Fuzzy Chance Constraints Research Articles

Robust Optimization Research of Cyber-Physical Power System Considering Wind Power Uncertainty and Coupled Relationship.

To mitigate the impact of wind power uncertainty and power-communication coupling on the robustness of a new power system, a bi-level mixed-integer robust optimization strategy is proposed. Firstly, a coupled network model is constructed based on complex network theory, taking into account the coupled relationship of energy supply and control dependencies between the power and communication networks. Next, a bi-level mixed-integer robust optimization model is developed to improve power system resilience, incorporating constraints related to the coupling strength, electrical characteristics, and traffic characteristics of the information network. The upper-level model seeks to minimize load shedding by optimizing DC power flow using fuzzy chance constraints, thereby reducing the risk of power imbalances caused by random fluctuations in wind power generation. Furthermore, the deterministic power balance constraints are relaxed into inequality constraints that account for wind power forecasting errors through fuzzy variables. The lower-level model focuses on minimizing traffic load shedding by establishing a topology-function-constrained information network traffic model based on the maximum flow principle in graph theory, thereby improving the efficiency of network flow transmission. Finally, a modified IEEE 39-bus test system with intermittent wind power is used as a case study. Random attack simulations demonstrate that, under the highest link failure rate and wind power penetration, Model 2 outperforms Model 1 by reducing the load loss ratio by 23.6% and improving the node survival ratio by 5.3%.

Configuration and optimisation of a green closed-loop supply chain with delivery time and green investment considering government subsidy under meta-heuristics algorithms

Consumers' environment consciousness, together with the fierce rivalries among companies, has attracted increasing attention from scholars and industries to the configuration and optimisation of green closed-loop supply chain. However, current studies of closed-loop supply chains rarely consider both the uncertainty from the market and the quality from the recycled used products. Therefore, this article constructs a green closed-loop supply chain total profit model that considers government subsidies, delivery times, green investment and the above uncertainties. Fuzzy chance constraint method is used to solve the uncertain demand; genetic and particle swarm optimisation algorithms are adopted to calculate the model. This study finds (1) enterprises can formulate flexible and adaptable strategies based on different demands; (2) their delivery time decisions rely on both consumer demand and their logistics development levels; (3) enterprises should adjust their delivery times and green investment decisions according to consumers' different sensitivities; (4) enterprises can increase profits by recycling products within a certain range of greenness; (5) for improving recycling and remanufacturing rates, the government can provide secondary subsidies to enterprises for product recycling within the green range of 0.35 ∼ 0.40. This study contributes to the optimisation of strategies developed by enterprises under conditions of subsidies and multiple uncertainties.

Research on optimization of multimodal hub-and-spoke transport network under uncertain demand

In the cargo transportation system, the hub-and-spoke transport network can make full use of the scale effect between logistics hubs and reduce logistics costs. Joint transportation of multiple modes of transportation can give full play to the advantages of different modes of transportation, which not only reduces logistics costs but also improves transportation efficiency. Therefore, this paper combines the advantages of multimodal transportation and hub-and-spoke network, and establishes an optimization model of multimodal hub-and-spoke transport network under demand uncertainty. The model takes into account hub capacity constraints and customer satisfaction with respect to transportation time, and to facilitate the model solution, we utilize the fuzzy expected value method and the fuzzy chance constraints based on credibility to clarify the uncertain variables in the model. We use mixed coding to describe the selection of hubs, assignment of nodes, and choice of transportation modes in this study and use the NSGA-II algorithm with local reinforcement to solve the problem. Finally, numerical experiments are designed to verify the validity of the model and algorithm through sensitivity analysis of relevant parameters, determine the reasonable number of hubs and confidence level, and obtain the influence of the change of hub capacity limit and the ratio of single and double hub transit on the research objectives. The results show that: the NSGA-II algorithm with local reinforcement can significantly improve the convergence speed of the algorithm; There is benefit inversion between economic cost and time cost, and the pursuit of economic cost minimization and time cost minimization, respectively, will lead to different choices of the number of hubs; Increasing the ratio of goods transfer between hubs is beneficial for fully utilizing the scale effect between hubs, achieving the goal of reducing economic costs, but at the same time, it will increase time costs.

