Set Of Noninferior Solutions Research Articles

Multiobjective Heuristic Search in AND/OR Graphs

The multiobjective search model is a framework for solving multi-criteria optimization problems using heuristic search techniques. In this framework, the different non-commensurate optimization criteria are mapped into distinct dimensions of a vector valued cost structure and partial order search techniques are used to determine the set of non-inferior solutions. Multiobjective state space search has been studied and generalizations of algorithms such asA* to the multiobjective framework have been considered. In this paper we address the problem of multiobjective heuristic (best-first) search of acyclic additive AND/OR graphs. We establish two results which show that in the multiobjective framework, the task of identifying a non-dominated cost potential solution graph is NP-hard in general. This indicates that by extending popular AND/OR graph search algorithms such asAO* to the multiobjective framework we will not be able to preserve some of their desirable properties. Under such circumstances, we investigate the task of developing effective algorithms for the multiobjective problem and present a linear space AND/OR graph search algorithm calledMObj*. Upper bounds on time and space complexities of this algorithm have been presented. It has also been established that when applied to OR graphs, the proposed algorithm is superior to the algorithm proposed by Stewart and White (MultiobjectiveA*,J. Assoc. Comput. Mech.38, No. 4 (1991), 775–814) in terms of the worst case set of nodes expanded.

Analyzing the interaction of design and control—1. A multiobjective framework and application to binary distillation synthesis

The design of a process determines the properties of variable interaction, disturbance rejection, and set-point tracking which comprise its inherent controllability. Process synthesis usually focuses on optimizing an economic objective without consideration of controllability. This paper outlines and illustrates a systematic procedure for incorporating open-loop steady-state controllability measures within the mathematical programming approach of process synthesis. This approach translates a superstructure of design alternatives into a multiobjective mixed-integer nonlinear (MINLP) optimization problem. With the flowsheet unknown a priori, the formulation requires expressions for the steady-state economic and control objectives in terms of the unknown design variables. Use of the ε-constraint method within the framework of the Generalized Benders Decomposition algorithm generates the noninferior solution set. A multiobjective optimization algorithm based on cutting planes determines a best-compromise solution among the various design and control objectives using as trade-off weights the partial derivative information from the noninferior set. The proposed procedure is then applied in binary distillation synthesis to consider multiple objectives for the RV control configuration, to examine different control configurations, and to deal with heat-integrated columns. Closed-loop dynamic simulations compare the responses of columns idenfied in the optimization.

Analyzing the interaction of design and control—2. reactor-separator-recycle system

Part 1 presented a systematic procedure to analyze the interaction of design and control at the process synthesis stage by incorporating both steady-state economic and open-loop controllability measures within a multiobjective mixed-integer nonlinear optimization problem. Part 2 applies this framework to study the design of a system consisting of an isothermal reactor followed by a distillation column, whose distillate stream recycles back to the reactor. For this particular flowsheet, the resulting optimization can be written as a nonlinear programming (NLP) problem. A comparison of alternative designs shows that leaving the reactor and recycle stream compositions as variables in the NLP significantly reduces the cost compared with the cost of designs when these compositions are fixed. Economic considerations drive the optimal solution toward a larger reactor with lower reactant composition and toward a stripper column with lower boilup. The derivation of steady-state gain expressions relating the product compositions in the column to manipulated and disturbance variables allows the incorporation of controllability measures within the NLP. Generation of the noninferior solution sets shows the quantitative trade-off between the steady-state economic and controllability objectives. In general, process controllability measures improve as the reactor holdup decreases and as the number of trays in the column increases, but the opposite changes improve the disturbance controllability measures. Closed-loop dynamic simulations compare the responses of several designs identified in the optimization.

