MATHEMATICAL PROGRAMMING ANALYSIS OF BILINEAR RHEOLOGICAL LUBRICATION MECHANICS

Bilinear theological lubrication mechanics provides an important basis for the designs of re- cently developed electrorheological(ER)'smart'journal bearings and those lubricated by mixed fluid-solid lubri- cants.But there is not yet a reliable and efficient numerical method for such a problem of non-Newtonian flu- id mechanics.In the present paper,a finite element method(FEM)together with mat hematical programming solution is successfully used to solve such a problem.A reliable and generalized numerical method for the designs of electrorheological 'smart' journal bearings and the bearings lubricated by mixed fluid- solid lubri- cant is presented.

Neutrosophic Mathematical Programming for Optimization of Multi-Objective Sustainable Biomass Supply Chain Network Design

In this paper,a multi-objective sustainable biomass supply chain network under uncertainty is designed by neutrosophic programming method.In this method,for each objective function of the problem,three functions of truth membership,non-determination and falsehood are considered.Neutrosophic programming method in this paper simultaneously seeks to optimize the total costs of the supply chain network,the amount of greenhouse gas emissions,the number of potential people hired and the time of product transfer along the supply chain network.To achieve the stated objective functions,strategic decisions such as locating potential facilities and tactical decisions such as optimal product flow allocation and vehicle routing must be made.The results of the implementation of neutrosophic programming method show the high efficiency of this method in achieving the optimal values of each objective function.Also,by examining the rate of uncertainty,it was observed that with increasing this rate,the total costs of supply chain network design,greenhouse gas emissions and product transfer times have increased,and in contrast,the potential employment rate of individuals has decreased.

MATHEMATICAL PROGRAMS WITH SYSTEM OF GENERALIZED VECTOR QUASI-EQUILIBRIUM CONSTRAINTS IN FC-SPACES

In this article, four new classes of systems of generalized vector quasi-equilibrium problems are introduced and studied in FC-spaces without convexity structure. The notions of Ci（x）-FC-partially diagonally quasiconvex, Ci（x）-FC-quasiconvex, and Ci（x）-FC- quasiconvex-like for set-valued mappings are also introduced in FC-spaces. By applying these notions and a maximal element theorem, the nonemptyness and compactness of solution sets for four classes of systems of generalized vector quasi-equilibrium problems are proved in noncompact FC-spaces. As applications, some new existence theorems of solutions for mathematical programs with system of generalized vector quasi-equilibrium constraints are obtained in FC-spaces. These results improve and generalize some recent known results in literature.

OPTIMIZATION OF THE TAKE-OFF MOVEMENT OF SKI JUMPING WITH THE METHOD OF MATHEMATICAL PROGRAMMING

This paper is based on the finite and dispersed data which were obtained from the experiments of the wind tunnel and of the force measurement and from the high-speed photography. It analyses and optimizes the take-off movement of ski jumping with the theory of dynamics of systems of rigid bodies and with the method of mathematical programming. The paper describes the optimal take-off movement of ski jumping. Furthermore, it presents an example and compares the result with those of other papers published at home and abroad. The comparison shows that our computation and optimization are reasonable and well-grounded.

An SQP algorithm for mathematical programs with nonlinear complementarity constraints

In this paper, we describe a successive approximation and smooth sequential quadratic programming （SQP） method for mathematical programs with nonlinear complementarity constraints （MPCC）. We introduce a class of smooth programs to approximate the MPCC. Using an 11 penalty function, the line search assures global convergence, while the superlinear convergence rate is shown under the strictly complementary and second-order sufficient conditions. Moreover, we prove that the current iterated point is an exact stationary point of the mathematical programs with equilibrium constraints （MPEC） when the algorithm terminates finitely.

A new smoothing technique for mathematical programs with equilibrium constraints

A kind of mathematical programs with equilibrium constraints （MPEC） is studied. By using the idea of successive approximation, a smoothing nonlinear programming, which is equivalent to the MPEC problem, is proposed. Thereby, it is ensured that some classical optimization methods can be applied for the MPEC problem. In the end, two algorithm models are proposed with the detail analysis of the global convergence.

A FINITE ELEMENT—MATHEMATICAL PROGRAMMING METHOD FOR ELASTOPLASTIC PROBLEMS BASED ON THE PRINCIPLE OF VIRTUAL WORK

By expanding the yielding function according toTaylor series and neglecting the high order terms, the elastoplastic constitutive equation is written in a linear complementary form. Based on this linear complementary form and the principle of virtual work, a finite element-complementary method is derived for elastoplastic problem. This method is available for materials which satisfy either associated or nonassociated flow rule. In addition, the existence and uniqueness oj solution for the method are also discussed and some useful conclusions are given.

