Multi-choice programming: an overview of theories and applications

This paper presents an exhaustive literature review on theories and applications in the field of multi-choice programming (MCP). The increasing competition in the business world has given rise to the situations where decision-makers are offered with multiple options/information to optimally decide on a single task. Under such circumstances a number of decision-making problems are falling into the scenario of MCPs. Thus, ever-increasing applicability of the MCPs is making more and more Operations Research practitioners to focus on them. This paper analyses all the related work and presents the same by classifying the research problems into two broad categories: theories and applications. As an aid to assist future researchers and practitioners, key insights on the evolution of various variants of MCPs have also been systematically delineated. The paper also throws light on some research gaps and concludes with the scope of future research in this area.     

Multi-choice mixed integer goal programming optimization for real problems in a sugar and ethanol milling company

Goal Programming (GP) is an important analytical approach devised to solve many realworld problems. The first GP model is known as Weighted Goal Programming (WGP). However, Multi-Choice Aspirations Level (MCAL) problems cannot be solved by current GP techniques. In this paper, we propose a Multi-Choice Mixed Integer Goal Programming model (MCMI-GP) for the aggregate production planning of a Brazilian sugar and ethanol milling company. The MC-MIGP model was based on traditional selection and process methods for the design of lots, representing the production system of sugar, alcohol, molasses and derivatives. The research covers decisions on the agricultural and cutting stages, sugarcane loading and transportation by suppliers and, especially, energy cogeneration decisions; that is, the choice of production process, including storage stages and distribution. The MCMIGP allows decision-makers to set multiple aspiration levels for their problems in which “the more/higher, the better” and “the less/lower, the better” in the aspiration levels are addressed. An application of the proposed model for real problems in a Brazilian sugar and ethanol mill was conducted; producing interesting results that are herein reported and commented upon. Also, it was made a comparison between MCMI GP and WGP models using these real cases.     

Multi-choice goal programming with utility functions

Goal programming (GP) has been, and still is, the most widely used technique for solving multiple-criteria decision problems and multiple-objective decision problems by finding a set of satisfying solutions. However, the major limitation of goal programming is that can only use aspiration levels with scalar value for solving multiple objective problems. In order to solve this problem multi-choice goal programming (MCGP) was proposed by Chang (2007a). Following the idea of MCGP this study proposes a new concept of level achieving in the utility functions to replace the aspiration level with scalar value in classical GP and MCGP for multiple objective problems. According to this idea, it is possible to use the skill of MCGP with utility functions to solve multi-objective problems. The major contribution of using the utility functions of MCGP is that they can be used as measuring instruments to help decision makers make the best/appropriate policy corresponding to their goals with the highest level of utility achieved. In addition, the above properties can improve the practical utility of MCGP in solving more real-world decision/management problems.     

Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach

In this paper, two new algorithms are presented to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. The membership functions for the defined fuzzy goals of all objective functions at all levels are developed in the model formulation of the problem; so also are the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels. Then the fuzzy goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtain the most satisfactory solution for all decision makers.     

An efficient method for solving linear goal programming problems

This note proposes a solution algorithm for linear goal programming problems. The proposed method simplifies the traditional solution methods. Also, the proposed method is computationally efficient.     

A fuzzy goal programming approach to multi-objective optimization problem with priorities

In goal programming problem, the general equilibrium and optimization are often two conflicting factors. This paper proposes a generalized varying-domain optimization method for fuzzy goal programming (FGP) incorporating multiple priorities. According to the three possible styles of the objective function, the varying-domain optimization method and its generalization are proposed. This method can generate the results consistent with the decision-maker (DM)’s expectation, that the goal with higher priority may have higher level of satisfaction. Using this new method, it is a simple process to balance between the equilibrium and optimization, and the result is the consequence of a synthetic decision between them. In contrast to the previous method, the proposed method can make that the higher priority achieving the higher satisfactory degree. To get the global solution of the nonlinear nonconvex programming problem resulting from the original problem and the varying-domain optimization method, the co-evolutionary genetic algorithms (GAs), called GENOCOPIII, is used instead of the SQP method. In this way the DM can get the optimum of the optimization problem. We demonstrate the power of this proposed method by illustrative examples.     