Optimal allocation method for MIES-based shared energy storage using cooperative game theory and CSP

To further promote the efficient use of energy storage and the local consumption of renewable energy in a multi-integrated energy system (MIES), a MIES model is developed based on the operational characteristics and profitability mechanism of a shared energy storage station (SESS), considering concentrating solar power (CSP), integrated demand response, and renewable energy output uncertainty. We propose a corresponding MIES model based on co-operative game theory and the CSP and an optimal allocation method for MIES shared energy storage. The model considers the maximum operating benefit of the SESS as the upper objective function and the minimum operating cost of the MIES as the lower objective function. First, the Karush–Kuhn–Tucker conditions of the lower-layer model are transformed into constraints of the upper-layer model, and the Big-M method is used to linearize the nonlinear problem and convert the two-layer nonlinear model into a single-layer linear model. Second, based on the Nash negotiation theory, the benefits of each IES in the MIES are allocated. Finally, the fuzzy chance constraints are used to relax the power balance constraints, and the trapezoidal fuzzy numbers are transformed into a deterministic equivalence class to assess the impact of renewable energy output uncertainty on system operation. The validity and rationality of the proposed two-layer model are verified through simulation, and the results demonstrate that the proposed shared storage capacity leasing model can effectively reduce the total operation cost, increase the profitability of the shared storage operator, and increase the utilization rate of the SESS.

A multi-period multi-objective closed-loop blood supply chain configuration and optimisation under disruption

The COVID-19 pandemic has highlighted the stability of the blood supply chain in case of interruption and other emergencies. Blood is a precious and perishable substance, and any disruption in its supply can cause serious consequences. Current research on the blood supply chain mostly focuses on a multi-objective perspective, and rarely involves both multi-objective and multi-period methods simultaneously. Therefore, we develop a multi-period multi-objective closed-loop blood supply chain model that considers disruptions and uncertainties of demand and recovery, aiming at achieving a balance among various objectives, and coordination of various periods. Fuzzy chance-constraint, meta-heuristic approaches and exact solution (CLPEX) are adopted to mitigate the impact of uncertainties and calculate the optimal solution of a calculating case. The calculation outcomes indicate that: With regard to economic costs and social benefit, the multi-period system is better than the single-period system; the comprehensive performance of the multi-target system is better than that of the single-target system; and the multi-period multi-objective system is more in line with the real situation. Meanwhile, the mitigation strategies, such as maintaining emergency inventory and implementing lateral transshipment, can significantly enhance the performance of the system in disruptions. This paper mainly contributes the follows: Firstly, it provides a reference for configuration and optimisation of a cross-region closed-loop supply chain under disruptions and uncertainties. Secondly, the comprehensive trade-off of multi-objective and multi-period in this paper is especially suitable for perishable production with short cycles, large fluctuations of supply and demand.

Water–Food Nexus System Management under Uncertainty through an Inexact Fuzzy Chance Constraint Programming Method

