Multi-objective Mathematical Programming Problem Research Articles

Constraint Qualifications and Optimality Conditions for Multiobjective Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds

In this paper, we investigate constraint qualifications and optimality conditions for multiobjective mathematical programming problems with vanishing constraints (MOMPVC) on Hadamard manifolds. The MOMPVC-tailored generalized Guignard constraint qualification (MOMPVC-GGCQ) for MOMPVC is introduced in the setting of Hadamard manifolds. By employing MOMPVC-GGCQ and the intrinsic properties of Hadamard manifolds, we establish Karush–Kuhn–Tucker (KKT)-type necessary Pareto efficiency criteria for MOMPVC. Moreover, we introduce several MOMPVC-tailored constraint qualifications and develop interrelations among them. In particular, we establish that the MOMPVC-tailored constraint qualifications introduced in this paper turn out to be sufficient conditions for MOMPVC-GGCQ. Suitable illustrative examples are furnished in the framework of well-known Hadamard manifolds to validate and demonstrate the importance and significance of the derived results. To the best of our knowledge, this is the first time that constraint qualifications, their interrelations, and optimality criteria for MOMPVC have been explored in the framework of Hadamard manifolds.

Constraint qualifications and optimality conditions for nonsmooth multiobjective mathematical programming problems with vanishing constraints on Hadamard manifolds via convexificators

In this paper, we introduce the notion of convexificators in the framework of Hadamard manifolds. We derive some calculus rules for convexificators on Hadamard manifolds and employ them to investigate nonsmooth multiobjective programming problems with vanishing constraints on Hadamard manifolds (abbreviated as, (NMPVC)). The Abadie constraint qualification (abbreviated as, (ACQ)), as well as the generalized (ACQ) (abbreviated as, (GACQ)), are introduced for (NMPVC) in terms of convexificators. Further, (GACQ) is employed to establish the well-known Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary Pareto efficiency criteria for (NMPVC). Moreover, we introduce several other constraint qualifications for (NMPVC) in the Hadamard manifold setting using convexificators and establish interrelations among them. We have furnished non-trivial examples in the framework of some well-known Hadamard manifolds to illustrate the significance of our results. As far as the best of our knowledge is concerned, constraint qualifications and optimality conditions for (NMPVC) have not yet been studied in the framework of Hadamard manifolds via convexificators.

Duality for Multiobjective Programming Problems with Equilibrium Constraints on Hadamard Manifolds under Generalized Geodesic Convexity

This article is devoted to the study of a class of multiobjective mathematical programming problems with equilibrium constraints on Hadamard manifolds (in short, (MPPEC)). We consider (MPPEC) as our primal problem and formulate two different kinds of dual models, namely, Wolfe and Mond-Weir type dual models related to (MPPEC). Further, we deduce the weak, strong as well as strict converse duality relations that relate (MPPEC) and the corresponding dual problems employing geodesic pseudoconvexity and geodesic quasiconvexity restrictions. Several suitable numerical examples are incorporated to demonstrate the significance of the deduced results. The results derived in this article generalize and extend several previously existing results in the literature.

Optimality Conditions for Multiobjective Mathematical Programming Problems with Equilibrium Constraints on Hadamard Manifolds

In this paper, we consider a class of multiobjective mathematical programming problems with equilibrium constraints on Hadamard manifolds (in short, (MMPEC)). We introduce the generalized Guignard constraint qualification for (MMPEC) and employ it to derive Karush–Kuhn–Tucker (KKT)-type necessary optimality criteria. Further, we derive sufficient optimality criteria for (MMPEC) using geodesic convexity assumptions. The significance of the results deduced in the paper has been demonstrated by suitable non-trivial examples. The results deduced in this article generalize several well-known results in the literature to a more general space, that is, Hadamard manifolds, and extend them to a more general class of optimization problems. To the best of our knowledge, this is the first time that generalized Guignard constraint qualification and optimality conditions have been studied for (MMPEC) in manifold settings.

