Goal programming

Most of the models developed for the cash management problem have focused upon the optimization of a single objective. This approach is unnecessarily limiting. These models ignore multiple objectives of business and additional concerns such as organizational values and environmental constraints. These additional concerns can greatly influence the actual decision process.  This paper presents a goal programming formulation for a cash management problem in which multiple goals are considered during the solution process. Priorities are given to each goal so that a hierarchical goal structure is included in this optimization.

In this paper, a Goal Programming (GP) model is converted into a multi-objective optimization problem (MOO) of minimizing deviations from fixed goals. To solve the resulting MOO problem, a hybrid metaheuristic with two steps is proposed to find the Pareto set's solutions. First, a Record-to-Record Travel with an adaptive memory is used to find first non-dominated Pareto frontier solutions preemptively. Second, a Variable Neighbour Search technique with three transformation types is used to intensify every non dominated solution found in the first Pareto frontier to produce the final Pareto frontier solutions. The efficiency of the proposed approach is demonstrated by solving two nonlinear GP test problems and three engineering design problems. In all problems, multiple solutions to the GP problem are found in one single simulation run. The results prove that the proposed algorithm is robust, fast and simply structured, and manages to find high-quality solutions in short computational times by efficiently alternating search diversification and intensification using very few user-defined parameters.

Purpose -The purpose of this paper is to review the literature which focuses on four major higher education decision problems. These are: resource allocation; performance measurement; budgeting; and scheduling. Design/methodology/approach -Related articles appearing in the international journals from 1996 to 2005 are gathered and analyzed so that the following three questions can be answered: "What kind of decision problems were paid most attention to?"; "Were the multiple criteria decision-making techniques prevalently adopted?"; and "What are the inadequacies of these approaches?" Findings -Based on the inadequacies, some improvements and possible future work are recommended, and a comprehensive resource allocation model is developed taking account of these factors. Finally, a new knowledge-based goal programming technique which integrates some operations of analytic hierarchy process is proposed to tackle the model intelligently. Originality/value -Higher education has faced the problem of budget cuts or constrained budgets for the past 30 years. Managing the process of the higher education system is, therefore, a crucial and urgent task for the decision makers of universities in order to improve their performance or competitiveness.

The success indicator of an institution and company is when the available resources have been used effectively and efficiently. The limited resources available can cause delays in ongoing business processes. The delay in a business process is always closely related to financing. The slower the business process runs, the more the expenses that must be incurred for labor and material costs. Therefore, it is necessary to optimize the cost and time for an organization. It can be minimized by using goal programming. In goal programming, weights are given to make sense of which activities to be watched. In this research using lingo software to calculate goal programming.

This paper presents a preemptive goal programming model for multi-objective nutrient management problem by determining the optimum fertilizer combination for chilli plantation in Sungai Buloh Malaysia. Application of nutrients to the soil is commonly done by using fertilizers. A fertilizer is said to be a complete or mixed fertilizer when it contains nitrogen, phosphorus and potassium (N-P-K). A set of data have been used to test the effectiveness and efficiency of the proposed model. Results of the model indicate that all objectives have been achieved. Moreover with the fertilizer combination, the current cost of fertilizer used for chilli plantation can be reduced. The flexibility of the model can be done by adjusting the goal priorities with respect to the importance of each objective.

Combined methods for modeling in simulation involve an integration of di erent model types. Often, discrete event methods are combined with continuous modeling techniques to model a complex environment that does not yield to formal description with only one model type. It is essential that, when de ning a method for combined modeling, the coupling and decomposition methods are well de ned. We demonstrate with an example model of boiling water that combined modeling is just a special case of using a heterogeneous approach to model composition. Using the heterogeneous approach provides a rich modeling environment where highly complex systems containing many distinct phases and multiple models can be conveniently represented.

In the production/operations management environment it is common for decision makers to be faced with problems involving multiple objectives or goals, which cannot be satisfied simultaneously. The conflicting nature of these objectives results in solutions that involve tradeoffs or compromise.

