Multiobjective Programming

This paper introduces an interactive approach to support multi-criteria decision analysis of project portfolios. In high-scale strategic decision domains, scientific studies suggest that the Decision Maker (DM) can find help by using many-objective optimisation methods, which are supposed to provide values in the decision variables that generate highquality solutions. Even so, DMs usually wish to explore the possibility of reaching some levels of benefits in some objectives. Consequently, they should repeatedly run the optimisation method. However, this approach cannot perform well-in an interactive way-for large instances under the presence of many objective functions. We present a mathematical model that is based on compromise programming and fuzzy outranking to aid DMs in analysing multi-criteria project portfolios on the fly. This approach allows relaxing the problem of rapidly optimising portfolios while preserving the beneficial properties of the DM's preferences expressed by outranking relations. Our model supports the decision analysis on two instance benchmarks: for the first one, a better compromise solution was generated 84% of the runs; for the second one, this ranged from 93% to 97%. Our model was also applied to a real-world problem involving social projects, obtaining satisfactory results.

Multiobjective combinatorial optimization problems have received increasing attention in recent years. Nevertheless, many algorithms are still restricted to the bicriteria case. In this paper we propose a new algorithm for computing all Pareto optimal solutions. Our algorithm is based on the notion of level sets and level curves and contains as a subproblem the determination of K best solutions for a single objective combinatorial optimization problem. We apply the method to the Multiobjective Quadratic Assignment Problem (MOQAP). We present two algorithms for ranking QAP solutions and nally give computational results comparing the methods.

A class of second order (F, α, ρ, d)-convex functions and their generalizations is introduced. Using the assumptions on the functions involved, weak, strong and strict converse duality theorems are established for a second order Mond–Weir type multiobjective dual.

The potential conflict between protection of water quality and economic development by different uses of land within river basins is a common problem in regional planning. Many studies have applied multiobjective decision analysis under uncertainty to problems of this ...

In this paper, we introduce a new class of generalized (F,α,ρ,θ)–d–V(F,α,ρ,θ)–d–V-univex functions for a nonsmooth multiobjective programming problem. Sufficient optimality conditions under generalized (F,α,ρ,θ)–d–V(F,α,ρ,θ)–d–V-univex functions are established for a feasible solution to be an efficient solution. Appropriate duality theorems for a Mond–Weir-type dual are also presented under the aforesaid assumptions.

Goal programming (GP) is perhaps one of the most widely used approaches in the field of multicriteria decision making. The major advantage of the GP model is its great flexibility which enables the decision maker to easily incorporate numerous variations on constraints and goals. Romero provides a general structure, extended lexicographic goal programming (ELGP) for GP and some multiobjective programming approaches. In this work, we propose the extension of this unifying framework to fuzzy multiobjective programming. Our extension is carried out by several methodologies developed by the authors in the fuzzy GP approach. An interval GP model has been constructed where the feasible set has been defined by means of a relationship between fuzzy numbers. We will apply this model to our fuzzy extended lexicographic goal programming (FELGP). The FELGP is a general primary structure with the same advantages as Romero’s ELGP and moreover it has the capacity of working with imprecise information. An example is given in order to illustrate the proposed method.

This paper is on the portfolio optimization problem for which two generic models are presented in the context of a proprietary solver called GENO:  the first is a pseudo-dynamic model with a single state variable that is meant for the single holding-period case; the second is a truly dynamic model with several state variables that applies to both single and multi-period scenarios. Both models can handle practical restrictions such as exposure, cardinality and round-lot constraints; both can accommodate non-traditional risk measures; and the usual two-element criterion set comprising ‘portfolio reward’ and ‘portfolio risk’ may be augmented by any number of sub-objectives deemed necessary. The paper also presents a ‘robust optimization’ model of the portfolio problem that explicitly accounts for data uncertainty; the robust model can also accommodate sophisticated risk measures, and one such function is presented in Appendix B. The compromise solution concept is used to compute portfolios that are Pareto-efficient; several numerical examples show that the portfolios thus found are not only optimal in the compromise sense, but they also have a competitive (and often the highest) reward-risk ratio—a proxy for the Sharpe ratio. A limited empirical analysis shows that the robust model is superior to its non-robust counterpart in terms of nominal performance and the potential for avoiding “opportunity costs” due to data uncertainty.

This work proposes a multiobjective approach for transmission expansion planning considering security constraints (N-1 criterion) and minimum cost. Elitist NSGA-II multiobjective algorithm is used and the operative problem is solved trhough high order interior point method. The methodology is tested on 24 and 6 bus IEEE systems.

A pair of Wolfe type multiobjective second order symmetric dual programs with cone constraints is formulated and usual duality results are established under second order invexity assumptions. These results are then used to investigate symmetric duality for minimax version of multiobjective second order symmetric dual programs wherein some of the primal and dual variables are constrained to belong to some arbitrary sets, i.e., the sets of integers. This paper points out certain omissions and inconsistencies in the earlier work of Mishra [S.K. Mishra, Multiobjective second order symmetric duality with cone constraints, European Journal of Operational Research 126 (2000) 675–682] and Mishra and Wang [S.K. Mishra, S.Y. Wang, Second order symmetric duality for nonlinear multiobjective mixed integer programming, European Journal of Operational Research 161 (2005) 673–682].

This paper introduces a new class of non-convex vector functions strictly larger than that of P-quasiconvexity, with P⊆ m being the underlying ordering cone, called semistrictly ( m \ −int P)-quasiconvex functions. This notion allows us to unify various results on existence of weakly efficient (weakly Pareto) optima. By imposing a coercivity condition we establish also the compactness of the set of weakly Pareto solutions. In addition, we provide various characterizations for the non-emptiness, convexity and compactness of the solution set for a subclass of quasiconvex vector optimization problems on the real-line. Finally, it is also introduced the notion of explicit ( m \ −int P)-quasiconvexity (equivalently explicit (int P)-quasiconvexity) which plays the role of explicit quasiconvexity (quasiconvexity and semistrict quasiconvexity) of real-valued functions.

This paper uses the grey fuzzy multiobjective programming to aid in decision making for the allocation of waste load in a river system under versatile uncertainties and risks. It differs from previous studies by considering a multicriteria objective function with combined grey and fuzzy messages under a cost benefit analysis framework. Such analysis technically integrates the prior information of water

This paper presents four optimization models for solving over-specified or under-specified nonlinear equation systems. The presentation includes a 'variable endogenization' technique that patently enhances computational efficiency of the methods proffered; a basic comparative study attests to the effectiveness of the methods presented.

In this work, we establish a strong duality theorem for Mond–Weir type multiobjective higher-order nondifferentiable symmetric dual programs. This fills some gaps in the work of Chen [X. Chen, Higher-order symmetric duality in nondifferentiable multiobjective programming problems, J. Math. Anal. Appl. 290 (2004) 423–435].

This paper introduces a new class of non-convex vector functions strictly larger than that of P-quasiconvexity, with P⊆ m being the underlying ordering cone, called semistrictly ( m \ −int P)-quasiconvex functions. This notion allows us to unify various results on existence of weakly efficient (weakly Pareto) optima. By imposing a coercivity condition we establish also the compactness of the set of weakly Pareto solutions. In addition, we provide various characterizations for the non-emptiness, convexity and compactness of the solution set for a subclass of quasiconvex vector optimization problems on the real-line. Finally, it is also introduced the notion of explicit ( m \ −int P)-quasiconvexity (equivalently explicit (int P)-quasiconvexity) which plays the role of explicit quasiconvexity (quasiconvexity and semistrict quasiconvexity) of real-valued functions.