Semidefinite Multiobjective Mathematical Programming Problems with Vanishing Constraints Using Convexificators

In this paper, we establish Fritz John stationary conditions for nonsmooth, nonlinear, semidefinite, multiobjective programs with vanishing constraints in terms of convexificator and introduce generalized Cottle type and generalized Guignard type constraints qualification to achieve strong S—stationary conditions from Fritz John stationary conditions. Further, we establish strong S—stationary necessary and sufficient conditions, independently from Fritz John conditions. The optimality results for multiobjective semidefinite optimization problem in this paper is related to two recent articles by Treanta in 2021. Treanta in 2021 discussed duality theorems for special class of quasiinvex multiobjective optimization problems for interval-valued components. The study in our article can also be seen and extended for the interval-valued optimization motivated by Treanta (2021). Some examples are provided to validate our established results.

Solving multi-objective chance constrained programming problem involving three parameters log normal distribution

The main point of this paper is to find the compromise solution of multi-objective chance constrained mathematical programming problem by considering the parameters of the right hand side restrictions as uncertain variables having three parameters log normal distribution. In the first step, we convert the multi-objective chance constrained mathematical programming problem to a deterministic equivalent multi-objective mathematical programming problem. After converting the multi-objective chance constrained programming problem into deterministic equivalent multi-objective programming problem, fuzzy programming method is used to find the solution which is closest to ideal solution. Numerical example is presented to illustrate the technique. Finally, case study on crop land and water management for irrigation is presented.

Second-order KKT optimality conditions for multiobjective discrete optimal control problems

This paper deals with second-order necessary and sufficient optimality conditions of Karush–Kuhn–Tucker-type for local optimal solutions in the sense of Pareto to a class of multi-objective discrete optimal control problems with nonconvex cost functions and state-control constraints. By establishing an abstract result on second-order optimality conditions for a multi-objective mathematical programming problem, we derive second-order necessary and sufficient optimality conditions for a multi-objective discrete optimal control problem. Using a common critical cone for both the second-order necessary and sufficient optimality conditions, we obtain “no-gap” between second-order optimality conditions.

Solving Multi-Objective Chance Constrained Programming Problem involving Three Parameters Log Normal Distribution

The main point of this paper is to find the compromise solution of multi-objective chance constrained mathematical programming problem by considering the parameters of the right hand side restrictions as uncertain variables having three parameters log normal distribution. In the first step, we convert the multi-objective chance constrained mathematical programming problem to a deterministic equivalent multi-objective mathematical programming problem. After converting the multi-objective chance constrained programming problem into deterministic equivalent multi-objective programming problem, fuzzy programming method is used to find the solution which is closest to ideal solution. Numerical example is presented to illustrate the technique. Finally, case study on crop land and water management for irrigation is presented.

MULTI-OBJECTIVE PROBABILISTIC FRACTIONAL PROGRAMMING PROBLEM INVOLVING TWO PARAMETERS CAUCHY DISTRIBUTION

The paper presents the solution methodology of a multi-objective probabilistic fractional programming problem, where the parameters of the right hand side constraints follow Cauchy distribution. The proposed mathematical model can not be solved directly. The solution procedure is completed in three steps. In first step, multi-objective probabilistic fractional programming problem is converted to deterministic multi-objective fractional mathematical programming problem. In the second step, it is converted to its equivalent multi-objective mathematical programming problem. Finally, ε -constraint method is applied to find the best compromise solution. A numerical example and application are presented to demonstrate the procedure of proposed mathematical model.

A novel multi-objective optimization approach for sustainable supply chain: A case study in packaging industry

This study presents a novel two-stage hybrid solution method designed for sustainable supply chain systems and applied in packaging industry. The proposed solution algorithm includes two main stages; Analytic Hierarchy Process (AHP) method that is applied to select the most efficient suppliers by considering different performance indicators of company and a mixed-integer linear multi-objective mathematical model that is proposed to optimize the design of the sustainable supply chain in terms of total cost, time and social factors. Mathematical modelling approach and data analysis are presented to help decision makers. Minimization of cost, time; and maximization of sustainability by using augmented ϵ-constraint method are outlined by using proposed model. Pareto solution sets of the multi-objective mathematical programming problem are obtained by using GAMS language and environment. Sensitivity analyses are made to highlight the details of illustrative cases based on real-life data from packaging industry.