This study discusses the planning of a regional-scale water–food nexus (WFN) system using an inexact fuzzy chance constraint programming (IFCCP) method. The IFCCP approach can handle uncertainties expressed as interval and fuzzy parameters, as well as the preferences of decision makers. An inexact fuzzy chance constraint programming-based water–food nexus (IFCCP-WFN) model has been developed for the City of Jinan with the consideration of various restrictions related to water and land availability, as well as food and vegetable demands. Solutions for the planting areas for different crops in different periods have been generated under the different preferences of decision makers. The water resource availability would be the priority factor affecting the WFN system under demanding conditions, in which wheat cultivation would be dominated by this factor under fuzzy confidence levels of 0.2 and 0.5, and the planting area of corn would be determined by this factor under high fuzzy confidence levels (e.g., 0.8). In addition, the reliability of irrigation would decrease with increasing fuzzy confidence levels under demanding conditions, limiting the planting areas for crops and leading to a decreasing trend of the system benefit. Adequate water resources would be available for irrigation under optimistic conditions, implying no significant contributions to the planting schemes. Nevertheless, increasing food loss rates would result in more planting areas to satisfy food requirements and thus a greater system benefit under advantageous conditions. Compared with the developed IFCCP-WFN model, the interval-linear-programming-based water–food nexus (ILP-WFN) model can merely reflect the lower and upper bounds of uncertain parameters and neglects the inherent distributional information within the fuzzy parameters. Thus, the ILP-WFN model is unable to reveal the inherent impacts of the fuzzy parameters on the resulting planting strategies.

Stackelberg-Nash asymmetric bargaining-based scheduling optimization and revenue-allocation for multi-operator regional integrated energy system considering competition-cooperation relationship and source-load uncertainties

The integration of source-grid-load-storage greatly improves the energy managing flexibility of regional integrated energy systems (RIES). Considering the conflicting interests of multi-operator, this paper divides RIES into 1) energy supply center (ESC); 2) load aggregator (LA); 3) energy storage aggregator (ESA), and establishes a multi-operator two-layer optimal scheduling model by combining 1) the Stackelbelberg-Nash asymmetric bargaining game based on the genetic algorithm; and 2) the alternating direction method of multipliers algorithm (GA-ADMM). Firstly, considering the absence of historical data, trapezoidal fuzzy parameters are introduced to construct fuzzy chance constraints based on credibility theory to simulate the source-load uncertainties. Secondly, the uncertainty of demand response behavior, real-time carbon credits trading within RIES, and cooperation-competition behavior of followers (LA and ESA) are considered to simulate the asymmetric relationship between followers and leaders (ESC) based on GA. Then, based on the ADMM algorithm, a Nash asymmetric bargaining game considering energy dependency and energy supply-reception status is proposed to realize the revenue allocation between followers. Finally, simulations are implemented to verify the validity and reliability of the proposed model. Results show that: 1) Cooperation improves the energy-managing flexibility of followers, reduces their dependence on ESC, and thus enhances their game ability. 2) The real-time carbon credits trading enables zero-carbon energy interactions and natural gas replenishment within RIES. 3) Increases in the price of various energy and carbon credits can enhance cooperation between followers.

Data-Driven State Transition Algorithm for Fuzzy Chance-Constrained Dynamic Optimization.

Many actual industrial production processes are dynamic and uncertain. When uncertain information are described by subjective experience and experts' knowledge based on scanty or vague information, fuzzy uncertainty exists. Fuzzy chance-constrained dynamic programming are applicable to industrial production modeling accompanied by fuzzy uncertainty and dynamics, where constraints need not or cannot be completely satisfied. In this article, a fuzzy chance-constrained dynamic optimization (FCCDO) formulation on the basis of credibility theory is established, in which, the credibility is used to measure the fuzzy uncertainty level of constraints. To solve the FCCDO problem (FCCDOP), an improved fuzzy simulation technique based on Hammersley sequence sampling is raised to transform fuzzy chance constraints to their deterministic equivalents, and then a data-driven state transition algorithm (DDSTA) using deep neural networks (DNNs) is put forward to achieve a stable, global and robust optimization performance. Finally, the successful applications of the FCCDO method to industrial studies demonstrate its advantages.