113 The construction and properties of the set of noninferior solutions of multiobjective convex programming

113 The construction and properties of the set of noninferior solutions of multiobjective convex programming

Multiobjective pricing of bus‐subway services in Santiago, Chile

AbstractPresently, the surface transit system in Santiago, Chile is private and atomized. It operates in competition with the subway, covering long distances and causing congestion in the central business district. As the subway operates below capacity, integrated services with a common fare have been identified as both operatively and financially possible, making better use of resources and avoiding negative externalities. A multi‐objective approach is formulated and applied to determine the price of the combined service, accounting for both users' benefits and operators' profit. The role of different parameters in the financial and operative constraints is analyzed in particular, as well as the conflicting nature of the objective functions. The non‐inferior set of solutions is constructed and analyzed both in the decision variables space and in the objectives space, and solutions are compared against the present conditions.

The Construction and Properties of the Set of Noninferior Solutions of Multiobjective Convex Programming

The convexity assumption is usually used in the theory of multiobjective optimization and its algorithms. In this paper, the set of noninferior solutions is constructed and its geometric properties are studied under the convexity assumption. It is shown that in m dimensional value space, the set of noninferior solutions turns out to be a strictly decreasing, strictly concave, and no chinky block (or piece) of (hyper-)surface (or curve).

A Multiobjective Optimization Approach for Analyzing the Interaction of Design and Control

Abstract This work presents a systematic procedure for analyzing the interaction of design and control. The mathematical programming framework of process synthesis is used to formulate and solve an optimization problem with multiple objectives involving open-loop controllability measures and economics. A multiobjective optimisation algorithm based on cutting planes is applied to determine the optimal trade-off using partial derivative information from the noninferior solution set. A binary distillation column example illustrates the proposed approach.

Reliability‐Based Vector Optimization of Structural Systems

Virtually all structural optimizations based on system reliability were conducted considering only the ultimate state of plastic collapse initiated from the intact (i.e., undamaged) system. These single‐objective optimizations may result in decreasing the safety levels with respect to other limit states of concern, such as deformation, plastic hinge occurrences, and residual system reliability. Motivated by the need of a rational reliability‐based multilimit‐state optimization approach, where different conflicting requirements in the design of structural systems are considered simultaneously, a vector‐optimization approach is formulated. In this approach a vector‐valued objective function is examined, and the system safety is considered with regard to multiple reliability requirements imposed simultaneously. The solution to this problem is defined as the set of noninferior solutions in the space of decision variables. A three‐step reliability‐based vector‐optimization searching strategy is suggested, a de...

Multiple objectives and non-separability in stochastic dynamic programming

A general separable class of stochastic multiobjective optimization problems with perfect state information is considered. A generating approach using a stochastic multiobjective dynamic programming method is developed to find the set of non-inferior solutions. The results reveal the variation of the optimal weighting coefficient vector along a non-inferior trajectory. Non-separability is not an inherent property of dynamic programming. A general class of non-separable dynamic problems can be transformed into corresponding separable multiobjective dynamic programming problems. Multiobjective dynamic programming is shown to be a separation strategy to solve non-separable dynamic programming.

Multiobjective Decision‐Tree Analysis1

Single‐objective‐based decision‐tree analysis has been extensively and successfully used in numerous decision‐making problems since its formal introduction by Howard Raiffa more than two decades ago. This paper extends the traditional methodology to incorporate multiple noncommensurate objective functions and use of the conditional expected value of the risk of extreme and catastrophic events. The proposed methodology considers the cases where (a) a finite number of actions are available at each decision node and (b) discrete or continuous states of nature can be presented at each chance node. The proposed extension of decision‐tree analysis is introduced through an example problem that leads the reader step‐by‐step into the methodological procedure. The example problem builds on flood warning systems. Two noncommensurate objectives—the loss of lives and the loss of property (including monetary costs of the flood warning system)–are incorporated into the decision tree. In addition, two risk measures—the common expected value and the conditional expected value of extreme and catastrophic events—are quantified and are also incorporated into the decision‐making process. Theoretical difficulties associated with the stage‐wise calculation of conditional expected values are identified and certain simplifying assumptions are made for computational tractibility. In particular, it is revealed that decisions concerning experimentation have a very interesting impact on the noninferior solution set of options—a phenomenon that has no equivalence in the single‐objective case.