This note discusses the properties of solutions generated by the minmax models of goal programming (GP) and compromise programming (CP). GP approaches use a certain target point in the criterion (attribute) space to model decision maker's preferences. When the ideal (utopia) point is used as the target, the minmax GP model coincides with the minmax (Chebyshev) CP model. In a recent review of the current GP state-of-the-art, there have been included suggestions that the two equivalent models ensure Pareto eciency of solutions and they guarantee a perfectly balanced allocation among the achievement of the individual targets. In this note, it is shown that the models, in general, do not ensure the eciency of solutions and they do not guarantee the perfect equity among the individual achievements. Moreover, there are given sucient and necessary conditions clarifying when the discussed properties of minmax solutions do occur. Ó

Several paradigms can be used to analyse multiple-criteria decision-making problems. Of these goal programming is probably the most widely used one, at least in management science. Goal programming seems lo offer considerable potential f o r application to multiple-criteria problems in farm planning. However, its applications in agricultural economics have been few and far between. Even these attempfs seem to suffer from some serious misconceptions. In this paper an effort is made to explain the structure of a goal programming model by deriving it from the familiar paradigm of linear programming. This is done to put the porential usefulness of goal programming and its relationship to linear programming in perspective, and to encourage further applications to multiple-criteria decisionmaking in farm planning.

This paper presents an integrated computer simulation, goal programming, and design of experiment (DOE) algorithm to solve the jobshop scheduling problem with multiple objectives. The proposed algorithm is applied to an actual textile dyeing, printing, and finishing workshop for minimization of tardiness and makespan. The simulation model enables managers to find schedule of jobs on machines given a defined combination of decision parameters which provide the smallest completion time of jobs. We then utilize DOE for determining the decision parameters to estimate the effect of both qualitative and qualitative factors through metamodeling. A goal programming model is used to find the optimum values of decision variables subject to a set of technical and managerial constraints. The proposed algorithm concentrates on two main managerial objectives simultaneously whereas previous studies are able to consider a single criterion. Finally, distinctive features of the proposed approach are compared with previous studies and models.

Over a long and remarkably productive career, Professor William W. (Bill) Cooper has made many pioneering contributions to Operations Research and Management Science (OR/MS), with notable forays into the areas of (a) linear and non-linear programming, (b) goal programming, (c) chance-constrained programming, (d) data envelopment analysis, and (e) manpower planning, among others. His legendary partnership with Abraham Charnes has provided results whose connections go back to the 18th century, bearing on problems conceived but left unsolved by Laplace and Gauss. We document cross-fertilizing links among Bill Cooper's multiple research focuses, and their impacts on other researchers. A trace of his work discloses a web of influence that has produced a wide range of advances in OR/MS by those who follow in his footsteps, representing a productive tour de force that shows no sign of abating.

In this paper a collective choice function (CCF), formulated within a p-metric distance function framework, is proposed as a generator of several compromise consensuses. Even though, the proposed CCF is not smooth, it is however demonstrated that it can be straightforwardly transformed into easily computable goal programming models. Finally, several cases of individual preferences aggregation are obtained by providing different interpretations of the CCF parameters: ordinal and complete information, ordinal and partial information and a cardinal case through 'pairwise' comparison matrices.

This paper will investigate the optimum portfolio for an investor, taking into account 5 criteria. The mean variance model of portfolio optimization that was introduced by Markowitz includes two objective functions; these two criteria, risk and return do not encompass all of the information about investment; information like annual dividends, S&P star ranking and return in later years which is estimated by using data from a longer history. Thus portfolio selection is a typical multi-objective decision making (MODM) problem. It is well known that Goal Programming (GP), based on preemptive priorities and target values, has been successful in solving MODM problems. In this paper we rank objectives of the MODM model according to weights elicited from Decision Maker’s (DM) preferences. Then we obtain goals from DM’s opinion. As a guidance for DM, we revise these goals consistent with ranking of objectives by a Linear Programming model in a way that new goals remain as close as possible t...

In this paper, the provision of bus services along different routes that comprise a public transit network is assessed taking into consideration, the service providers, the users and the societal perspectives. This model is based on Data Envelopment Analysis DEA) [Charnes, A., Cooper W.W., Rhodes, E., 1978. Measuring the efficiency of decision-making units. European Journal of Operational Research 2, 429-444] and derives from the Network Model in DEA [Färe, R., Grosskopf, S., 2000. Network DEA. Socio-Economic Planning Sciences 34, 35-49] and Goal Programming in DEA [Athanassopoulos, A., 1995. Goal programming and data envelopment analysis (GoDEA) for target-based multi-level planning: allocating central grants to the Greek local authorities. European Journal of Operational Research 87, 535-550]

