Posynomial geometric programming with interval exponents and coefficients

Geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by an exponential or power function. In the real world, many applications of geometric programming are engineering design problems in which some of the problem parameters are estimates of actual values. This paper develops a solution method when the exponents in the objective function, the cost and the constraint coefficients, and the right-hand sides are imprecise and represented as interval data. Since the parameters of the problem are imprecise, the objective value should be imprecise as well. A pair of two-level mathematical programs is formulated to obtain the upper bound and lower bound of the objective values. Based on the duality theorem and by applying a variable separation technique, the pair of two-level mathematical programs is transformed into a pair of ordinary one-level geometric programs. Solving the pair of geometric programs produces the interval of the objective value. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering optimization areas.     

Posynomial geometric programming with parametric uncertainty

Geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by exponential or power function. This paper develops a procedure to derive the lower and upper bounds of the objective of the posynomial geometric programming problem when the cost and constraint parameters are uncertain. The imprecise parameters are represented by ranges, instead of single values. An imprecise geometric program is transformed to a family of conventional geometric programs to calculate the objective value. The derived result is also in a range, where the objective value would appear. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering design areas.     

Cost efficiency measures in data envelopment analysis with data uncertainty

This paper extends the classical cost efficiency (CE) models to include data uncertainty. We believe that many research situations are best described by the intermediate case, where some uncertain input and output data are available. In such cases, the classical cost efficiency models cannot be used, because input and output data appear in the form of ranges. When the data are imprecise in the form of ranges, the cost efficiency measure calculated from the data should be uncertain as well. So, in the current paper, we develop a method for the estimation of upper and lower bounds for the cost efficiency measure in situations of uncertain input and output data. Also, we develop the theory of efficiency measurement so as to accommodate incomplete price information by deriving upper and lower bounds for the cost efficiency measure. The practical application of these bounds is illustrated by a numerical example.     

Posynomial Parametric Geometric Programming with Interval Valued Coefficient

The article presents solution procedure of geometric programming with imprecise coefficients. We have considered problems with imprecise data as a form of an interval in nature. Many authors have solved the imprecise problem by geometric programming technique in a different way. In this paper, we introduce parametric functional form of an interval number and then solve the problem by geometric programming technique. The advantage of the present approach is that we get optimal solution of the objective function directly without solving equivalent transformed problems. Numerical examples are presented to support of the proposed approach.     

A deteriorating multi-item inventory model with fuzzy costs and resources based on two different defuzzification techniques

Normally inventory models of deteriorating items, such as food products, vegetables, etc. involve imprecise parameters, like imprecise inventory costs, fuzzy storage area, fuzzy budget allocation, etc. In this paper, we aim to provide two defuzzification techniques for two fuzzy inventory models using (i) extension principle and duality theory of non-linear programming and (ii) interval arithmetic. On the basis of Zadeh’s extension principle, two non-linear programs parameterized by the possibility level α are formulated to calculate the lower and upper bounds of the minimum average cost at α-level, through which the membership function of the objective function is constructed. In interval arithmetic technique the interval objective function has been transformed into an equivalent deterministic multi-objective problem defined by the left and right limits of the interval. This formulation corresponds to the possibility level, α = 0.5. Finally, the multi-objective problem is solved by a multi-objective genetic algorithm (MOGA). The model has been illustrated through a numerical example and solved for different values of possibility level, α through extension principle and for α = 0.5 via MOGA. As a particular case, the results have been obtained for the inventory model without deterioration. Results from two methods for α = 0.5 are compared.     

Fractional Goal Programming for Fuzzy Solid Transportation Problem with Interval Cost

In this paper, we study a solid transportation problem with interval cost using fractional goal programming approach (FGP). In real life applications of the FGP problem with multiple objectives, it is difficult for the decision-maker(s) to determine the goal value of each objective precisely as the goal values are imprecise, vague, or uncertain. Therefore, a fuzzy goal programming model is developed for this purpose. The proposed model presents an application of fuzzy goal programming to the solid transportation problem. Also, we use a special type of non-linear (hyperbolic) membership functions to solve multi-objective transportation problem. It gives an optimal compromise solution. The proposed model is illustrated by using an example.     

An integer linear programming approach for a class of bilinear integer programs

We propose an Integer Linear Programming (ILP) approach for solving integer programs with bilinear objectives and linear constraints. Our approach is based on finding upper and lower bounds for the integer ensembles in the bilinear objective function, and using the bounds to obtain a tight ILP reformulation of the original problem, which can then be solved efficiently. Numerical experiments suggest that the proposed approach outperforms a latest iterative ILP approach, with notable reductions in the average solution time.     