New approach for nonseparable dynamic programming problems

A general class of nonseparable dynamic problems is studied in a dynamic programming framework by introducingkth-order separability. The solution approach uses multiobjective dynamic programming as a separation strategy forkth-order separable dynamic problems. The theoretical grounding on which the optimal solution of the original nonseparable dynamic problem can be attained by a noninferior solution of the corresponding multiobjective dynamic programming problem is established. The relationship between the overall optimal Lagrangian multipliers and the stage-optimal Lagrangian multipliers and the relationship between the overall weighting vector and the stage weighting vector are explored, providing the basis for identifying the optimal solution of the original nonseparable problem from among the set of noninferior solutions generated by the envelope approach.

An interactive approach to identify the best compromise solution for two objective shortest path problems

In recent years there has been a growing interest in multiobjective path problems. Although efficient algorithms exist for solving single objective shortest path problems, the same is not true for generating the noninferior solution set for multiobjective problems, as the number of noninferior solutions may grow exponentially with the number of nodes. Hansen (1980) [ Lecture Notes in Economics and Mathematical Systems 177 (Edited by M. Beckmann and H.P. Kunzi),pp. 109–127. Springer, Berlin], Climaco and Martins (1982) [ Eur. J. Opl Res. 11, 399–404], Martins (1984) [ Eur. J. Opl. Res. 16, 236–245], and Henig (1985) [ Eur. J. Opl. Res. 25, 281–291] present exact algorithms for generating the entire noninferior solution set for a two objective shortest path problem. All of the above authors acknowledge the computational complexity of the problem and consequently, with the exception of Martins (1984), propose algorithms to generate an approximation of the noninferior solution set. None of these authors, however, recommend direct interaction with the decision makers. In this article, we propose an interactive method to generate an approximation of the noninferior solution set for two objective shortest path problems. The goal of this approach is to assist the decision maker in selecting the preferred or best compromise solution from among the noninferior solutions.

Expression of TLX antigen recognized by GB24 on trophoblast and leukocytes

In recent years there has been a growing interest in multiobjective path problems. Although efficient algorithms exist for solving single objective shortest path problems, the same is not true for generating the noninferior solution set for multiobjective problems, as the number of noninferior solutions may grow exponentially with the number of nodes.Hansen (1980) [Lecture Notes in Economics and Mathematical Systems 177 (Edited by M. Beckmann and H.P. Kunzi),pp. 109–127. Springer, Berlin], Climaco and Martins (1982) [Eur. J. Opl Res. 11, 399–404], Martins (1984) [Eur. J. Opl. Res. 16, 236–245], and Henig (1985) [Eur. J. Opl. Res. 25, 281–291] present exact algorithms for generating the entire noninferior solution set for a two objective shortest path problem. All of the above authors acknowledge the computational complexity of the problem and consequently, with the exception of Martins (1984), propose algorithms to generate an approximation of the noninferior solution set. None of these authors, however, recommend direct interaction with the decision makers. In this article, we propose an interactive method to generate an approximation of the noninferior solution set for two objective shortest path problems. The goal of this approach is to assist the decision maker in selecting the preferred or best compromise solution from among the noninferior solutions.

A Consistent and Cost-Effective Quantification of Containment Performance Criteria

Containment performance criteria (CPC) are derived systematically, given top level safety goals related to public risk. The main focus is on the relationships between the top level safety goals and lower level design objectives, and the way in which the latter are determined. A set of CPC is identified in terms of the reliabilities of the systems that perform various containment functions. The multiobjective optimization approach is used as a method for deriving a finite manageable set of self-consistent relations between the top level safety goals and specific containment performance. As a global set of measures of plant performance or objective functions, acute and latent fatalities and the total reliability cost are chosen. The latter is included because it represents both technical and economic limitations in achieving a certain level of the first two members of the global set. A specific application is made to a large dry containment. A set of noninferior solutions (optimized solutions in a multi-objective optimization problem) is shown and discussed.

