Weighted Objective Function Research Articles

An Interactive Approach to the Multicriteria Decision Making

The main purpose of this paper is to introduce a solution method for multicriteria problems: The method offers a practical solution by combining judgement with an automatic optimization technique in decision making. This is realised by using the basic ideas behind the method of compromise programming. The "nearly equal" and the "closest" sets of optimal weights of objective functions are established by using the maximum entropy (MEP) and minimum squared deviation principle (MSDP) respectively. The solut1ons in terms of deviation measure from the ideal point are presented to the decisionmaker (OM), and he is only asked to specify the amount by which specific alternatives should exceed the "optimal" alternatives. Such interactive methods allow a re-adjustment of optimal weights, displacement of the ideal point and new pre-decision conflict formation, and leads to a decision in a finite number of iterations. A numerical example is included to illustrate the proposed technique.

Optimum design of actively controlled structures

AbstractThe optimum design of seismic structures equipped with active controls is presented. The design variables are structural parameters, such as structural stiffness or moment of inertia. The optimization procedure is carried out in two stages. First, the optimal control forces are expressed in closed form, as implicit functions of the design variables. In the second stage, the structural parameters are varied in order to minimize a structural weight objective function. The algorithm is applied to structural systems under earthquake excitations, for displacement, frequency and control force constraints.

Network reoptimization procedures for multiobjective network probelems

Network models are attractive because of their computational efficiency. Network applications can involve multiple objective analysis. Multiple objective analysis requires generating nondominated solutions in various forms. Two general methods exist to generate new solutions in continuous optimization: changing objective function weights and inserting objective bounds through constraints. In network flow problems, modifying weights is straightforward, allowing use of efficient network codes. Use of bounds on objective attainment levels can provide a more controlled generation of solutions reflecting tradeoffs among objectives. To constrain objective attainment, however, would require a side constrained network code, sacrificing some computational efficiency for greater model flexibility. We develop reoptimization procedures for the side constrained problem and use them in conjunction with simplex-based techniques. Our approach provides a useful tool for generating solutions allowing greater decision maker control over objective attainments, allowing multiobjective analysis of large-scale problems. Results are compared with solutions obtained from the computationally more attractive weighting technique. Reoptimization procedures are discussed as a means of more efficiently conducting multiple objective network analyses.

An Interactive Entropy Based Method for Solving the Multi-criteria Problems

The main purpose of this paper is to introduce a solution method for multi-criteria linear programming (MCLP) problems. The method offers a practical solution to MCLP problems by combining judgements with an automatic optimization technique in decision making. This is realised by using the ideas underlying the method of compromise programming and the maximum entropy principle (MEP) in an iterative way. The "nearly equal" optimal weights of objective functions in an MCLP problem are established by using the MEP. The solutions in terms of deviation measurement from an ideal point are presented to the decisionmaker, and he has only to specify the amount by which specific alternatives should exceed the “optimal” ones. In general, a nearly-equal weights procedure may incorporate additional constraints for eliciting weights. Such interactive method for solving MCLP allows re-adjustment of optimal weights, displacement of the ideal point and new pre-decision conflict formation, and leads to a decision in a finite number of iterations. A numerical example is included to illustrate the proposed technique.

Weighting factor extensions for finite multiple objective vector minimization problems

This paper develops two procedures for finding the efficient set of finite action multiple-objective problems. The first gives an exact characterization of the efficient set in terms of the weighted q-th-powers of the objective functions, for a fixed q, and gives a lower bound for q. The second gives an exact characterization of the efficient set in the t-th-order minima of the weighted objective functions. Some consideration is given to problems whose feasible action sets correspond to the vertices of certain polytopes and connectedness conjectures are given for which no counterexamples have yet been found.

The setting of weights in linear goal-programming problems

It is the author's view that preemptive linear-goal programming and its associated simplex-based allgorithm is too restrictive in its ability to search out an acceptable solution and imposes an unrealistic burden on the decision maker by requiring a statement of strict preemptive priorities. Also, the question of goal norming is one that has little relevance in terms of what the basic problem really is: i.e. finding a suitable set of objective function weights that can be applied to the over- and under-achievement deviation variables. This paper describes why the author has reached these conclusions, and his approach for solving large-scale goal programs.

The location-allocation problem using multi-objectives

The paper presents a model for the location-allocation problem which incorporates both qualitative and quantitative factors. A normalized composite measure incorporating all the qualitative factors is obtained. The qualitative factor's are weighted on the basis of the relative importance assigned to each factor. An overall weighted objective function is then developed using the qualitative and quantitative measures. The values of the weights are influenced by the desirable emphasis on the qualitative and quantitative factors. The transportation method of linear programming is used to obtain optimal solutions to the above-mentioned problem. These solutions are compared with those obtained from using only the quantitative factor in the objective function. The sensitivity of the solution to the degree of emphasis on the quantitative factor is studied through an example.

