Interval Goal Programming Research Articles

Accurate preference-based method to obtain the deterministically optimal and satisfactory fairness-efficiency trade-off

The resource allocation problem is a classic multi-objective challenge, particularly when balancing the fairness-efficiency trade-off. To achieve a deterministically optimal and satisfactory solution, researchers frequently employ preference-based methods, including selecting among Pareto solutions based on the decision-maker's a posteriori preference and using deterministic models incorporating a priori preferences. In this study, we address two main challenges—specifically, (1) the limitations in measuring the abstract concepts of fairness and efficiency and (2) finding a deterministically optimal and satisfactory balance between fairness and efficiency. We apply a Gini impurity index derived from the classification and regression tree to calculate fairness, ensuring the Gini index function's differentiability. Additionally, we unify the scales of fairness and efficiency to facilitate calculation. Using accurate preference information, we employ the extended interval goal programming method to solve the model and achieve a deterministically optimal and satisfactory solution. The comparative analysis results demonstrate that our model (1) efficiently addresses the real-world water resource allocation problem concerning the fairness-efficiency trade-off; and (2) generates fewer penalties, with an average improvement ratio of 8% in the case study, using more refined penalty functions that align closer to the decision-maker's real and nonlinear preferences.

A novel behavioral penalty function for interval goal programming with post-optimality analysis

Goal programming (GP) is a multi-objective extension of linear programming. Interval GP (IGP) is one of the earliest methods to expand the range of preferred structures in GP. The decision maker’s (DM’s) utility or preference in IGP is investigated by incorporating a widening range of underlying utility functions, commonly known as penalty functions. The basic idea of these functions is that undesirable deviations from the target levels of the goals are penalized regarding a constant or variable penalty value. The main concern with introducing the penalty functions is providing a wide range of a priori preference structures. Yet, the evaluation of how undesirable deviations are penalized based on DM’s behavioral preferences is not sufficiently addressed in the penalty function types developed in the GP literature. In real-world scenarios involving risk, the achievement levels of decision-making attributes are typically associated with the behavior of the DM. In such scenarios, the DM’s unavoidable attitude toward risk should be integrated into the decision-making process. We introduce the concept of behavioral penalty functions into the IGP approach, incorporating a risk aversion parameter tailored to the nature of each attribute to address this gap. This concept offers an innovative framework for capturing the preferences of the DMs and their various attitudes toward risk within the IGP approach. In this paper, we first introduce the concept of behavioral penalty functions. Next, we develop a behavioral utility-based IGP model. Finally, we present a portfolio selection case study to demonstrate the applicability and efficacy of the proposed procedure, followed by a post-optimality analysis and comparisons with other GP approaches.

Renewable energy portfolio optimization with public participation under uncertainty: A hybrid multi-attribute multi-objective decision-making method

With the demand for energy increasing rapidly, the public is increasingly concerned about renewable energy (RE) planning. However, existing studies on RE planning rarely involved public opinions in the decision-making process. This study devotes to introducing a multi-attribute multi-objective decision-making model for RE portfolio selection with public participation. First, to reduce the complexity of the problem resulted from many participators, a linguistic risk appetite-based method is adopted to classify the public into subgroups. Considering that public opinions may be incomplete, the evidential reasoning approach is then used to aggregate the opinions of individuals within a subgroup. Next, the stochastic multi-attribute acceptability analysis method is applied to aggregate the preferences among subgroups and further derive a robust result regarding the public’s social acceptance of RE technologies. On the basis of the derived social acceptance, a risk-based fuzzy interval goal programming model is proposed to derive the optimal RE portfolio. Finally, an illustrative case of optimizing RE portfolios is given to demonstrate the applicability of the proposed model.

Optimizing renewable energy portfolios with a human development approach by fuzzy interval goal programming