The envelope approach for multiobjeetive optimization problems

A multiobjective optimization problem is usually solved by finding the set of all noninferior solutions to the problem. A methodology termed the envelope approach is presented for generating the set of noninferior solutions. The relationship between the envelope approach and multiobjective optimization is explored. Investigation of the use of the envelope approach in multiobjective dynamic programming and in the parametric decomposition method shows that this approach is very suitable for solving certain classes of multiobjective optimization problems by decomposition and coordination.

Reliability and Risk Allocation in Nuclear Power Plants: A Decision-Theoretic Approach

A methodology is described for allocating reliability to various nuclear reactor systems, subsystems, components, operations, and structures consistent with a set of global safety criteria that are not rigid. The problem is formulated as a multiattribute decision analysis paradigm; the multiobjective optimization, which is performed on a probabilistic risk assessment model and reliability cost functions, serves as the guiding principle for reliability and risk allocation. The concept of ''noninferiority'' is used in the multiobjective optimization problem. Finding the noninferior solution set is the main theme of the current approach. The assessment of the decision-maker's preferences could then be performed more easily on the noninferior solution set.

Multi‐Objective Decision‐Making in Waste Disposal Planning

A multiple objective programming model of the Boston sludge disposal problem was formulated in which the objectives of net economic benefits, environmental impact, and variability of impacts were included. A generating technique was used to solve for the noninferior solutions. To reduce information overloading on the part of decision-makers and redundancy among solutions, cluster analysis was used to prune the noninferior set of solutions.

Designing the optimal structure of the petrochemical industry for minimum cost and least gross toxicity of chemical production

The factor of toxicity associated with chemical production is incorporated into the simulation of the industrial development of the petrochemical industry which was previously studied for production economics. Choosing threshold limit value as chemical toxicity index, a toxicity model is formulated capable of structuring the industry for the least amount of toxicity resulting from the emission of organic pollutants. The optimal industry also points out the industrial sectors which contribute more to the industry-wide toxicity of chemical production. The industrial structure corresponding to the least gross toxicity is observed to be in conflict with that of the minimum production cost in terms of process selection. Using the theory of multiobjective programming this conflict is then explicitly displayed as a noninferior solution set which comprises a host of optimal structures each representing a particular value judgment between the cost of chemical production and its associated toxicity. Further, based on a balanced view on the two objectives, a small solution set is proposed as the best compromise. Within this solution region, the industrial structures corresponding to both design objectives are observed to be practically identical. This can be seen as a flexible as well as a reasonable compromise solution to the dual-objective industrial problem.

Water Conservation and Capacity Expansion

AS water becomes relatively more scarce and as the cost of developing new sources of water increases, projects designed to reduce the quantities of water consumed (i.e, demand management projects) become more attractive alternatives to supply augmentation projects. A framework is presented that allows planners of water supply systems to make efficient use of water resources by explicitly considering the management of the demand for water as a supplement to traditional supply augmentation projects. The framework includes a dynamic programming algorithm to determine the optimal combination of supply augmentation and demand management projects. Two optimization criteria are considered: (1) Minimization of the present value of the cost of implementing the projects; and (2) minimization of the expected value of the costs to cope with emergencies in the supply of water. The tradeoffs between the two objectives are examined through the development of the set of noninferior solutions using the Non‐Inferior Set Es...

A heuristic weight determination procedure for goal programs used for harvest scheduling models

A heuristic, "unstructured" weight determination procedure was developed for harvest scheduling models in which goal programming algorithms are employed. Six noninferior solution sets that included four goals, total volume harvested, total discounted cash flow, total undiscounted cash flow, and total discounted costs, were generated using a complementary linear and goal programming approach. Representatives from a pulp and paper company (The Company) located in the southeastern United States ranked the solution sets from most to least preferred and provided a statement of why they ranked the sets the way they did. Additional noninferior solutions were generated to improve the achievement levels of the initial preferred solution. The procedure contributes to improved forest management decisions by providing the optimal harvest scheduling plan under each of the four goals individually and by providing goal achievement level trade offs under different weight structures. The procedure was well received and instructive to The Company representatives since it was easy to understand and required minimal analytic expertise and time on the part of the decision makers.