Multiobjective Linear Programming Algorithm Research Articles

Comparison of Multi-Object Control Methods Using Multi-Objective Optimization

The aim of this work is to obtain multi-objective linear programming algorithms that can be used to solve the global problem of multi-object safety control processes in order to minimize the risk of collisions. In multi-objective linear optimization models, satisfactory trade-off assesses and resolves the conflict between different control objectives. A comparison of single-, bi-, and tri-objective linear programming algorithms allows us to adapt the appropriate optimization method to the conditions of the control process. An important outcome of the present research is the demonstration of the greater effectiveness of bi- and tri-objective optimization compared to single-objective optimization, reflecting the compromises taken into account when choosing between objects and achieving a minimum risk of collision when passing them.

A Closure Jacking Force Calculation Algorithm for Curved Prestressed Concrete Continuous Rigid-Frame Bridges with Asymmetric Cantilevers and Piers

Concrete continuous rigid-frame bridges are the most commonly used to cross exciting highway and railway. Before the bridge closure, the closure jacking force (CJF) needs to be determined firstly, which plays an important role in mid-span deflection controlling and the whole bridge stress state. This study proposed a closure jacking force calculation algorithm for curved prestressed concrete continuous rigid-frame bridges with asymmetric cantilevers and piers. Firstly, the mechanics method was used to analyze the relationship between the CJF and normal stress of main-girder and the pier. In addition, an actual bridge was taken as a numerical simulation case to determine the influence of the deformation and the stress induced by concrete shrinkage and creep of the main-girder and the jacking force. Finally, in order to obtain the optimal stress state of bridge piers, the multi-objective linear programming method was proposed to calculate the CJF. The results show that curved prestressed concrete continuous rigid-frame bridges usually suffered longitudinal and transverse bending and torsion under the concrete shrinkage and creep of the main-girder and the CJF, and the CJF significantly affected the normal stress of the pier with a low anti-push rigidity. The normal stress of piers was obviously linear with the CJF during whole construction and operation state of bridges at the following three states: the jacking process, bridge completion, and ten years of bridge operation. The multi-objective linear programming algorithm was an effective way to determine the optimal CJF and can fully reflect both longitudinal and transverse bending of piers.

Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes

Multiplicative programming problems (MPPs) are global optimization problems known to be NP-hard. In this paper, we employ algorithms developed to compute the entire set of nondominated points of multi-objective linear programmes (MOLPs) to solve linear MPPs. First, we improve our own objective space cut and bound algorithm for convex MPPs in the special case of linear MPPs by only solving one linear programme in each iteration, instead of two as the previous version indicates. We call this algorithm, which is based on Benson’s outer approximation algorithm for MOLPs, the primal objective space algorithm. Then, based on the dual variant of Benson’s algorithm, we propose a dual objective space algorithm for solving linear MPPs. The dual algorithm also requires solving only one linear programme in each iteration. We prove the correctness of the dual algorithm and use computational experiments comparing our algorithms to a recent global optimization algorithm for linear MPPs from the literature as well as two general global optimization solvers to demonstrate the superiority of the new algorithms in terms of computation time. Thus, we demonstrate that the use of multi-objective optimization techniques can be beneficial to solve difficult single objective global optimization problems.

Implementing and testing a complex interactive MOLP algorithm

Many business decisions can be modeled as multiobjective linear programming (MOLP) problems. MOLP algorithms seek solutions to these problems by interacting with decision makers to arrive at an acceptable solution. However, due in part to the increasing complexity of these algorithms, and in part to the failure of developers to use graphical user interfaces, testing and comparison of competing algorithms has been minimal. We present herein results of research designed to address this circumstance. Using widely available microcomputer tools, we designed and built a Decision Support System (DSS) capable of running MOLP algorithms, and conducted a field test which asked 98 decision makers to solve a business case using the system. Two algorithms were programmed into the DSS, one a new and more mathematically complex algorithm, and one a previously used benchmark. Results demonstrate that the more complex algorithm was preferred as a decision-making aid over the benchmark. Additionally, results show that users found the DSS equally easy to work with for both algorithms, suggesting that the graphical user interface sufficiently masked the complexity of the new algorithm. This result is encouraging for the possibility of the implementation and testing of increasingly sophisticated MOLP algorithms.