Quantitative Analysis of Mineral Mixtures Using Linear Programming

AbstractThe approximate maximum and minimum amounts of any phase in a complex mineral mixture can be determined by solving a linear programming problem involving chemical mass balance and X-ray powder diffraction (XRD) data. The chemical information necessary is the bulk composition of the mixture and an estimation of the compositional range of each of the minerals in the mixture. Stoichiometric constraints for the minerals may be used to reduce their compositional variation. If only a partial chemical analysis for the mixture is available, the maximum amounts of the phases may still be estimated; however, some or all of the stoichiometric constraints may not apply. XRD measurements (scaled using an internal standard) may be incorporated into the linear programming problem using concentration-intensity relations between pairs of minerals. Each XRD constraint added to the linear programming problem, in general, reduces the difference between the calculated maximum and minimum amounts of each phase. Because it is necessary to define weights in the objective function of the linear programming problem, the proposed method must be considered a model. For many mixtures, however, the solution is relatively insensitive to the objective function weights.An example consisting of a mixture of montmorillonite, plagioclase feldspar, quartz, and opal-cristobalite illustrates the linear programming approach. Chemical information alone was used to estimate the mineral abundances. Because quartz and opal-cristobalite are not chemically distinct, it was only possible to determine the sum, quartz + opal-cristobalite, present in the mixture.

The location-allocation problem using multi-objectives

The paper presents a model for the location-allocation problem which incorporates both qualitative and quantitative factors. A normalized composite incorporating all the qualitative factors is obtained. The qualitative factors are weighted on the basis of the relative importance assigned to each factor. An overall weighted objective function is then developed using the qualitative and quantitative measures. The values of the weights are influenced by the desirable emphasis on the qualitative and quantitative factors. The transportation method of linear programming is used to obtain optimal solutions to the above-mentioned problem. These solutions are compared with those obtained from using only the quantitative factor in the objective function. The sensitivity of the solution to the degree of emphasis on the quantitative factor is studied through an example.

A multiple criteria approach to the location-allocation problem

This paper presents a model for the location-allocation problem which considers both qualitative and quantitative factors. The proposed model is based on a weighted objective function. The weights are determined on the basis of the relative importance assigned to the qualitative factors. Applications of the proposed method to real-world problems are discussed and two examples are presented to describe the proposed method. The familiar transportation method of linear programming is used to obtain optimal solutions. Solutions for the examples were compared with those obtained from using only the quantitative factor (transportation cost) and the corresponding results are discussed.

On the set partitioning type formulation for the discrete location problem

An approach to the discrete location problem is presented. The formulation is based on the set partitioning problem with the emphasis being placed on the weighted objective function and the importance of the “value” or “attractiveness” of the potential “covers”, i.e. the number of centres served by a specific location, FORTRAN programs were written to generate the potential subsets of centres to be served, for a specific county in northern Greece, and a mathematical programming package (APEX III) was used in its integer programming mode to provide solutions for various sets of subsets of centres which were generated according to certain criteria.

Three-way metric unfolding via alternating weighted least squares

Three-way unfolding was developed by DeSarbo (1978) and reported in DeSarbo and Carroll (1980, 1981) as a new model to accommodate the analysis of two-mode three-way data (e.g., nonsymmetric proximities for stimulus objects collected over time) and three-mode, three-way data (e.g., subjects rendering preference judgments for various stimuli in different usage occasions or situations). This paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis. We begin with an introduction of the problem and a review of three-way unfolding. The three-way unfolding model, weighted objective function, and new algorithm are presented. Monte Carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance. Finally, three applications of the methodology are reported illustrating the flexibility and robustness of the procedure.

A single facility location problem with a weighted maximin-minimax rectilinear distance

This paper provides an algorithm for locating a single facility in a region, where the objective function is composed of the weighted maximin and minimax rectilinear distances from a set of given demand points. This weighted objective function is applicable when the facility to be located is somewhat desirable but it should not be too close to the demand points, since it also has some undesirable effects. It has been proven in this paper, that it is enough to test for optimality all the intersection points of any two lines forming the equirectilinear distances between any pair of demand points or boundary lines of the region. The algorithm developed here tests these intersection points. The efficient set of points and their optimality range are found. This parametric form of the solution provides an optimal solution for any desired weight.

Smoothing of the pareto optimal solutions set in linear multiobjective optimization problems

Optimal solutions of linear programming problems, if exist, are either extreme points of the feasible region or are not unique. This is also true for linear problems of multiobjective optimization when the utility function is defined as a linear combination of partial objective functions. Moreover in this case continious changes of weighting coefficients of the utility function result in discontinuity of Pareto optimal solutions. In some practical cases, e.g. in the case of water distribution ¦2¦, it is undesirable. To avoid the mentioned disadvantages of linear optimization it is proposed to introduce into the utility function a quadratic term equivalent to a prize given for being away from some “undesirable” constraints. For such a non-linear utility function the Pareto set is smooth. Moreover the weights of partial objective functions are no longer constant as in the case of linear utility function, but depend on the level attained by this objectives. Since there is no reason to assume that the introduced quadratic term is sing-definite the resulting problem is not a convex one. Some possibilities of solving such a problem in the considered case are analysed.