Bi-matrix Games with Fuzzy Strategies and Fuzzy Payoffs and Their Mathematical Programming Equivalents

A fuzzy bi-matrix game（FBG）,namely a two-person non-zero-sum game with fuzzy strategies and fuzzy payoffs is proposed.We have defined and analyzed the optimal strategies of this FBG,and shown that it can be transformed into a corresponding fuzzy mathematical programming issue,for which a ranking function approach can be applied.In addition,optimal strategies of FBG for both Player I and Player II can be gotten.

A MATHEMATICAL PROGRAMMING ALGORITHM FOR LIMIT ANALYSIS

This paper deals with the limit analyses of perfect rigid-plastic continua.Based on the kinematic theorem of the limit analysis theory,a mathematical programming finite element formula for determining the upper bound load multiplier has been established,and an iteration algorithm proposed accordingly.In this algorithm the plastic and rigid zones are distinguished for every iteration step,and the goal function is modified gradually.The difficulties caused by the nonsmoothness of the goal function are over- come.Some examples solved by this algorithm are presented.

Optimal Solution of Multi-Choice Mathematical Programming Problem Using a New Technique

The study deals with the multi-choice mathematical programming problem, where the right hand side of the constraints is multi-choice in nature. However, the problem of multi-choice linear programming cannot be solved directly by standard linear or nonlinear programming techniques. The aim of this paper is to transform such problems to a standard mathematical linear programming problem. For each constraint, exactly one parameter value is selected out of a multiple number of parameter values. This process of selection can be established in different ways. In this paper, we present a new simple technique enabling us to handle such problem as a mixed integer linear programming problem and consequently solve them by using standard linear programming software. Our main aim depends on inserting a specific number of binary variables and using them to construct a linear combination which gives just one parameter among the multiple choice values for each choice of the values of the binary variables. A numerical example is presented to illustrate our analysis.

Some Convexificators-Based Optimality Conditions for Nonsmooth Mathematical Program with Vanishing Constraints

In this paper, by using the notion of convexificator, we introduce the generalized standard Abadie constraint qualification and the generalized MPVC Abadie constraint qualification, and define the generalized stationary conditions for the nonsmooth mathematical program with vanishing constraints (MPVC for short). We show that the generalized strong stationary is the first order necessary optimality condition for nonsmooth MPVC under the generalized standard Abadie constraint qualification. Sufficient conditions for global or local optimality for nonsmooth MPVC are also derived under some generalized convexity assumptions.

Combining reinforcement learning with mathematical programming:An approach for optimal design of heat exchanger networks

Heat integration is important for energy-saving in the process industry.It is linked to the persistently challenging task of optimal design of heat exchanger networks(HEN).Due to the inherent highly nonconvex nonlinear and combinatorial nature of the HEN problem,it is not easy to find solutions of high quality for large-scale problems.The reinforcement learning(RL)method,which learns strategies through ongoing exploration and exploitation,reveals advantages in such area.However,due to the complexity of the HEN design problem,the RL method for HEN should be dedicated and designed.A hybrid strategy combining RL with mathematical programming is proposed to take better advantage of both methods.An insightful state representation of the HEN structure as well as a customized reward function is introduced.A Q-learning algorithm is applied to update the HEN structure using theε-greedy strategy.Better results are obtained from three literature cases of different scales.

Two-Stage Mathematical Programming Approach for Steelmaking Process Scheduling Under Variable Electricity Price

The steelmaking process scheduling problem by considering variable electricity price （SMSPVEP） was in- vestigated. A decomposition approach was proposed for the SMSPVEP. At the first stage, mathematical program-ming was utilized to minimize the maximum completion time for each cast without considering variable electricity price. At the second stage, based on obtained relative schedules of all casts, a mathematical model was formulated with an objective of minimizing the energy cost for all casts scheduling problem. The two-stage models were tested on randomly generated instances based on the practical process in a Chinese steelmaking plant. Computational results demonstrate the effectiveness of the proposed approach.

A Partial Parallel Airport Gate Assignment Supported by a Knowledge－based System Combined with Mathematical Programming

The gate assignment at an airport is one of the major activities in airport operations.With the increase of passenger traffic volumes and the number of flights, the complexity of this task and the factors to be considered have increased significantly, and an efficient gate utilizationhas received considerable attention. For overcoming the shortcomings of previous gate assignmentapproaches, this paper presents a partial parallel gate assignment approach, by which more factorsconcerning aircraft and gates can be collsidered at the same time. This paper also presents themethod of using a knowledge-based system combined with a mathematical programming method forgetting an optimized feasible assignment solution. By this way, it is more easily to get the solutionthat satisfies both the static and dynamic situations,and thus it may adapt well to meet the needsof actual use to rea-time operations. An experimental prototype has been implemented, and a casestudy is presented at the end of the paper.