In this article, we propose a model that incorporates the preferences of multiple decision makers into a decison-making process using (1) The analytical hierarchy process (AHP); and (2) multiple criteria and multiple constraint levels (MC 2) linear programming in a capital budgeting context. Our model can foster strategic and nonfinaneial factors that are important in the capital budgeting problems of the current business environment. The two-phased solution framework proposed in this article is sufficiently flexible to reach a compromise among decision makers. Our method also facilitates collection of decision makers' preferences to minimize suboptimization of overall company's goals. In addition, application of AHP to derive weights to decision makers' preferences for resource availability decreases the solution complexity. All these characteristics represent a significant improvement compared with previous linear or goal programming approaches to capital budgeting problems.

Performance appraisal is a process used by some firms to evaluate their employees' efficiency and productivity in order to plan their promotion policy, salary policy, layoffs policy, etc. Initially this process was just carried out by the executive staff, but recently it has evolved into an evaluation process based on the opinion of different reviewers, supervisors, collaborators, customers and the employees themselves (360-degree method). In such an evaluation process the reviewers evaluate some indicators related to employees performance appraisal. In this paper we propose an evaluation framework where there are different sets of reviewers taking part in the evaluation process. Since reviewers have a different knowledge about the evaluated employee, it seems suitable to offer a flexible framework in which different reviewers can express their assessments in different finite scales according to their knowledge. The final aim is to compute a global evaluation for each employee, that can be used by the management team to make their decisions regarding their human resources policy. In this way, to obtain a global evaluation for each employee, we propose a methodology able to aggregate individual valuation in a metric L p framework. In this context, the associated optimization problems can be reduced to an Extended Goal Programming formulation that is very easy to compute.

Les développements récents de la théorie de la décision ont fourni divers modèles de préférences sophistiqués pour la prise de décision en environnement complexe (décision dans l'incertain et le risque, décision multicritère, décision collective). Dans ce domaine, l'essentiel de l'effort a porté sur l'accroissement du pouvoir descriptif des modèles pour rendre compte de la diversité des comportements observés, mais aussi sur l'analyse axiomatique et la justification théorique des critères de décision employés. Cependant, lorsqu'il s'agit de problèmes d'optimisation combinatoire, la définition d'un modèle de préférences sur les solutions ne suffit plus pour déterminer le(s) meilleur(s) choix puisque l'ensemble des alternatives (solutions réalisables) est défini en compréhension et sa taille empêche toute énumération explicite. De plus les approches constructives simples de type programmation dynamique ou algorithme glouton ne s'étendent pas naturellement en présence de préférences complexes et l'on a besoin d'algorithmiques spécifiques pour la recherche des solutions préférées. Dans ce contexte, l'objet de cette présentation est tout d'abord, en partant de problèmes d'optimisation classiques dans les graphes (chemins, arbres couvrants, affectation/transport) d'introduire des problèmes d'optimisation combinatoire à base de préférences complexes, et ce dans divers contextes tels que l'optimisation robuste dans l'incertain ou le risque, l'optimisation équitable en décision multiagent et l'optimisation multiobjectif. Nous évoquerons ensuite la complexité des problèmes introduits puis présenterons quelques voies possibles pour la résolution exacte et/ou approchée de tels problèmes. Les quelques thèmes abordés dans cet exposé illustrent en fait un programme de recherche plus vaste, au carrefour de la théorie de la décision et de l'algorithmique, que nous esquisserons en guise de conclusion.

ln this paper we develop a series of nonlinear time/cost tradeoff models for project management. A base model with quadratic cost relations is developed and then extended to include other operational considerations. The use of nonlinear functions to describe time/cost relationships in project scheduling environments permits a more accurate representation of the operational setting than is possible with linear or piecewise linear approximations. Examples of the models presented are solved directly with commercially available nonlinear programming codes.