Aggregate production planning using chance-constrained goal programming

Multiple objective models have frequently been proposed to assist in solving aggregate production planning problems. Although such models are an improvement over those with single objectives, demand is usually considered deterministic. For this reason, previous attempts at solving production problems have often lacked realism and could not be successfully applied in many real decision environments. This paper suggests a chance-constrained goal programming (CCGP) approach to production planning which allows the decision maker to specify both probabilistic product demands and production line operating characteristics more in keeping with actual situations. The CCGP approach is based on the sequential solution of a linear programming formulation, allowing efficient solution of large-scale real-world problems using commercially available LP codes. The procedure is demonstrated with a hypothetical example, and proper interpretation of goal achievement is discussed. The findings in the paper are applicable whether preemptive goal programming or a weighted objective function approach is used.

A Goal Programming Example in Public Health Resource Allocation

A model developed to aid a state-level resource allocation process in the United States Department of Agriculture Special Supplement Food Program for Women, Infants, and Children (WIC) is presented. The model is formulated as a linear integer goal program, utilizing service levels for six categories of WIC participants as goals. The model employs a methodical, consistent approach to the allocation process, yet allows ample flexibility for consideration of state-specific issues. By allowing the user to adjust the relative importance of each goal, the model can incorporate subjective attitudes of state-level WIC administrators. This is extremely important as these attitudes are the result of the administrators' familiarity with the WIC program in their states, and the significance of input of this nature should not be neglected. An application of the model to the Indiana WIC program is presented. Sensitivity of the model to changes in the objective function weights and in the target values for the goals is explored. Although orginally developed to allocate budget increases, the model may be used in allocating budget cuts as well; this form of the model is also presented. The model can be extended hierarchically to incorporate resource allocation at the federal level. This modeling approach should be useful in all public sector programs characterized by multiple objective and hierarchical decisionmaking.

On the optimal design of multi-speed gear trains

An efficient method for analyzing multi-speed gear trains is presented and used in this paper. This new technique obtains equations for the diameters of all the gears used in the transmission based on information contained in the speed diagram. The form of the equations is such that they can be generated by the computer automatically. Further, the equations are applicable to the general arrangement, the single composite arrangements, and the double composite arrangements. As a result, all of the promising kinematic arrangements possible for a given number of speeds can be easily studied. By promoting different constraints and objective functions, trace-offs between different parameters can easily be examined. The technique is illustrated using a case study of a 9 speed gear train. A multi-parameter optimization technique is used to solve 19 different arrangements for a weighted objective function minimizing volume and maximizing stiffness.

Parameter estimation of rainfall-runoff models with heteroscedastic streamflow errors — The noninformative data case

The use of the weighted least-squares criterion for parameter estimation of rainfall-runoff models has been restricted because of the practical difficulties encountered in obtaining the adequate weight estimates. In this paper an estimation procedure capable of correcting the effect of heteroscedastic (changing variance) streamflow errors is presented. The noninformative data case is assumed and for that a Bayesian formulation is used to automatically estimate the proper weights. The weights are subsequently used in the maximum likelihood estimator to evaluate the model parameters. The new method was tested on a four-parameter model and compared with the simple least-squares criterion and the weighted objective function of the HEC-1 of the U.S. Army Corps of Engineers, using synthetically generated data. The self-correcting ability of the new procedure was extremely effective in filtering the inconsistencies caused by the heteroscedastic errors. As a result, the parameter estimates were improved and better representative of their true underlying values. The implementation of the new procedure was found to be easy and no convergence problems were experienced.

Optimum shape design of rotating disks

This paper deals with optimum shape design of the rotating disks by nonlinear programming method. The shape of the cross section is defined by 5th degree polynomial which is completely determined by the boundary conditions and four design variables. The stress analysis of the disk is carried out by finite element method using isoparametric elements. The optimization technique used is with improved movelimit method of sequential linear programming. Progress of optimization is investigated with three different objective functions. After preliminary studies a weighted objective function is selected for detailed investigation. Optimum shapes are obtained for different speeds and for different fit pressures from hub. The results are presented in non-dimensionalised form.

Trade‐offs for multiple objective planning through linear programing

Benefit transformation curves are derived from a multiple objective linear programing model. These transformation curves are used to assess the relationship between objectives. The model consists of a weighted objective function which can be parametized. Procedures are suggested to narrow the search for an efficient management strategy on the transformation curve. However, the validity of the transformation curve depends on how non‐commensurables are valued and so different methods of valuation are presented and used. The model is illustrated through application to a policy problem in northern New South Wales, Australia.