CSP-IES economic dispatch strategy with generalized energy storage and a conditional value-at-risk model

To promote the efficient use of energy storage and renewable energy consumption in the integrated energy system (IES), an economic dispatch strategy for the concentrating solar power (CSP)-IES with generalized energy storage and a conditional value-at-risk (CVaR) model is proposed. First, considering the characteristics of energy storage and distributed power supply timing, a CSP-IES configuration is established by using a CSP plant to achieve thermal decoupling of the combined heat and power unit and by defining the thermal storage system of the CSP plant and the battery as the actual energy storage. Second, the fuzzy response of the logistic function is used to optimize the time-of-use tariff to guide load shifting, and the load shifting is defined as virtual energy storage. Third, the CSP-IES economic dispatch model is established to consider the carbon emission allowance model. Finally, considering the system uncertainty, a fuzzy chance constraint is used to relax the system power balance constraint, and then the trapezoidal fuzzy number is transformed into a deterministic equivalence class, and the CVaR model is used as a risk assessment index to quantify the risk cost of the system due to uncertainty. The CSP-IES economic dispatch model with CVaR is constructed. The feasibility and effectiveness of the proposed optimization model are verified by comparing various scenarios.

Project Portfolio Selection Considering the Fuzzy Chance Constraint of Water Environmental Restoration

The water environment restoration project portfolio (WERP) selection is discussed in this paper. By complying with the analysis of the project’s multidimensional property and operation mode, this paper develops the chance constraint and the management constraint of the WERP from the perspectives of public service and enterprise operation. In addition, the multi-objective mixed integer linear programming model is constructed by combining the expectation method and the fuzzy chance constraint programming method. The results demonstrate that: (1) Our proposed method successfully circumvents the occurrence of local objective optimization within a specific confidence interval, thereby achieving a balance between economic and water environment restoration objectives; (2) including fuzzy chance constraints in our proposed method significantly diminishes the risk of exceeding the WERP capacity, thereby ensuring the effectiveness of water environment restoration by adopting a market-based approach. However, further examination of the impact of various sub-projects in WERP is necessary, along with the integration of novel evolutionary algorithms to enhance the efficiency of our model.

Optimal Modeling of Sustainable Land Use Planning under Uncertain at a Watershed Level: Interval Stochastic Fuzzy Linear Programming with Chance Constraints

In this paper, an uncertain interval stochastic fuzzy chance constraint land use optimal allocation method is proposed and applied to solve the problem of land use planning in river basins. The UISFCL-LUP method is an aggregation of interval parametric programming, fuzzy linear programming and chance constraint programming which can cope with uncertain problems such as interval value, fuzzy set and probability. In this paper, the uncertain mathematical method is explored and studied in the optimal allocation of land use in the next two planning periods of Nansihu Lake Basin in China. Moreover, it was proved that ISFCL-LUP can deal with the uncertainty of interval, membership function and probability representation and can also be used to solve the land use planning and land use strategy analysis under uncertain conditions. On the basis of model calculations, we obtained the optimal allocation results for six types of land use in four regions over two planning periods based on different environmental constraints. The results show that the optimized λ value (that is, the degree of satisfaction with all the model conditions) is in the range of [0.54, 0.79] and the corresponding system benefits are between [18.4, 20.4] × 1012 RMB and [96.7, 109.3] × 1012 RMB. The results indicate that land managers can make judgments based on the different socio-economic development needs of different regions and determine strategic land use allocation plans under uncertain conditions. At the same time, the model obtained interval solutions under different system satisfaction and constraint violation probabilities, which helps land managers to analyze the importance of land system optimization and sustainable development more deeply.

Credibility based chance constrained programming for parallel machine scheduling under linear deterioration and learning effects with considering setup times dependent on past sequences

The industry has expressed significant concern regarding the issue of parallel machines and the influence of learning and deterioration. This research investigates non-identical parallel machine scheduling, taking into account the simultaneous consideration of learning effects, deterioration, and past-sequence-dependent setup times. Due to the existence of uncertain parameters in real-world scenarios, the processing times and due dates are assumed to be triangular fuzzy numbers. A fuzzy nonlinear mathematical model with two objective functions is presented and solved using the fuzzy Chance Constraint Programming approach. The two objectives are the summation of earliness and tardiness, as well as makespan. To achieve an efficient near-optimal Pareto front for the problem, a hybrid NSGA-II and VNS multi-objective meta-heuristic is proposed and the results are discussed. Finally, the augmented ε-constraint method is utilized to address issues with small dimensions. The computational analysis demonstrates the effectiveness of this proposed algorithm in tackling problems, especially those with substantial dimensions.