• Developing a systematic assessment framework for shaping a RE portfolio. • Considering human development as a primary parameter in RE strategic planning. • Extending the classic FIGP to provide the setting of interval coefficients. • Illustrating the proposed model's capability using a real-world scenario in China. Human development (HD) is considered one of the main metrics for determining the development level of a region. The central motivation of this research is establishing a strategic decision-making paradigm for optimizing renewable energy project (REP) portfolios, with a particular application to east China to quantify how HD influences a REP portfolio selection. The REP portfolio selection is subject to many unclear criteria and various sustainability goals. Fuzzy interval goal programming (FIGP) models are used to deal with the imprecision and uncertainty of the models' goals and coefficients. In this study, firstly, four objectives of social, economic, environmental, and technical criteria, as well as 12 sub-criteria were weighted by fuzzy ANP. Secondly, the potential REPs were assessed on these criteria to deal with the underlying ambiguities. Thirdly, the performance of potential projects was evaluated by the MABAC technique. Fourthly, a mixed-integer linear programming model was constructed and the FIGP is employed to form an optimal portfolio. Eventually, a sensitivity analysis was done to verify the validity of the suggested system. Based on this study's findings, HD is the most significant criterion. These innovative results will assist policymakers in better understanding the role of HD in the REP portfolio selection process.

Optimizing sustainable and renewable energy portfolios using a fuzzy interval goal programming approach

Determining the most sustainable renewable energy (RE) source portfolio that meets decision-maker preferences is a complicated and uncertain multi-criteria decision-making (MCDM) problem. The RE selection process involves meeting decision-makers’ (DMs) preferences wherein several conflicting criteria are present, such as environmental, societal, and economic. Fuzzy goal programming (FGP) is one of the most well-known techniques for dealing with uncertainty existing in MCDM problems. However, conventional FGP techniques suppose only one single coefficient (or parameter) for each decision variable. This paper proposes a novel multi-objective decision-making model called fuzzy interval goal programming (FIGP) to release the restrictions of FGP with single-coefficient modeling. The proposed model can formulate an interval coefficient for each decision variable. To formulate such a model, this study adopts the concept of multi-choice aspiration levels (MCALs) from the revised multi-choice goal programming (RMCGP) technique. Specifically, the integrated model considers various types of fuzzy goals in real-world problems and offers DMs more flexibility to express and formulate their preferences in terms of fuzzy interval goals. The proposed method is illustrated by selecting the optimal RE portfolio for electricity generation in Italy. The relevant renewables are biomass, solar photovoltaic (PV), tidal currents, and wind energy. An empirical analysis shows that the proposed methodology is capable of assisting the DMs in ascertaining the optimal portfolio of RE under a high level of uncertainty and in imprecise environments.

Sustainability prioritization framework of biorefinery: A novel multi-criteria decision-making model under uncertainty based on an improved interval goal programming method

Abstract As a sustainable substitute of traditional petroleum refining technology, the biorefinery, which has advantages of higher resource utilization efficiency, better sustainability and lower negative environmental impacts, have developed and applied. However, the variety of developed biorefinery system makes it difficult for decision-makers to determine the most sustainable biorefinery system. How to deal with the uncertainties in the decision-making process is another issue existing in biorefinery prioritization. In order to solve the decision-making problem under uncertainties in the selection process of biorefineries, a novel decision-making framework based on an improved interval goal programming was developed. In this study, a case study, which is to select the most sustainable biorefinery technology from grain ethanol (GE), cellulosic ethanol (CE), and Fischer-Tropsch diesel (FTD), was adopted to practice the proposed method. Three other methods were also employed to validate the results determined by the proposed method in this study. The results revealed that FTD was recognized as the most sustainable scenario by the proposed method and the proposed method was validated and recognized as the most robust method in uncertain decision-making problem.

Multiobjective bilevel optimisation method to solve environmental-economic power generation and dispatch problem with interval data uncertainty

This paper presents a multiobjective bilevel programming (MOBLP) based interval goal programming (IVGP) method for modelling and solving to solve environmental-economic power generation and dispatch (EEPGD) problem using genetic algorithm (GA) in an inexact environment. In the proposed method, the objective functions to be optimised are represented as goals by introducing target intervals for the achievement of objective values in decision horizon. Then, using interval arithmetic rules, the defined planned – interval goals are converted into standard goals as in conventional goal programming (GP) to formulate minsum IVGP model of the problem. In the solution search process, a genetic algorithm (GA) computational scheme is employed in an interactive way to reach the solution of EEPGD problem. The effective use of the approach is illustrated through a case example of the standard IEEE 30-bus 6-generator test system. The model solution is also compared with the solutions obtained by using other approaches studied previously.

A multiobjective interval goal programming method using penalty function

This article demonstrates how the penalty function can efficiently be used for solving multiobjective linear programming problem with interval valued objective functions. In the proposed approach, first the target intervals are obtained for the defined objectives. Then, using interval arithmetic and the concept of interval goal programming (IGP) the interval goals are transformed into crisp goals. In the process of solution, the goal achievement function is constructed as a convex combination of the sum of total penalty and the maximum penalty associated with unwanted deviations of the respective goals. Numerical examples are provided to illustrate the proposed approach.

