Canada.     

An exact penalty function method for nonlinear mixed discrete programming problems

In this paper, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additional linear and quadratic constraints. Then, an exact penalty function is employed to construct a sequence of unconstrained optimization problems, each of which can be solved effectively by unconstrained optimization techniques, such as conjugate gradient or quasi-Newton methods. It is shown that any local optimal solution of the unconstrained optimization problem is a local optimal solution of the transformed nonlinear constrained continuous optimization problem when the penalty parameter is sufficiently large. Numerical experiments are carried out to test the efficiency of the proposed method.     

Impulsively-controlled systems and reverse dwell time: A linear programming approach

We present a receding horizon algorithm that converges to the exact solution in polynomial time for a class of optimal impulse control problems with uniformly distributed impulse instants and governed by so-called reverse dwell time conditions. The cost has two separate terms, one depending on time and the second monotonically decreasing on the state norm. The obtained results have both theoretical and practical relevance. From a theoretical perspective we prove certain geometrical properties of the discrete set of feasible solutions. From a practical standpoint, such properties reduce the computational burden and speed up the search for the optimum thus making the algorithm suitable for the on-line implementation in real-time problems. Our approach consists in approximating the optimal impulse control problem via a binary linear programming problem with a totally unimodular constraint matrix. Hence, solving the binary linear programming problem is equivalent to solving its linear relaxation. Then, given the feasible solution from the linear relaxation, we find the optimal solution via receding horizon and local search. Numerical illustrations of a queueing system are performed.