Multiobjective linear programming model on injection oilfield recovery system

This paper proposes a Multiobjective Linear Programming (MLP) model on injection oilfield recovery system. A modified interior-point algorithm to MLP problems has been constructed by using concepts of Kamarkar's interior point algorithm and the Analytic Hierarchy Process (AHP). This algorithm is shown to likely be more efficient than other MLP's algorithms in the application of decision making on the petroleum industry through the demonstration of a numerical example. The MLP model's optimal solution allows decision makers to optimally design the developing plan of the injection oilfield recovery system.

An experimental evaluation of the efficacy of four multi-objective linear programming algorithms

Four multi-objective linear programming algorithms are implemented on microcomputer software packages and a large field experiment is conducted using the implemented algorithms. Two new algorithms which incorporate formal models of decision maker behavior are tested along with two established algorithms which include no formal models of decision maker behavior. The new algorithms are shown to outperform the established algorithms.

Multiobjective linear programming with context-dependent preferences

Multiobjective linear programming algorithms are typically based on value maximization. However, there is a growing body of experimental evidence showing that decision maker behavior is inconsistent with value maximization. Tversky and Simonson provide an alternative model for problems with a discrete set of choices. Their model, called the componential context model, has been shown to capture observed decision maker behavior. In this paper, an interactive multiobjective linear programming algorithm is developed which follows the rationale of Tversky and Simonson. The algorithm is illustrated with an example solved using standard linear programming software. Finally, an interactive decision support system based on this algorithm is developed to field test the usefulness of the algorithm. Results show that this algorithm compares favorably with an established algorithm in the field.

Using aspiration levels in an interior primal‐dual multiobjective linear programming algorithm

Using aspiration levels in an interior primal‐dual multiobjective linear programming algorithm

Using aspiration levels in an interior primal-dual multiobjective linear programming algorithm

Using aspiration levels in an interior primal-dual multiobjective linear programming algorithm

Using aspiration levels in an interactive interior multiobjective linear programming algorithm

Using aspiration levels in an interactive interior multiobjective linear programming algorithm

Using approximate gradients in developing an interactive interior primal-dual multiobjective linear programming algorithm

Using approximate gradients in developing an interactive interior primal-dual multiobjective linear programming algorithm

Using approximate gradients in developing an interactive interior primal-dual multiobjective linear programming algorithm

We present a new interactive multiobjective linear programming algorithm that is based on one variant of Karmarkar's algorithm known as the path-following primal-dual algorithm. The modification of this single-objective linear programming algorithm to the multiobjective case is done by deriving an approximate gradient to the implicitly-known utility function. By interacting with the decision maker, locally-relevant preference information are elicited and the approximated gradient can therefore be continuously updated. The interior step direction is then generated by projecting the approximate gradient and taking an interior step from the current iterate to the new one along this projection.

Using aspiration levels in an interactive interior multiobjective linear programming algorithm

In this paper, we propose the use of an interior-point linear programming algorithm for multiple objective linear programming (MOLP) problems. At each iteration, a Decision Maker (DM) is asked to specify aspiration levels for the various objectives, and an achievement scalarizing function is applied to project aspiration levels onto the nondominated set. The interior-point algorithm is used to find an interior solution path from a starting solution to a nondominated solution corresponding to the optimum of the achievement scalarizing function. The proposed approach allows the DM to re-specify aspiration levels during the solution process and thus steer the interior solution path toward different areas in objective space. We illustrate the use of the approach with a numerical example.

A multiobjective interior primal-dual linear programming algorithm

This paper presents an interior Multiobjective Linear Programming (MOLP) algorithm that is based on a variant of Karmarkar's interior-point algorithm known as the path-following primal-dual algorithm. The modification is accomplished by combining the single step direction vectors generated by the single-objective algorithm into a combined vector along which we step from the current iterate to the next iterate. Combining step direction vectors into a single step can be done by either approximating the gradient of an implicit utility function at the current iterate, or by creating convex combinations of single directions. We discuss both approaches and use the latter to develop an interior algorithm directed at handling MOLP problems. We refer to the resulting class of MOLP algorithms resulting from this variant as Primal-Dual Interior Multiobjective Linear Programming (PDIMOLP) algorithm.