A method for solving fuzzy goal programming problems based on MINMAX approach

Narasimhan incorporated fuzzy set theory within goal programming formulation in 1980. Since then numerous research has been carried out in this field. One of the well-known models for solving fuzzy goal programming problems was proposed by Hannan in 1981. In this paper the conventional MINMAX approach in goal programming is applied to solve fuzzy goal programming problems. It is proved that the proposed model is an extension to Hannan model that deals with unbalanced triangular linear membership functions. In addition, it is shown that the new model is equivalent to a model proposed in 1991 by Yang et al. Moreover, a weighted model of the new approach is introduced and is compared with Kim and Whang’s model presented in 1998. A numerical example is given to demonstrate the validity and strengths of the new models.     

Pattern classification by goal programming and support vector machines

Support Vector Machines (SVMs) are now very popular as a powerful method in pattern classification problems. One of main features of SVMs is to produce a separating hyperplane which maximizes the margin in feature space induced by nonlinear mapping using kernel function. As a result, SVMs can treat not only linear separation but also nonlinear separation. While the soft margin method of SVMs considers only the distance between separating hyperplane and misclassified data, we propose in this paper multi-objective programming formulation considering surplus variables. A similar formulation was extensively researched in linear discriminant analysis mostly in 1980s by using Goal Programming(GP). This paper compares these conventional methods such as SVMs and GP with our proposed formulation through several examples.Received: September 2003, Revised: December 2003,     

A chance constraints goal programming model for the advertising planning problem

This paper presents a model which has been designed to decide the number of advertisement in different advertising media and the optimal allocation of the budget assigned to the different media. The main objective of this problem is to maximize the reach to the desired section of people for different media within their maximum allowable budget without violating the max and min number of advertisement goals. The media have been considered as different newspapers and different channels in Televisions. Here in this article the model has been formulated in such a way that the advertisement should reach to those who are suitable for the product instead of going to those section who are not considered suitable for the product as well. A chance constrained goal programming model has been designed after considering the parameter corresponding to reach for different media as random variables. The random variables in this case has been considered as values which have known mean and standard deviations. A case for an upcoming institution who are interested to advertise for its two years Post Graduate Diploma in Management (PGDM) programme to the different newspapers and television channels has been designed to illustrate the solution methodology.     

A multi-objective programming approach to 1.5-dimensional assortment problem

In this paper we study a 1.5-dimensional cutting stock and assortment problem which includes determination of the number of different widths of roll stocks to be maintained as inventory and determination of how these roll stocks should be cut by choosing the optimal cutting pattern combinations. We propose a new multi-objective mixed integer linear programming (MILP) model in the form of simultaneously minimization two contradicting objectives related to the trim loss cost and the combined inventory cost in order to fulfill a given set of cutting orders. An equivalent nonlinear version and a particular case related to the situation when a producer is interested in choosing only a few number of types among all possible roll sizes, have also been considered. A new method called the conic scalarization is proposed for scalarizing non-convex multi-objective problems and several experimental tests are reported in order to demonstrate the validity of the developed modeling and solving approaches.     

A note on article “A tolerance approach to the fuzzy goal programming problems with unbalanced triangular membership function”

Kim and Whang use a tolerance approach for solving fuzzy goal programming problems with unbalanced membership functions [J.S. Kim, K. Whang, A tolerance approach to the fuzzy goal programming problems with unbalanced triangular membership function, European Journal of Operational Research 107 (1998) 614–624]. In this note it is shown that some results in that article are incorrect. The necessary corrections are proposed.     

Interval goal programming for S-shaped penalty function

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.     

Binary fuzzy goal programming

Goal programming is an important technique for solving many decision/management problems. Fuzzy goal programming involves applying the fuzzy set theory to goal programming, thus allowing the model to take into account the vague aspirations of a decision-maker. Using preference-based membership functions, we can define the fuzzy problem through natural language terms or vague phenomena. In fact, decision-making involves the achievement of fuzzy goals, some of them are met and some not because these goals are subject to the function of environment/resource constraints. Thus, binary fuzzy goal programming is employed where the problem cannot be solved by conventional goal programming approaches. This paper proposes a new idea of how to program the binary fuzzy goal programming model. The binary fuzzy goal programming model can then be solved using the integer programming method. Finally, an illustrative example is included to demonstrate the correctness and usefulness of the proposed model.     

