New Optimal Pivot Rule for the Simplex Algorithm

The purpose of this paper is to introduce a new pivot rule of the simplex algorithm. The simplex algorithm first presented by George B. Dantzig, is a widely used method for solving a linear programming problem (LP). One of the important steps of the simplex algorithm is applying an appropriate pivot rule to select the basis-entering variable corresponding to the maximum reduced cost. Unfortunately, this pivot rule not only can lead to a critical cycling (solved by Bland’s rules), but does not improve efficiently the objective function. Our new pivot rule 1) solves the cycling problem in the original Dantzig’s simplex pivot rule, and 2) leads to an optimal improvement of the objective function at each iteration. The new pivot rule can lead to the optimal solution of LP with a lower number of iterations. In a maximization problem, Dantzig’s pivot rule selects a basis-entering variable corresponding to the most positive reduced cost;in some problems, it is well-known that Dantzig’s pivot rule, before reaching the optimal solution, may visit a large number of extreme points. Our goal is to improve the simplex algorithm so that the number of extreme points to visit is reduced;we propose an optimal improvement in the objective value per unit step of the basis-entering variable. In this paper, we propose a pivot rule that can reduce the number of such iterations over the Dantzig’s pivot rule and prevent cycling in the simplex algorithm. The idea is to have the maximum improvement in the objective value function: from the set of basis-entering variables with positive reduced cost, the efficient basis-entering variable corresponds to an optimal improvement of the objective function. Using computational complexity arguments and some examples, we prove that our optimal pivot rule is very effective and solves the cycling problem in LP. We test and compare the efficiency of this new pivot rule with Dantzig’s original pivot rule and the simplex algorithm in MATLAB environment.

A Study of Optimization and Rule/Goal Graph for a Logical Query

Static optimization of logical queries is, in substance, to move selections down as far as possible in evaluating logical queries. This paper extends Ullman's RGG (Rule/Goal Graph) and introduces P- graph, with which a wide range of recursive logical queries can be statically optimized top-down and evaluated bottom-up, some of which are usually optimized by dynamic approaches. The paper also shows that for some logical queries the complexity of pushing selections down and computing bottom-up is related to the complexity of base relation in the queries.

A SECRETARY PROBLEM ON FUZZY SETS

This paper deals with a secretary problem on fuzzy sets, which allows both the recall of applicants and the uncertainty of a current applicant receiving an offer of employmellt. A new decision criterion is given to select a satisfactory applicant. This result extends the works of M.C.K. Yang and M.H. Smith.

Recent advances in system reliability optimization driven by importance measures

System reliability optimization problems have been widely discussed to maximize system reliability with resource constraints.Bimbaum importance is a wellknown method for evaluating the effect of component reliability on system reliability.Many importance measures(IMs)are extended for binary,multistate,and continuous systems from different aspects based on the Bimbaum importance.Recently,these IMs have been applied in allocating limited resources to the component to maximize system performance.Therefore,the significance of Bimbaum importance is illustrated from the perspective of probability principle and gradient geometrical sense.Furthermore,the equations of various extended IMs are provided subsequently.The rules for simple optimization problems are summarized to enhance system reliability by using ranking or heuristic methods based on IMs.The importance-based optimization algorithms for complex or large-scale systems are generalized to obtain remarkable solutions by using IM-based local search or simplification methods.Furthermore,a general framework driven by IM is developed to solve optimization problems.Finally,some challenges in system reliability optimization that need to be solved in the future are presented.

Assessing Flood Risk Using Reservoir Flood Control Rules

The application of conventional flood operation regulation is restricted due to insufficient description of flood control rules for the Pubugou Reservoir in southern China. Based on the requirements of different flood control objects, this paper proposes to optimize flood control rules with punishment mechanism by defining different parameters of flood control rules in response to flood inflow forecast and reservoir water level. A genetic algorithm is adopted for solving parameter optimization problem. The failure risk and overflow volume of the downstream insufficient flood control capacity are assessed through the reservoir operation policies. The results show that an optimised regulation can provide better performance than the current flood control rules.

Optimal Portfolio and Consumption Rule with a CIR Model Under HARA Utility

In the real-world environments,different individuals have different risk preferences.This paper investigates the optimal portfolio and consumption rule with a Cox–Ingersoll–Ross(CIR)model in a more general utility framework.After consumption,an individual invests his wealth into the financial market with one risk-free asset and multiple risky assets,where the short-term rate is driven by the CIR model and stock price dynamics are simultaneously influenced by random sources from both stochastic interest rate and stock market itself.The individual hopes to optimize their portfolios and consumption rules to maximize expected utility of terminal wealth and intermediate consumption.Risk preference of individual is assumed to satisfy hyperbolic absolute risk aversion(HARA)utility,which contains power utility,logarithm utility,and exponential utility as special cases.By using the principle of stochastic optimality and Legendre transform-dual theory,the explicit expressions of the optimal portfolio and consumption rule are obtained.The sensitivity of the optimal strategies to main parameters is analysed by a numerical example.In addition,economic implications are also presented.Our research results show that Legendre transform-dual theory is an effective methodology in dealing with the portfolio selection problems with HARA utility and interest rate risk can be completely hedged by constructing specific portfolios.