A cross-country pipeline construction project is exposed to an uncertain environment due to its enormous size (physical, manpower requirement and financial value), complexity in design technology and involvement of external factors. These uncertainties can lead to several changes in project scope during the process of project execution. Unless the changes are properly controlled, the time, cost and quality goals of the project may never be achieved. A methodology is proposed for project control through risk analysis, contingency allocation and hierarchical planning models. Risk analysis is carried out through the analytic hierarchy process (AHP) due to the subjective nature of risks in construction projects. The results of risk analysis are used to determine the logical contingency for project control with the application of probability theory. Ultimate project control is carried out by hierarchical planning model which enables decision makers to take vital decisions during the changing environment of the construction period. Goal programming (GP), a multiple criteria decision-making technique, is proposed for model formulation because of its flexibility and priority-base structure. The project is planned hierarchically in three levels—project, work package and activity. GP is applied separately at each level. Decision variables of each model are different planning parameters of the project. In this study, models are formulated from the owner's perspective and its effectiveness in project control is demonstrated.

This paper analyzes a situation where a decision-maker provides cardinal and ordinal information about her/his preferences towards m objects (criteria, alternatives, etc.). However, the two rankings derived from her/his preferences are different. This situation of ''inconsistent preferences'' is formalized. Moreover, in order to deal with this type of conflictive situation, a satisficing model based on the minimization of p-metric distance functions is proposed. In this way, a final set of weights associated with the m objects, by using the cardinal and the ordinal information, is obtained. The final model corresponds to a linear extended goal programming, so the proposed approach is operationally and computationally efficient, since the computations are reduced to solving a small number of linear programming models of a moderate size.

We examined the psychometric properties of the Self-Regulation Scale (SRS; , a measure of attention control in goal pursuit, in 2 independent studies. Study 1 included young adults (N = 443), whereas Study 2 included young, middle-aged, and older adults (N = 330). In both studies, the SRS showed good internal consistency. In Study 1, the SRS also showed satisfactory test-retest reliability over a 6-week period. We found support for the criterion validity of the SRS in terms of positive correlations with measures of general and domain-specific self-efficacy, proactive coping, and positive affect and in terms of negative correlations with depressive symptoms and negative affect. Hierarchical regression analyses showed that attention control accounted for unique portions of variance in relevant outcome variables above and beyond measures of self-efficacy and proactive coping.

Supplier selection is a strategic decision-making task for any company, as it engages multiple criteria for the evaluation and selection of alternate suppliers. So, multi-criteria decision making (MCDM) is a necessary approach in this domain. When both the qualitative and quantitative criteria are available and required to be evaluated simultaneously in decision making process for the supplier selection, a proper integrated approach is necessary to be adopted in order to select the best supplier and to find out the interrelationship structure within criteria. To address these issues, this paper proposes a unique integrated multi-attribute decision making (MADM) and mathematical programming (MP)-based model in a mixed environment (i.e., considering qualitative and quantitative criteria together) by combining decision making trial and evaluation laboratory (DEMATEL)-based on analytic network process (ANP), i.e., DANP, fuzzy technique for order of preference by similarity to ideal solution, i.e., FTOPSIS and multiple segment goal programming (MSGP). Network relationship map (NRM) is used to outline the interrelationships among the dimensions and criteria. The novelty of the proposed supplier selection approach lies with the fact that both the qualitative and quantitative criteria have been considered in it. Sensitivity analysis has also been carried out to validate the result of our proposed methodology. In addition, the result of our proposed method is supported by DANP-based fuzzy VIseKriterijumska Opti-mizacija I Kompromisno Resenje (FVIKOR) technique and the best supplier has been identified. Delivery schedule and environmental collaboration are revealed to be the most and least important dimensions, respectively.

In the last 40 years, there has been a marked transformation in the development of new methodologies to assist the decision-making process, especially in the development of procedures in multi-criterion decision-making and in multiobjective programming (MOP). Goal programming (GP) is the most commonly known model of MOP and it is today alive more than ever, supported by a network of researchers and practitioners continually feeding it with theoretical developments and applications, all of these with resounding success. This paper paints a picture summarizing the history of GP and suggests a few areas of research in this era of globalization. Ó

There have been signi®cant advances in the theory of goal programming (GP) in recent years, particularly in the area of intelligent modelling and solution analysis. The intention of this paper is to provide an overview of these developments, to detail and assess the current state-of-the-art in the subject, and to highlight areas which seem promising for future research. Modelling techniques such as detection and restoration of pareto eciency, normalisation, redundancy checking, and non-standard utility function modelling are overviewed. The connection between GP and other multiobjective-programming techniques as well as a utility interpretation of GP are examined. The rationality of ranking Multi-Criteria Decision Making techniques, and of placing GP in such a ranking, is discussed. Ó

Laporan Praktikum Riset Operasi: Model goal programming sudah sering dipergunakan dalam penelitianpenelitian terdahulu untuk pemodelan masalah multi sasaran. Goal programming merupakan salah satu model matematis yang dapat dipergunakan sebagai dasar dalam pengambilan keputusan untuk menganalisis dan membuat solusi persoalan yang melibatkan banyak sasaran sehingga diperoleh solusi yang optimal.

