Use of goal programming and integer programming for water quality management—A case study of Gaza Strip

This paper describes a project dealing with achieving an optimum mix of water from different underground wells, each having different amounts of nitrates and chlorides. The amounts of chlorides and nitrates in each of the wells may be higher or lower than the World Health Organization (WHO) standards. Therefore, the optimum mix would be the one that meets WHO standard which is 250 mg/l for chlorides and 50 mg/l for nitrates. A goal programming model was developed to identify the combination of wells along with the amounts of water from each well that upon mixing would result in minimizing the deviation of the amounts of chlorides and nitrates from the standards set by WHO. The output of the goal programming model along with the coordinates of the wells identified above was then used for a second model that determines the locations of the mixing points “reservoirs” in such a way that minimizes the total weighted distances from the corresponding wells. Finally, an easy-to-use pumping schedule was developed using integer programming. Results indicate that for the case study, there exist several optima which make it easier for the decision maker to consider other intangible factors if there are any.     

Integer Programming to Minimize Labour Costs

The main purpose of this paper is to demonstrate a real-world application of pure integer programming to find the optimum solution to a labour cost problem. The length of a daily working shift is defined as an integer variable and several shift strategies are analysed to determine the optimum length and shift combinations that satisfy a predicted demand at minimum cost. The state-space model has been used to predict the stochastic behaviour of monthly demands for beer and soft drink. Savings of about 7% of the annual sales have been obtained as a result of implementing the integer programming approach. A numerical example shows that the solution obtained by rounding off the continuous optimal solution does not match with the integer optimal solution. It was also noted that if a rounded-off solution is feasible, then it provides an initial integer solution for the branch-and-bound algorithm that may reduce the computational time.     

Compromise allocation in multivariate stratified sample surveys under two stage randomized response model

A general model for the randomized response (RR) method was introduced by Warner (J. Am. Stat. Assoc. 60:63–69, 1965) when a single-sensitive question is under study. However, since social surveys are often based on questionnaires containing more than one sensitive question, the analysis of multiple RR data is of considerable interest. In multivariate stratified surveys with multiple RR data the choice of optimum sample sizes from various strata may be viewed as a multiobjective nonlinear programming problem. The allocation thus obtained may be called a “compromise allocation” in sampling literature. This paper deals with the two-stage stratified Warner’s RR model applied to multiple sensitive questions. The problems of obtaining compromise allocations are formulated as multi-objective integer non linear programming problems with linear and quadratic cost functions as two separate problems. The solution to the formulated problems are achieved through goal programming technique. Numerical examples are presented to illustrate the computational details.     

Goal Programming and Agricultural Planning

Goal Programming is similar in structure to linear programming, but offers a more flexible approach to planning problems by allowing a number of goals which are not necessarily compatible to be taken into account, simultaneously. The use of linear programming in farm planning is reviewed briefly. Consideration is given to published evidence of the goals of farmers, and ways in which these goals can be represented. A goal programming model of a 600 acre mixed farm is described and evaluated. Advantages and shortcomings of goal programming in relation to linear programming are considered. It is found that goal programming can be used as a means of generating a range of possible solutions to the planning problem.     

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.     

Use of Mixed Integer Programming to Investigate Optimum Running Conditions for a Thermal Power Station and Possible Extension to Capacity

A procedure is described for investigating optimum running conditions for a thermal power station.The solution was obtained by the application of mixed integer programming: by the same method, various possible extensions to capacity were examined which involved evaluating twenty additional combinations of boilers and turbines.     

A bicriteria solid transportation problem with fixed charge under stochastic environment

In this paper, a bicriteria solid transportation problem with stochastic parameters is investigated. Three mathematical models are constructed for the problem, including expected value goal programming model, chance-constrained goal programming model and dependent-chance goal programming model. A hybrid algorithm is also designed based on the random simulation algorithm and tabu search algorithm to solve the models. At last, some numerical experiments are presented to show the performance of models and algorithm.     

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.     

计算机指令优化控制模型

为了对计算机指令进行最优控制设计 ,我们建立了解决最优控制的整数线性规划模型 .由于变量较多 ,约束条件全都是线性的 ,目标函数为一次 ,我们采用单纯形法对问题求解 ,整个算法都用 c语言实现 ,并对实例进行了求解 .本模型很好的解决了计算机指令优化控制的问题 ,也适用于其他类似问题 .     

An Algorithm for Solving the Linear Goal Programming Problem by Solving its Dual

Goal programming, and in particular lexicographic goal programming (i.e. goal programming within a so-called ‘pre-emptive priority’ structure or having non-Archimedean weights), has become one of the most widely used of the approaches for multi-objective mathematical programming. While also applicable to non-linear or integer models, most of the literature has considered the lexicographic linear goal-programming model and its solution via primal simplex-based methods. However, in many cases, enhanced efficiency (and significant additional flexibility) may be gained via an investigation of the dual of this problem. In this paper we consider an algorithm for solving such a dual and also indicate how it may be implemented on conventional (i.e. single objective) simplex software.     

A note on the multidimensional dual

Lexicographic linear goal programming (i.e., the specific form of goal programming wherein one seeks the lexicographic minimum of an ordered set of goal deviations) represents one of the most common and, evidently, widely used forms of goal programming. This form, also described as linear goal programming with a ‘preemptive priority’ structure, has also engendered considerable interest in the literature. However, the dual of the lexicographic linear goal programming model has, thus far, received only minimal exposure. This despite the fact that this dual was developed more than ten years ago. The purpose of this note is to provide a single, unified presentation of the fundamental notion of duality in lexicographic linear goal programming and to indicate existing as well as potential exploitations of this concept.     