A practical weight sensitivity algorithm for goal and multiple objective programming

This paper presents a weight sensitivity algorithm that can be used to investigate a portion of weight space of interest to the decision maker in a goal or multiple objective programme. The preferential information required from the decision maker is an initial estimate of their starting solution, with an equal weights solution being used as a default if this is not available, and preference information that will define the portion of weight space on which the sensitivity analysis is to be conducted. The different types of preferential information and how they are incorporated by the algorithm are discussed. The output of the algorithm is a set of distinct solutions that characterise the portion of weight space searched. The possible different output requirements of decision makers are detailed in the context of the algorithm.The methodology is demonstrated on two examples, one hypothetical and the other relating to predicting cinema-going behaviour. Conclusions and avenues for future research are given.     

Theorems of the alternative for conic integer programming

Farkas’ Lemma is a foundational result in linear programming, with implications in duality, optimality conditions, and stochastic and bilevel programming. Its generalizations are known as theorems of the alternative. There exist theorems of the alternative for integer programming and conic programming. We present theorems of the alternative for conic integer programming. We provide a nested procedure to construct a function that characterizes feasibility over right-hand sides and can determine which statement in a theorem of the alternative holds.     

Inverse conic programming with applications

Linear programming duality yields efficient algorithms for solving inverse linear programs. We show that special classes of conic programs admit a similar duality and, as a consequence, establish that the corresponding inverse programs are efficiently solvable. We discuss applications of inverse conic programming in portfolio optimization and utility function identification.     

Revised multi-choice goal programming

Chang [C.-T. Chang, Multi-choice goal programming, Omega, The Inter. J. Manage. Sci. 35 (2007) 389–396] has recently proposed a new method namely multi-choice goal programming (MCGP) for multi-objective decision problems. The multi-choice goal programming allows the decision maker to set multi-choice aspiration levels for each goal to avoid underestimation of the decision. However, to express the multi-choice aspiration levels, multiplicative terms of binary variables are involved in their model. This leads to difficult implementation and it is not easily understood by industrial participants. In this paper, we propose an alternative method to formulate the multi-choice aspiration levels with two contributions: (1) the alternative approach does not involve multiplicative terms of binary variables, this leads to more efficient use of MCGP and is easily understood by industrial participants, and (2) the alternative approach represents a linear form of MCGP which can easily be solved by common linear programming packages, not requiring the use of integer programming packages. In addition, a new concept of constrained MCGP is introduced for constructing the relationships between goals in this paper. Finally, to demonstrate the usefulness of the proposed method, an illustrate example is included.     

Cascading: an adjusted exchange method for robust conic programming

It is well known that the robust counterpart introduced by Ben-Tal and Nemirovski (Math Oper Res 23:769–805, 1998) increases the numerical complexity of the solution compared to the original problem. Kočvara, Nemirovski and Zowe therefore introduced in Kočvara et al. (Comput Struct 76:431–442, 2000) an approximation algorithm for the special case of robust material optimization, called cascading. As the title already indicates, we will show that their method can be seen as an adjustment of standard exchange methods to semi-infinite conic programming. We will see that the adjustment can be motivated by a suitable reformulation of the robust conic problem.      

Solving a multi-objective multi-skilled manpower scheduling model by a fuzzy goal programming approach

Manpower scheduling is an intricate problem in production and service environments with the purpose of generating fair schedules that consider employers’ objectives and employees’ preferences as much as possible. However, sometimes, vagueness of information related to employers’ objectives and employees’ preferences leads to the fuzzy nature of the problem. This paper presents a multi-objective manpower scheduling model regarding the lack of clarity on the target values of employers’ objectives and employees’ preferences. Hence, a fuzzy goal programming model is developed for the presented model. Afterwards, two fuzzy solution approaches are used to convert the fuzzy goal programming model to two single-objective models. Finally, the results obtained by both single-objective models are compared with each other to select the solution that has the greatest degree of the satisfaction level of employers’ objectives and employees’ preferences.     

Enhanced interactive satisficing method via alternative tolerance for fuzzy goal programming with progressive preference

This paper proposes an enhanced interactive satisficing method via alternative tolerance for fuzzy goal programming with progressive preference. The alternative tolerances of the fuzzy objectives with three types of fuzzy relations are used to model progressive preference of decision maker. In order to improve the dissatisficing objectives, the relaxed satisficing objectives are sacrificed by modifying their tolerant limits. By means of attainable reference point, the auxiliary programming is designed to generate the tolerances of the dissatisficing objectives for ensuring feasibility. Correspondingly, the membership functions are updated or the objective constraints are added. The Max–Min goal programming model (or the revised one) and the test model of the M-Pareto optimality are solved lexicographically. By our method, the dissatisficing objectives are improved iteratively till the preferred result is acquired. Illustrative examples show its power.