Multi-objective stochastic transportation problem involving three-parameter extreme value distribution

In this paper, we considered a multi-objective stochastic transportation problem where the supply and demand parameters follow extreme value distribution having three-parameters. The proposed mathematical model for stochastic transportation problem cannot be solved directly by mathematical approaches. Therefore, we converted it to an equivalent deterministic multi-objective mathematical programming problem. For solving the deterministic multi-objective mathematical programming problem, we used an ?-constraint method. A case study is provided to illustrate the methodology.

A multi-objective multi-echelon green supply chain network design problem with risk-averse retailers in an uncertain environment

In this paper, the multi-objective multi-echelon supply chain network design problem is investigated. The resulted problem considers uncertainty of single product demand and the downstream risk attitude simultaneously and integrate it into the greenness concept. That is, makes the model more realistic in comparison with the others. According to the related literature study, there is no specific study to design the green supply chain network based on the risk attitude of the retailer. This paper aims to deal with this research gap by formulating robust counterpart of the main proposed uncertain problem. To do this, the conditional value at risk (CVaR) method through the data-driven approach is applied to deal with demand uncertainty to makes the model to be convex and sensitive to the risk averseness level in a robust manner. In addition, uncertainty sets approach is employed to deal with the uncertainty of other three parameters according to CO2 emission. The augmented e-constraint method is used to transform the resulted multi-objective mathematical programming problem into the single objective one, to obtain the global optimal solution through the exact mathematical solution method. Finally, the model is successfully simulated and its results are illustrated through a numerical example.

Possibilistic Linear Programming Problems involving Normal Random Variables

A new solution procedure of possibilistic linear programming problem is developed involving the right hand side parameters of the constraints as normal random variables with known means and variances and the objective function coefficients are considered as triangular possibility distribution. In order to solve the proposed problem, convert the problem into a crisp equivalent deterministic multi-objective mathematical programming problem and then solved by using fuzzy programming method. A numerical example is presented to illustrate the solution procedure and developed methodology.

Valuing crop diversity in biodiesel production plans

The problem of defining efficient and environmentally compatible short-term agricultural plans for biodiesel exploitation is dealt with in this paper with a multi-objective modelling framework. To optimally use local resources, the first phase of the plan consists in the analysis of land and climate features in order to evaluate which energy crop can be successfully grown. This phase is performed at local scale using GIS (geographic information system) data and software. The second phase consists in the formulation of a multi-objective mathematical programming problem. Using the land to be cultivated in each parcel with each crop as decision variables, we solve a three objectives problem: the maximization of the net energy produced, of the greenhouse gases avoided with respect to conventional fossil fuels and of the diversity of the energy crop mix. The last is quantitatively measured using a well-known biodiversity index, which allows to study the trade-off between a more varied crop mix and the other two objectives along the frontier of Pareto efficient solutions. The proposed methodology is applied to a region of Mato Grosso, Brazil, where biodiesel is produced from oleaginous crops.

A Maximum Entropy Fixed-Point Route Choice Model for Route Correlation

In this paper we present a stochastic route choice model for transit networks that explicitly addresses route correlation due to overlapping alternatives. The model is based on a multi-objective mathematical programming problem, the optimality conditions of which generate an extension to the Multinomial Logit models. The proposed model considers a fixed point problem for treating correlations between routes, which can be solved iteratively. We estimated the new model on the Santiago (Chile) Metro network and compared the results with other route choice models that can be found in the literature. The new model has better explanatory and predictive power that many other alternative models, correctly capturing the correlation factor. Our methodology can be extended to private transport networks.