Preference modelling by estimating local utility functions for multiobjective optimization

This paper is intended to design goal programming models for capturing the decision maker's (DM's) preference information and for supporting the search for the best compromise solutions in multiobjective optimization. At first, a linear goal programming model is built to estimate piecewise linear local utility functions based on pairwise comparisons of efficient solutions as well as objectives. The interactive step trade-off method (ISTM) is employed to generate a typical subset of efficient solutions of a multiobjective problem. Another general goal programming model is then constructed to embed the estimated utility functions in the original multiobjective problem for utility optimization using ordinary nonlinear programming algorithms. This technique, consisting of the ISTM method and the newly investigated search process, facilitates the identification and elimination of possible inconsistent information which may exist in the DM's preferences. It also provides various ways to carry out post-optimality analysis to test the robustness of the obtained best solutions. A modified nonlinear multiobjective management problem is taken as example to demonstrate the technique.     

When to rush a ‘behind’ in Australian rules football: a dynamic programming approach

In Australian rules football, points are scored when the ball passes over the goal line. Six points are awarded for a goal when the ball passes between the two centre posts, and one point for a ‘behind’, when the ball passes between a centre post and an adjacent outer post. After a behind, the defending team has a free kick from the goal line. It may be worthwhile, particularly in the closing stages of a game, for a defending team voluntarily to concede a behind, by themselves passing the ball between the two outer posts, either to avert the possibility of an imminent goal or to increase the probability of scoring a goal themselves. A dynamic programming model is used to analyse this situation.     

Comparison of Four Goal Programming Algorithms

A major limitation in the use of goal programming has been the lack of an efficient algorithm for model solution. Schniederjans and Kwak recently published a proposal for more efficient solution of goal programming models utilizing dual simplex procedures. A goal programming algorithm based upon that method has been coded, as well as a revised, simplex-based algorithm. These algorithms are compared in terms of accuracy and time requirements with algorithms previously presented by Lee and by Arthur and Ravindran. Solution times for a series of 12 goal programming models are presented. The dual simplex method appears to have superior computational times for models with a large proportion of positive deviational variables in the solution. The revised simplex algorithm appears more consistent in time and accuracy for general goal programming models.     

Optimum control laws in a non-autonomous dynamic model of a gas field

A model of gas field development described as a nonlinear optimum control problem with an infinite planning horizon is considered. The Pontryagin maximum principle is used to solve it. The theorem on sufficient optimumity conditions in terms of constructions of the Pontryagin maximum principles is used to substantiate the optimumity of the extremal solution. A procedure for constructing the optimum solution by dynamic programming is described and is of some methodological interest. The obtained optimum solution is used to construct the Bellman function. Reference is made to a work containing an economic interpretation of the problem.     

Estimation of more than one parameters in stratified sampling with fixed budget

In a multivariate stratified sampling more than one characteristic are defined on every unit of the population. An optimum allocation which is optimum for one characteristic will generally be far from optimum for others. A compromise criterion is needed to work out a usable allocation which is optimum, in some sense, for all the characteristics. When auxiliary information is also available the precision of the estimates of the parameters can be increased by using it. Furthermore, if the travel cost within the strata to approach the units selected in the sample is significant the cost function remains no more linear. In this paper an attempt has been made to obtain a compromise allocation based on minimization of individual coefficients of variation of the estimates of various characteristics, using auxiliary information and a nonlinear cost function with fixed budget. A new compromise criterion is suggested. The problem is formulated as a multiobjective all integer nonlinear programming problem. A solution procedure is also developed using goal programming technique.     

A Linear Goal Programming Model for Planning the Exports of Emerging Countries

A linear goal programming model for planning the exports of emerging countries is formulated in this paper. The model includes some economic factors which may affect the exports, e.g. industrial structure, labour force, value added in exports, capital efficiency, imports of raw material used for manufacturing goods designated for exports, investment plan, etc. An illustrative example is presented. A sensitivity analysis was performed in order to examine the effect of various changes made in the model on the results. It is felt that the model presented in this study would be useful to economic planners in emerging countries.     

A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making

In decision making problems, there may be the cases where the decision makers express their judgements by using preference relations with incomplete information. Then one of the key issues is how to estimate the missing preference values. In this paper, we introduce an incomplete interval multiplicative preference relation and give the definitions of consistent and acceptable incomplete ones, respectively. Based on the consistency property of interval multiplicative preference relations, a goal programming model is proposed to complement the acceptable incomplete one. A new algorithm of obtaining the priority vector from incomplete interval multiplicative preference relations is given. The goal programming model is further applied to group decision-making (GDM) where the experts evaluate their preferences as acceptable incomplete interval multiplicative preference relations. An interval weighted geometric averaging (IWGA) operator is proposed to aggregate individual preference relations into a social one. Furthermore, the social interval multiplicative preference relation owns acceptable consistency when every individual one is acceptably consistent. Two numerical examples are carried out to show the efficiency of the proposed goal programming model and the algorithms.     

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.     

运筹学中若干线性目标规划和线性规划的人工智能-代数解法

运筹学中的线性目标规划和线性规划问题一直分别采用修正单纯形法和单纯形法求解.当变量稍多时计算还是有些繁琐、费时,最近作者通过研究发现,可应用人工智能-代数方法求得这两类问题的解,而且具有相当广泛的适用性.若干例题说明,本法的结果和传统方法的结果由于传统算法在计算中发生的错误,除少数例外大都是一致的.本文的一个 重要目的是希望和广大读者一起研究该方法是否具有通用性.