A multiobjective interval goal programming method using penalty function

This article demonstrates how the penalty function can efficiently be used for solving multiobjective linear programming problem with interval valued objective functions. In the proposed approach, first the target intervals are obtained for the defined objectives. Then, using interval arithmetic and the concept of interval goal programming (IGP) the interval goals are transformed into crisp goals. In the process of solution, the goal achievement function is constructed as a convex combination of the sum of total penalty and the maximum penalty associated with unwanted deviations of the respective goals. Numerical examples are provided to illustrate the proposed approach.

On interval goal programming switching surface robust design for Integral Sliding Mode Control

This paper presents a systematic technique for Integral Sliding Mode Control design applied to linear time-invariant systems, subject to disturbances, whose uncertain parameters belong to closed intervals. This approach considers a computational optimization-based procedure for the switching surface design, formulated as an interval goal programming problem, equivalent to the interval Ackermann׳s equation for the robust eigenvalue placement problem. The design objective is the minimization of the overall deviation from the specified performance for the transient response of the controlled system, which is represented by a polytope of characteristic polynomials. The controllability of interval systems is also numerically treated within an interval analysis framework. An experimental implementation of this switching surface design technique was compared with a classic robust LMI-based design procedure of the switching surface in the position control of a DC motor, coupled to different loads, to illustrate its main features and efficiency.

Interval Goal Programming Approach to Multiobjective Fuzzy Goal Programming Problem with Interval Weights

Abstract This article presents interval goal programming (IGP) approach for solving linear multiobjective fuzzy goal programming problem with interval weights. In the proposed approach, interval weights for achievement of fuzzy goals to their aspired levels on the basis of their relative importance are considered in an uncertain environment. In the model formulation of the problem, the membership functions for each of the fuzzy goals are defined first. Then, the membership functions are transformed into membership goals by assigning the highest membership value (unity) and introducing under-and over-deviational variables to each of them. In the solution process, the interval weights (derived from pairwise interval judgment matrix) associated with the unwanted deviational variables is introduced in the goal achievement function for minimizing them to reach the aspired goal levels of the problem. To illustrate the proposed approach, a numerical example is solved.

Efficient model for interval goal programming with arbitrary penalty function

Penalty function is a key factor in interval goal programming (IGP), especially for decision makers weighing resources vis-a-vis goals. Many approaches (Chang et al. J Oper Res Soc 57:469–473, 2006; Chang and Lin Eur J Oper Res 199, 9–20, 2009; Jones et al. Omega 23, 41–48, 1995; Romero Eur J Oper Res 153, 675–686, 2004; Vitoriano and Romero J Oper Res Soc 50, 1280–1283, 1999)have been proposed for treating several types of penalty functions in the past several decades. The recent approach of Chang and Lin (Eur J Oper Res 199, 9–20, 2009) considers the S-shaped penalty function. Although there are many approaches cited in literature, all are complicated and inefficient. The current paper proposes a novel and concise uniform model to treat any arbitrary penalty function in IGP. The efficiency and usefulness of the proposed model are demonstrated in several numeric examples.

An Interval-Index-Oriented Intelligent Optimization Method and its Industrial Application

Based on the strong coupling and interval requirement of multiple quality indices, the interval-index-oriented optimization method is proposed to effectively realize the optimal control of alumina blending process. Firstly, the lexicographic interval goal programming model is built to describe the process requirements for quality indices. Then, based on the characteristics of the programming model, a kind of classificatory knowledge base is constructed by using the empirical knowledge accumulated in long-term production and the expert reasoning strategy is proposed to realize the optimal control of quality indexes with interval constraints. The results of industrial application shows that the proposed method can realize the optimal control of quality indices. It provides a good optimization mode for other blending processes of nonferrous metal production.