Pengerjaan Kasus ini bisa dilihat di link berikut:
https://safeku.com/vuaf7sr

PT. Model 2004 adalah perusahaan yang memproduksi dua jenis produk yaitu Oscilloscope (O) dan Voltmeter (V). Proses produksi untuk kedua produk tersebut melibatkan tiga Stasiun Kerja (SK) yaitu SK Circuit Board (CB), SK Chasis (CH) dan SK Final Assembly (Ass) (Lihat Gambar 1). Data waktu proses produksi pada masing-masing SK untuk ke dua produk tersebut dapat dilihat pada Tabel 1(a) dan Tabel 1(b), dan data ongkos produksinya dapat dilihat pada Tabel 2. Gambar 1: Tahapan proses produksi untuk produk O dan V Proyeksi permintaan pasar dari ke dua produk di atas untuk 4 minggu ke depan diberikan pada Tabel 3. Dari pengalaman masa lalu kesalahan pada proyeksi ramalan permintaan relatif kecil, yaitu 3-4% (ini merupakan Laporan dari Manajer Produksi). Gudang untuk menyimpan produk O dan V memiliki kapasitas yang cukup untuk memenuhi permintaan untuk satu minggu produksi, yaitu 3000 unit produk O dan 5000 unit produk V. Ongkos inventori per minggu diestimasi sebesar 2% dari harga barang. Dari laporan gudang diketahui bahwa persediaan pada awal minggu 1 untuk produk O dan V masing-masing 600 unit dan 1800 unit. Data masa lalu menunjukkan bahwa rata-rata inventori per minggu relatif tinggi, 300-700 unit untuk produk O dan 1000-2000 unit untuk produk V. Laporan dari bagian pengendalian kualitas menunjukkan bahwa persentase produk cacat minimal 0,5% dan maksimal 1,5% dari jumlah produksi per minggu. Bahan baku dan komponen untuk produk tersebut selama ini mudah diperoleh dan dapat memenuhi kebutuhan produksi, dengan harga yang sedikit mengalami kenaikan dari minggu ke minggu, yaitu 2-5% dari biaya pada Tabel 2. Pangsa pasar perusahaan saat ini adalah 55,3%. Dalam rangka memperkuat posisi sebagai market leader, perusahaan akan memenuhi permintaan pasar dengan harga dan kualitas yang bersaing. Namun perusahaan juga bertujuan untuk mencari keuntungan agar dapat terus tumbuh dan berkembang. Dengan memperhatikan tujuan perusahaan tersebut, manajer produksi sekarang ini sedang merencanakan produksi pabriknya untuk 4 minggu ke depan. Manager Pemasaran menekankan perlunya memaximasi layanan kepada konsumen atau dengan kata lain permintaan produk untuk setiap minggu harus dapat dipenuhi pada minggu tersebut. Karena hal ini merupakan faktor kunci agar perusahaan dapat mempertahankan pangsa pasarnya. Namun upaya itu harus juga memperhatikan ongkos produksi dan inventori. Apabila permintaan tidak dapat dipenuhi maka akan terjadi kehilangan penjualan karena konsumen akan membeli produk O maupun V, yang diproduksi oleh perusahaan lain. Dan ini akan menyebabkan perusahaan akan mengalami tidak saja kehilangan keuntungan per unit tapi juga reputasi (besar kehilangan ini diperkirakan (setara dengan) 3-4 kali dari keuntungan per unit). Manager Pemasaran mendapat laporan dari unit penjualannya bahwa terjadinya permintaan yang tidak dapat dipenuhi sebesar 5-7% per minggu, dan angka ini cukup besar mengingat besarnya kehilangan yang diakibatkannya. Isu ini kemudian mendapatkan perhatian serius dari Direktur Pemasaran dan menyampaikannya dalam rapat direksi. Disamping itu, munculnya beberapa perusahaan pesaing yang menawarkan harga yang lebih murah juga menjadi ancaman bagi perusahaan. Hal ini ditunjukkan dengan permintaan yang menurun dalam beberapa minggu terakhir ini. Selanjutnya direksi perusahaan CB CH Ass