Using Efficient Anchoring Points for Generating Search Directions in Interior Multiobjective Linear Programming

This paper modifies the affine-scaling primal algorithm to multiobjective linear programming (MOLP) problems. The modification is based on generating search directions in the form of projected gradients augmented by search directions pointing toward what we refer to as anchoring points. These anchoring points are located on the boundary of the feasible region and, together with the current, interior, iterate, define a cone in which we make the next step towards a solution of the MOLP problem. These anchoring points can be generated in more than one way. In this paper we present an approach that generates efficient anchoring points where the choice of termination solution available to the decision maker at each iteration consists of a set of efficient solutions. This set of efficient solutions is being updated during the iterative process so that only the most preferred solutions are retained for future considerations. Current MOLP algorithms are simplex-based and make their progress toward the optimal solution by following an exterior trajectory along the vertices of the constraints polytope. Since the proposed algorithm makes its progress through the interior of the constraints polytope, there is no need for vertex information and, therefore, the search for an acceptable solution may prove less sensitive to problem size. We refer to the resulting class of MOLP algorithms that are based on the affine-scaling primal algorithm as affine-scaling interior multiobjective linear programming (ASIMOLP) algorithms.

Using Efficient Anchoring Points for Generating Search Directions in Interior Multiobjective Linear Programming

This paper modifies the affine-scaling primal algorithm to multiobjective linear programming (MOLP) problems. The modification is based on generating search directions in the form of projected gradients augmented by search directions pointing toward what we refer to as anchoring points. These anchoring points are located on the boundary of the feasible region and, together with the current, interior, iterate, define a cone in which we make the next step towards a solution of the MOLP problem. These anchoring points can be generated in more than one way. In this paper we present an approach that generates efficient anchoring points where the choice of termination solution available to the decision maker at each iteration consists of a set of efficient solutions. This set of efficient solutions is being updated during the iterative process so that only the most preferred solutions are retained for future considerations. Current MOLP algorithms are simplex-based and make their progress toward the optimal solution by following an exterior trajectory along the vertices of the constraints polytope. Since the proposed algorithm makes its progress through the interior of the constraints polytope, there is no need for vertex information and, therefore, the search for an acceptable solution may prove less sensitive to problem size. We refer to the resulting class of MOLP algorithms that are based on the affine-scaling primal algorithm as affine-scaling interior multiobjective linear programming (ASIMOLP) algorithms.

Anchoring Points and Cones of Opportunities in Interior Multiobjective Linear Programming

This paper presents a modification of one variant of Karmarkar's interior-point linear programming algorithm to Multiobjective Linear Programming (MOLP) problems. We show that by taking the variant known as the affine-scaling primal algorithm, generating locally-relevant scaling coefficients and applying them to the projected gradients produced by it, we can define what we refer to as anchoring points that then define cones in which we search for an optimal solution through interaction with the decision maker. Currently existing MOLP algorithms are simplex-based and make their progress toward the optimal solution by following the vertices of the constraints polytope. Since the proposed algorithm makes its progress through the interior of the constraints polytope, there is no need for vertex information and, therefore, the search for an optimal solution may prove less sensitive to problem size. We refer to the class of MOLP algorithms resulting from this variant as Affine-Scaling Interior Multiobjective Linear Programming (ASIMOLP) algorithms.

An interior multiobjective linear programming algorithm

This paper presents a multiobjective linear programming (MOLP) algorithm that is based on one variant of Karmarkar's interior-point algorithm known as the affine scaling primal algorithm. We show that by using convex combinations of individual projected gradients, we derive a combined direction along which we step toward the next iterate. We refer to the class of MOLP algorithms resulting from this variant as affine-scaling interior multiobjective linear programming (ASIMOLP) algorithms.

A weighted-gradient approach to multi-objective linear programming problems using the analytic hierarchy process

This paper presents a Multi-objective Linear Programming (MOLP) algorithm that is using the AHP to generate locally-relevant scaling coefficients and applying them to the projected gradients produced by a variant of Karmarkar's interior-point algorithm known as the affine-scaling primal algorithm. We refer to the class of MOLP algorithms resulting from this variant as Affine-Scaling Interior Multi-objective Linear Programming (ASIMOLP) algorithms.

Generating interior search directions for multiobjective linear programming using approximate gradients and efficient anchoring points

We present in this paper a new multiobjective linear programming (MOLP) algorithm. The algorithm is based on modifying a variant of Karmarkar's algorithm known as the affine-scaling primal algorithm to multiobjective linear programming problems. This modification is accomplished by combining approximate gradients of the multiple objective functions together with what we refer to as anchoring points that allow us to generate a search direction and move toward the solution through the interior of the constraints polytope. In contrast, current multiobjective linear programming algorithms are using the simplex algorithm to generate a sequence of steps that follow the exterior of the constraints polytope toward the optimal solution. As MOLP problems grow in size, following an exterior trajectory may become prohibitively costly in terms of the number of iterations and the required interaction cycles with the Decision Maker. Following an interior trajectory may prove less sensitive to problem size as vertex info...