Interval goal programming for S-shaped penalty function

Abstract The approach of Jones and Tamiz (1995) [Jones, D.F., Tamiz, M., 1995. Expanding the flexibility of goal programming via preference modeling techniques. Omega 23, 41–48] has been accepted as the most efficient approach in the field of interval goal programming (IGP). Although several modifications to the original approach have been proposed recently [Vitoriano, B., Romero, C., 1999. Extended interval goal programming. Journal of the Operational Research Society 50, 1280–1283; Chang, C.-T., 2006. Mixed binary interval goal programming. Journal of the Operational Research Society 35, 389–396], all of them cannot formulate IGP with an S-shaped penalty function. In order to improve the utility of IGP, we extend the model of Chang (2006) [Chang, C.-T., 2006. Mixed binary interval goal programming. Journal of the Operational Research Society 35, 389–396] to be able to model an S-shaped penalty function. The newly formulated model is more concise and compact than the method of Li and Yu (2000) and it can easily be applied to a decision problem with the S-shaped penalty function. Finally, an illustrative example (i.e. how to build an appropriate E-learning system) is included for demonstrating the usefulness of the proposed model.

EFFICIENT STRUCTURES OF ACHIEVEMENT FUNCTIONS FOR GOAL PROGRAMMING MODELS

Following the idea of Li (1996), Romero (2004) and Chang (2006), this paper proposes several efficient structures of achievement functions for goal programming and interval goal programming models. These proposed structures of achievement functions are more efficient than the traditional structures of achievement functions such as Lexicographic, Weight, MINMAX (Chebyshev) and interval achievement functions. The structures of achievement functions are comprehensive in terms of the breath of goal programming and interval goal programming. In order to demonstrate the superiority of the proposed methods in terms of the number of iterations and execution time, a computational experiment was conducted on a set of generated test examples.

Modelling aesthetic variables in the valuation of paintings: an interval goal programming approach

The economic valuation of works of art is a decisive subject in the general field of valuation. Unlike in other areas of valuation, the explanatory power of the directly observable and quantifiable variables is very low, therefore, aesthetic criteria must be used to obtain valuation models with a greater explanatory power. Frequently, these aesthetic criteria are not always precise, and experts usually express them as an interval of values. This paper describes different valuation models that use the goal programming optimisation method to include explanatory variables of the closing price in the form of intervals of values. We have also modelled the possibility that an expert can determine the relevance of each observation in the formation of the valuation function depending on the degree of precision with which the variables have been defined.

Mixed binary interval goal programming

This paper focuses on the mixed binary preferences decision problem associated with the use of penalty functions in goal programming. In this sense, a new formulation approach for interval goal programming is derived, which is more efficient than the model of Jones and Tamiz. In addition, to enhance the usefulness of the proposed model, binary variables subject to the environmental constraints are added. This leads to the model of binary interval goal programming. Finally, examples to illustrate these models are given.

Transitive approximation to pairwise comparison matrices by using interval goal programming

Paired comparison is a very popular method for establishing the relative importance of n objects, when they cannot be directly rated. The challenge faced by the pairwise comparison method stems from some missing properties in its associated matrix. In this paper, we focus on the following general problem: given a non-reciprocal and inconsistent matrix computing intransitivities, what is its associated ranking (defined by importance values)? We propose to use inconsistencies as a source of information for obtaining importance values. For this purpose, a methodology with a decomposition and aggregation phase is proposed. Interval Goal Programming will be a useful tool for implementing the aggregation process defined in the second phase.

Aggregation of partial ordinal rankings: an interval goal programming approach

This paper shows how interval goal programming can be a useful tool for aggregating incomplete individual patterns of preference in a group decision-making problem. The different consensus solutions obtained have a precise preferential meaning and hold interesting properties. The way in which the methodology functions is illustrated with the help of a numerical example. Scope and purpose This paper deals with group decision-making problems where decision makers are not able to give a complete ranking of alternatives but partial orders. The aggregation of these partial orders is addressed within a distance-based framework. The proposed approach is made operational with the help of an Interval Goal Programming formulation.

A dynamic interval goal programming approach to the regulation of a lake–river system

This paper describes a model and a decision support tool for the regulation of a lake–river system using both goal sets and goal points for the water level over the year. The inflow forecast is updated periodically, which results in a series of dynamic rolling horizon goal programming problems. This involves heavy computation, and yet it can be successfully done with the spreadsheet program. The related decision support tool with a graphical user interface is called Interactive analysis of dynamic water regulation Strategies by Multi-criteria Optimization (ISMO). The Finnish Environment Institute (FEI) actively uses it in the generation of regulation policy alternatives when considered from the different perspectives of the stakeholders. Copyright © 2001 John Wiley & Sons, Ltd.