This paper proposes a goal programming model for working capital management.Goal programming is necessary to model the working capital decision, as a balance has to be achieved between the conflicting objectives of liquidity and profitability. The model determines, for given working capital turnover and fixed assets turnover ratios, how funds should be maintained between working capital/current assets and fixed assets to achieve targeted levels of liquidity andprofitability, whilst minimizing the opportunity cost/loss of excess liquidity.

Generally, in the portfolio selection problem the Decision Maker (DM) considers simultaneously conflicting objectives such as rate of return, liquidity and risk. Multi-objective programming techniques such as goal programming (GP) and compromise programming (CP) are used to choose the portfolio best satisfying the DMÕs aspirations and preferences. In this article, we assume that the parameters associated with the objectives are random and normally distributed. We propose a chance constrained compromise programming model (CCCP) as a deterministic transformation to multi-objective stochastic programming portfolio model. CCCP is based on CP and chance constrained programming (CCP) models. The proposed program is illustrated by means of a portfolio selection problem from the Tunisian stock exchange market.

This paper contains a complete and updated review of the literature for multiobjective flowshop problems which are among the most studied environments in the scheduling research area. No previous comprehensive reviews exist in the literature.

There have been signi®cant advances in the theory of goal programming (GP) in recent years, particularly in the area of intelligent modelling and solution analysis. The intention of this paper is to provide an overview of these developments, to detail and assess the current state-of-the-art in the subject, and to highlight areas which seem promising for future research. Modelling techniques such as detection and restoration of pareto eciency, normalisation, redundancy checking, and non-standard utility function modelling are overviewed. The connection between GP and other multiobjective-programming techniques as well as a utility interpretation of GP are examined. The rationality of ranking Multi-Criteria Decision Making techniques, and of placing GP in such a ranking, is discussed. Ó

There have been signi®cant advances in the theory of goal programming (GP) in recent years, particularly in the area of intelligent modelling and solution analysis. The intention of this paper is to provide an overview of these developments, to detail and assess the current state-of-the-art in the subject, and to highlight areas which seem promising for future research. Modelling techniques such as detection and restoration of pareto eciency, normalisation, redundancy checking, and non-standard utility function modelling are overviewed. The connection between GP and other multiobjective-programming techniques as well as a utility interpretation of GP are examined. The rationality of ranking Multi-Criteria Decision Making techniques, and of placing GP in such a ranking, is discussed. Ó

In this paper, we consider a supply chain network design problem with popup stores which can be opened for a few weeks or months before closing seasonally in a marketplace. The proposed model is multi-period and multi-stage with multi-choice goals under inventory management constraints and formulated by 0-1 mixed integer linear programming. The design tasks of the problem involve the choice of the popup stores to be opened and the distribution network design to satisfy the demand with three multi-choice goals. The first goal is minimization of the sum of transportation costs in all stages; the second is to minimization of set up costs of popup stores; and the third goal is minimization of inventory holding and backordering costs. Revised multi-choice goal programming approach is applied to solve this mixed integer linear programming model. Also, we provide a realworld industrial case to demonstrate how the proposed model works.

Activity-Based Costing (ABC) makes use of multiple cost drivers. The selection of these cost drivers from a set of candidate cost drivers can pose a difficult selection problem. This paper proposes the use of the Analytic Hierarchy Process (AHP) and a multi-objective programming methodology as aids in making cost driver selections. A demonstration of the application of these methodologies in ABC cost driver selection is presented. The informational efficacy of the proposed combined methodologies is also discussed. © 1997 Elsevier Science B.V.

This paper illustrates the application of "fuzzy subsets" concepts to goal programming in a fuzzy environment. In contrast to a typical goal-programming problem, the goals are stated imprecisely when the decision environment is fuzzy. The paper first considers a fuzzy goal-programming problem with multiple goals having equal weights associated with them. A solution approach based on linear programming is developed. Next, the solution approach is extended to the case where unequal fuzzy weights are associated with multiple goals. Numerical examples are provided for both cases to illustrate the solution procedure.