EXPONENTIAL CONVERGENCE OF SAMPLE AVERAGE APPROXIMATION METHODS FOR A CLASS OF STOCHASTIC MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

In this paper, we propose a Sample Average Approximation （SAA） method for a class of Stochastic Mathematical Programs with Complementarity Constraints （SMPCC） recently considered by Birbil, G/irkan and Listes [3]. We study the statistical properties of obtained SAA estimators. In particular we show that under moderate conditions a sequence of weak stationary points of SAA programs converge to a weak stationary point of the true problem with probability approaching one at exponential rate as the sample size tends to infinity. To implement the SAA method more efficiently, we incorporate the method with some techniques such as Scholtes＇ regularization method and the well known smoothing NCP method. Some preliminary numerical results are reported.

Robust design of static synchronous series compensator-based stabilizer for damping inter-area oscillations using quadratic mathematical programming

This paper presents a procedure for designing a supplementary damping stabilizer for a static synchronous series compensator(SSSC) in multi-machine power systems.The objective is to shift the lightly damped inter-area modes toward the prescribed stability region.A lead-lag stabilizer is used to demonstrate this technique,in which a particular measure of stabilizer gain is considered as an objective function.Constraints of the problem for phase-lead and lag structures are derived.The objective function with the constraints is formed as a quadratic mathematical programming problem.For robust design,the parameters of the stabilizer are calculated under various operating conditions.Two types of SSSC-based stabilizer have been presented and designed.Numerical results including eigenvalue analysis and the nonlinear simulations on the 4-and 50-machine power systems are pre-sented to show the effectiveness of the proposed method.

A Quadratic Programming Model for Blast Scheduling

A quadratic programming model is established to choose the blocks to be blasted in a given period. The length of this period depends on the production planning requirements. During the given period, the blocks' parameters are available from the geological database of the mine. The objective is to minimize the deviation of the average ore grade of blasted blocks from the standard ore grade required by the mill. Transportation ability constraint. production quantity demand constraint. minimum safety bench constraint. block size constraint and block, bench precedence constraints are considered in forming the programming model. This model has more practical objective function and reasonable constraints compared with the existing model for this kind of problems.

Using mixed integer programming and airborne laser scanning to generate forest management units

Airborne laser scanning(ALS)has been widely applied to estimate tree and forest attributes,but it can also drive the segmentation of forest areas.Clustering algorithms are the dominant technique in segmentation but spatial optimization using exact methods remains untested.This study presents a novel approach to segmentation based on mixed integer programming to create forest management units(FMUs).This investigation focuses on using raster information derived from ALS surveys.Two mainstream clustering algorithms were compared to the new MIP formula that simultaneously accounts for area and adjacency restrictions,FMUs size and homogeneity in terms of vegetation height.The optimal problem solution was found when using less than 150 cells,showing the problem formulation is solvable.The results for MIP were better than for the clustering algorithms;FMUs were more compact based on the intravariation of canopy height and the variability in size was lower.The MIP model allows the user to strictly control the size of FMUs,which is not possible in heuristic optimization and in the clustering algorithms tested.The definition of forest management units based on remote sensing data is an important operation and our study pioneers the use of MIP ALS-based optimal segmentation.

PARAMETRIC QUADRATIC PROGRAMMING METHOD FOR VISCOPLASTICITY

Perzyna model in viscoplasticity has been studied by the parametric variational principle, which could be transformed into solving the parametric quadratic programming problem. The FEM form of this problem and its implementation have also been discussed in the paper.

Optimal design of culling compensation policy under the African swine fever——Based on simulations of typical pig farms in China

This paper aims to optimize the culling compensation policy from a micro perspective through scenario simulation.Based on an investigation of 273 pig farms in eight regions,four typical pig farms were constructed according to farm size and breeding mode,representing the swine producers in China.Besides,a decision objective function of pig farms facing suspected African swine fever(ASF)outbreaks was constructed.This study used a mathematical programming model to design and simulate scenarios based on compensation standards and local implementation levels,aiming to incentivize pig farms to report epidemics.The results show that the optimal decisions on epidemic reports differed among typical farms and by herd daily age.The results suggest the following adjustments for optimizing culling compensation policies:(1)to set culling compensation standards based on the market value and(2)to maintain a high level of epidemic surveillance capability in the animal husbandry and veterinary sector.