A simple augmented ∊-constraint method for multi-objective mathematical integer programming problems

A simple augmented ∊-constraint (SAUGMECON) method is put forward to generate all non-dominated solutions of multi-objective integer programming (MOIP) problems. The SAUGMECON method is a variant of the augmented ∊-constraint (AUGMECON) method proposed in 2009 and improved in 2013 by Mavrotas et al. However, with the SAUGMECON method, all non-dominated solutions can be found much more efficiently thanks to our innovations to algorithm acceleration. These innovative acceleration mechanisms include: (1) an extension to the acceleration algorithm with early exit and (2) an addition of an acceleration algorithm with bouncing steps. The same numerical example in Lokman and Köksalan (2012) is used to illustrate workings of the method. Then comparisons of computational performance among the method proposed by Özlen and Azizogˇlu (2009), Özlen et al. (2012), the method developed by Lokman and Köksalan (2012) and the SAUGMECON method are made by solving randomly generated general MOIP problem instances as well as special MOIP problem instances such as the MOKP and MOSP problem instances presented in Table 4 in Lokman and Köksalan (2012). The experimental results show that the SAUGMECON method performs the best among these methods. More importantly, the advantage of the SAUGMECON method over the method proposed by Lokman and Köksalan (2012) turns out to be increasingly more prominent as the number of objectives increases.

Towards a multi-objective performance assessment and optimization model of a two-echelon supply chain using SCOR metrics

This paper aims at multi-objective performance assessment and optimization of a multi-period two-echelon supply chain consisting of a supplier and a manufacturer. On the basis of the assessment system of the supply-chain operations reference model, the supply chain’s performance is investigated with respect to costs, assets, agility, reliability and responsiveness. First, methods to quantify these five performance attributes are put forward. Then a multi-objective mathematical programming model is developed for production decision making of components and products so that the supply chain’s performance frontier formed with Pareto efficient performance values can be achieved. Thereafter a simple augmented $$\epsilon $$ -constraint method is proposed for searching for all Pareto efficient solutions of the multi-objective mathematical programming problem. Finally, efficiency of the method is demonstrated with a numerical example and a sensitivity analysis is implemented to reveal effects of capacity expansion on supply chains’ performance.

A Fixed Point Route Choice Model for Transit Networks that Addresses Route Correlation

In this paper we present a stochastic route choice model for transit networks, which explicitly addresses route correlation due to overlapping in the alternatives. The model is based on a multi-objective mathematical programming problem, which optimality conditions generate an extension to the Multinomial Logit models. The model proposed considers a fixed point problem for treating correlation between routes, which can be solved iteratively. We estimated the new model on the Santiago Metro network and compared the results with other route choice models that can be found in literature. The new model has better explanatory and predictive power, correctly capturing the correlation factor. Our methodology can be extended to private transport networks.

Strength Pareto Evolutionary Algorithm based Multi-Objective Optimization for Shortest Path Routing Problem in Computer Networks

Problem statement: A new multi-objective approach, Strength Pareto Evolutionary Algorithm (SPEA), is presented in this paper to solve the shortest path routing problem. The routing problem is formulated as a multi-objective mathematical programming problem which attempts to minimize both cost and delay objectives simultaneously. Approach: SPEA handles the shortest path routing problem as a true multi-objective optimization problem with competing and noncommensurable objectives. Results: SPEA combines several features of previous multi-objective evolutionary algorithms in a unique manner. SPEA stores nondominated solutions externally in another continuously-updated population and uses a hierarchical clustering algorithm to provide the decision maker with a manageable pareto-optimal set. SPEA is applied to a 20 node network as well as to large size networks ranging from 50-200 nodes. Conclusion: The results demonstrate the capabilities of the proposed approach to generate true and well distributed pareto-optimal nondominated solutions.

Multi-objective congestion management incorporating voltage and transient stabilities

Congestion in a power system is turned up due to operating limits. To relieve congestion in a deregulated power market, the system operator pays to market participants considering their bids to alter their active powers. After relieving congestion, the network may be operated with a reduced voltage or transient stability margin because of hitting security limits or increasing the contribution of risky participants. The proposed multi-objective framework for congestion management in this paper simultaneously optimizes competing objective functions of congestion management cost, voltage security, and dynamic security. The voltage stability margin and corrected transient energy margin are employed as indices to be incorporated into the multi-objective congestion management. A fuzzy decision maker is proposed to derive the most efficient solution among Pareto-optimal solutions of multi-objective mathematical programming problem. Results of testing the proposed method on the New-England test system elaborate the efficiency of the proposed method.