An interactive method for inventory control with fuzzy lead-time and dynamic demand

Normally, the real-world inventory control problems are imprecisely defined and human interventions are often required to solve these decision-making problems. In this paper, a realistic inventory model with imprecise demand, lead-time and inventory costs have been formulated and an inventory policy is proposed to minimize the cost using man–machine interaction. Here, demand increases with time at a decreasing rate. The imprecise parameters of lead-time, inventory costs and demand are expressed through linear/non-linear membership functions. These are represented by different types of membership functions, linear or quadratic, depending upon the prevailing supply condition and marketing environment. The imprecise parameters are first transformed into corresponding interval numbers and then following the interval mathematics, the objective function for average cost is changed into respective multi-objective functions. These functions are minimized and solved for a Pareto-optimum solution by interactive fuzzy decision-making procedure. This process leads to man–machine interaction for optimum and appropriate decision acceptable to the decision maker’s firm. The model is illustrated numerically and the results are presented in tabular forms.     

An interactive method of tackling uncertainty in interval multiple objective linear programming

Mathematical programming models for decision support must explicitly take account of the treatment of the uncertainty associated with the model coefficients along with multiple and conflicting objective functions. Interval programming just assumes that information about the variation range of some (or all) of the coefficients is available. In this paper, we propose an interactive approach for multiple objective linear programming problems with interval coefficients that deals with the uncertainty in all the coefficients of the model. The presented procedures provide a global view of the solutions in the best and worst case coefficient scenarios and allow performing the search for new solutions according to the achievement rates of the objective functions regarding both the upper and lower bounds. The main goal is to find solutions associated with the interval objective function values that are closer to their corresponding interval ideal solutions. It is also possible to find solutions with non-dominance relations regarding the achievement rates of the upper and lower bounds of the objective functions considering interval coefficients in the whole model.     

Minimizing loss probability bounds for portfolio selection

In this paper, we derive a portfolio optimization model by minimizing upper and lower bounds of loss probability. These bounds are obtained under a nonparametric assumption of underlying return distribution by modifying the so-called generalization error bounds for the support vector machine, which has been developed in the field of statistical learning. Based on the bounds, two fractional programs are derived for constructing portfolios, where the numerator of the ratio in the objective includes the value-at-risk (VaR) or conditional value-at-risk (CVaR) while the denominator is any norm of portfolio vector. Depending on the parameter values in the model, the derived formulations can result in a nonconvex constrained optimization, and an algorithm for dealing with such a case is proposed. Some computational experiments are conducted on real stock market data, demonstrating that the CVaR-based fractional programming model outperforms the empirical probability minimization.     

Stochastic optimization of an urban rail timetable under time‐dependent and uncertain demand

Urban rail planning is extremely complex, mainly because it is a decision problem under different uncertainties. In practice, travel demand is generally uncertain, and therefore, the timetabling decisions must be based on accurate estimation. This research addresses the optimization of train timetable at public transit terminals of an urban rail in a stochastic setting. To cope with stochastic fluctuation of arrival rates, a two‐stage stochastic programming model is developed. The objective is to construct a daily train schedule that minimizes the expected waiting time of passengers. Due to the high computational cost of evaluating the expected value objective, the sample average approximation method is applied. The method provided statistical estimations of the optimality gap as well as lower and upper bounds and the associated confidence intervals. Numerical experiments are performed to evaluate the performance of the proposed model and the solution method.     

Warp effects on calculating interval probabilities

In real-life decision analysis, the probabilities and utilities of consequences are in general vague and imprecise. One way to model imprecise probabilities is to represent a probability with the interval between the lowest possible and the highest possible probability, respectively. However, there are disadvantages with this approach; one being that when an event has several possible outcomes, the distributions of belief in the different probabilities are heavily concentrated toward their centres of mass, meaning that much of the information of the original intervals are lost. Representing an imprecise probability with the distribution’s centre of mass therefore in practice gives much the same result as using an interval, but a single number instead of an interval is computationally easier and avoids problems such as overlapping intervals. We demonstrate why second-order calculations add information when handling imprecise representations, as is the case of decision trees or probabilistic networks. We suggest a measure of belief density for such intervals. We also discuss properties applicable to general distributions. The results herein apply also to approaches which do not explicitly deal with second-order distributions, instead using only first-order concepts such as upper and lower bounds.     

Monotonic bounds in multistage mixed-integer stochastic programming

Multistage stochastic programs bring computational complexity which may increase exponentially with the size of the scenario tree in real case problems. For this reason approximation techniques which replace the problem by a simpler one and provide lower and upper bounds to the optimal value are very useful. In this paper we provide monotonic lower and upper bounds for the optimal objective value of a multistage stochastic program. These results also apply to stochastic multistage mixed integer linear programs. Chains of inequalities among the new quantities are provided in relation to the optimal objective value, the wait-and-see solution and the expected result of using the expected value solution. The computational complexity of the proposed lower and upper bounds is discussed and an algorithmic procedure to use them is provided. Numerical results on a real case transportation problem are presented.     

Sublinear upper bounds for stochastic programs with recourse

Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integrations in contrast with previous upper bounds that require a number of function evaluations that grows exponentially in the number of random variables. The sublinear bound can often be used when other suggested upper bounds are intractable. Computational results indicate that the sublinear approximation provides good, efficient bounds on the stochastic program objective value.This research has been partially supported by the National Science Foundation. The first author's work was also supported in part by Office of Naval Research Grant N00014-86-K-0628 and by the National Research Council under a Research Associateship at the Naval Postgraduate School, Monterey, California.     

Correlation propagation for uncertainty analysis of structures based on a non-probabilistic ellipsoidal model

Traditional non-probabilistic methods for uncertainty propagation problems evaluate only the lower and upper bounds of structural responses, lacking any analysis of the correlations among the structural multi-responses. In this paper, a new non-probabilistic correlation propagation method is proposed to effectively evaluate the intervals and non-probabilistic correlation matrix of the structural responses. The uncertainty propagation process with correlated parameters is first decomposed into an interval propagation problem and a correlation propagation problem. The ellipsoidal model is then utilized to describe the uncertainty domain of the correlated parameters. For the interval propagation problem, a subinterval decomposition analysis method is developed based on the ellipsoidal model to efficiently evaluate the intervals of responses with a low computational cost. More importantly, the non-probabilistic correlation propagation equations are newly derived for theoretically predicting the correlations among the uncertain responses. Finally, the multi-dimensional ellipsoidal model is adopted again to represent both uncertainties and correlations of multi-responses. Three examples are presented to examine the accuracy and effectiveness of the proposed method both numerically and experimentally.     

Intuitionistic fuzzy optimization technique for Pareto optimal solution of manufacturing inventory models with shortages

This paper discusses a manufacturing inventory model with shortages where carrying cost, shortage cost, setup cost and demand quantity are considered as fuzzy numbers. The fuzzy parameters are transformed into corresponding interval numbers and then the interval objective function has been transformed into a classical multi-objective EPQ (economic production quantity) problem. To minimize the interval objective function, the order relation that represents the decision maker’s preference between interval objective functions has been defined by the right limit, left limit, center and half width of an interval. Finally, the transformed problem has been solved by intuitionistic fuzzy programming technique. The proposed method is illustrated with a numerical example and Pareto optimality test has been applied as well.     

带上下界的双参与人双向委托代理模型及优化算法研究

考虑现实中双参与人同时具有委托人和代理人双重身份情形下,双参与人间的互为委托代理关系,设计虚拟委托人期望效用函数表达式,建立带上下界的双参与人双边约束双向委托代理模型,利用不动点定理确定参数的上下界,并运用不等式组的旋转算法并结合序列二次规划算法进行求解.通过算例分析表明,为达到联盟总效用最大化,需通过确定联盟成员各自合适的保留效用值,以平衡联盟成员的投资和回报,真正实现对联盟成员的激励.     

Error analysis for convex separable programs: Bounds on optimal and dual optimal solutions

Computable lower and upper bounds on the optimal and dual optimal solutions of a nonlinear, convex separable program are obtained from its piecewise linear approximation. They provide traditional error and sensitivity measures and are shown to be attainable for some problems. In addition, the bounds on the solution can be used to develop an efficient solution approach for such programs, and the dual bounds enable us to determine a subdivision interval which insures the objective function accuracy of a prespecified level. A generalization of the bounds to certain separable, but nonconvex, programs is given and some numerical examples are included.     

A branch and bound method for stochastic global optimization

A stochastic branch and bound method for solving stochastic global optimization problems is proposed. As in the deterministic case, the feasible set is partitioned into compact subsets. To guide the partitioning process the method uses stochastic upper and lower estimates of the optimal value of the objective function in each subset. Convergence of the method is proved and random accuracy estimates derived. Methods for constructing stochastic upper and lower bounds are discussed. The theoretical considerations are illustrated with an example of a facility location problem.     

New upper bounds for the two-dimensional orthogonal non-guillotine cutting stock problem

The two-dimensional orthogonal non-guillotine cutting stockproblem (NGCP) appears in many industries (e.g. the wood andsteel industries) and consists of cutting a rectangular mastersurface into a number of rectangular pieces, each with a givensize and value. The pieces must be cut with their edges alwaysparallel to the edges of the master surface (orthogonal cuts).The objective is to maximize the total value of the pieces cut. New upper bounds on the optimal solution to the NGCP are described.The new bounding procedures are obtained by different relaxationsof a new mathematical formulation of the NGCP. Various proceduresfor strengthening the resulting upper bounds and reducing thesize of the original problem are discussed. The proposed newupper bounds have been experimentally evaluated on test problemsderived from the literature. Comparisons with previous boundingprocedures from the literature are given. The computationalresults indicate that these bounds are significantly betterthan the bounds proposed in the literature.