Capacity Allocation Strategy Using Virtual Synchronous Compensator for Renewable Energy Stations Based on Fuzzy Chance Constraints

The uncertainty of high penetration of renewable energy brings challenges to the safe and stable operation of a power system; the virtual synchronous compensation (VSCOM) can shift the demand and compensate real-time discrepancy between generation and demand, and can improve the active support ability for the power system. This paper proposes a novel capacity allocation strategy using VSCOM for renewable energy stations based on fuzzy constraints. Firstly, the basic framework of the VSCOM is constructed with energy storage and reactive power generator (SVG) unit. Secondly, the inertia and standby capacity requirements of high penetration of renewable energy system are modeled; on this basis, a capacity allocation model of each sub unit of the VSCOM is developed, and the investment economy and stable support needs are considered. Thirdly, the uncertainty set of wind power output is defined based on the historical data to find a decision that minimizes the worst-case expected where the worst case should be taken. Finally, the simulation results show that the proposed optimal sizing strategy can effectively take advantage of stability and economy, and the VSCOM can meet the inertia support demand of 98.6% of a high proportion of renewable energy systems.

A Demand Response Transaction Method for Integrated Energy Systems with a Trigonometric Membership Function-Based Uncertainty Model of Costumers’ Responsive Behaviors

As an important regulation tool for power systems, demand response can greatly improve system flexibility and economy. However, when an integrated energy system with a large number of flexible loads is aggregated for a demand response transaction, the uncertainty in the amount of the load response should be considered. Therefore, a demand response transaction model for an integrated energy system that considers the uncertainty of customer demand responses is proposed in this paper. We first analyze the uncertainty of incentive-based demand responses. Next, we investigate the relationship between the incentive level and the fluctuation of customer response volume. The flexible loads are classified into curtailable loads, translatable loads, and replaceable loads. Fuzzy variables are then used to represent the response volume of users, and a trigonometric membership function is used to represent the degree of uncertainty in the response volume of different flexible loads. Finally, the objective functions and chance constraints containing fuzzy variables are converted into explicit equivalence classes for solving. In the case study, the impact of the uncertainty of the user response volume on the revenue of each transaction entity and the impact of the fuzzy chance constraint confidence level on the response revenue are investigated. The results show that the revenue of each transaction entity decreases to a certain extent under the consideration of the uncertainty of the user response volume; the social welfare of the whole transaction increases as the confidence level of the chance constraint changes from high to low.

Stability analysis with general fuzzy measure: An application to social security organizations.

An effective method for evaluating the efficiency of peer decision-making units (DMUs) is data envelope analysis (DEA). In engineering sciences and real-world management problems, uncertainty in input and output data always exists. To achieve reliable results, uncertainties must be taken into account. In this research, a General Fuzzy (GF) approach is designed to cope with uncertainty in the presence of fuzzy observations for categorizing and specifying stability radius and alterations ranges of efficient and inefficient DMUs, which is applicable to real-world decision-making problems. For this purpose, a DEA sensitivity analysis model is presented, which will be modeled by fuzzy sets. Then, by applying the General Fuzzy (GF) approach, the fuzzy DEA sensitivity analysis model is transformed into the equivalent crisp form of fuzzy chance constraints according to specific confidence levels. Finally, a numerical example and a case study of branches of the social security organization are presented to illustrate sensitivity and stability analysis in the presence of fuzzy data. The obtained results provide the input and output changes of the evaluated units according to the attitude and preference of the decision maker with different confidence levels so that the data changes in the fuzzy environment do not change the units’ classification from efficient to inefficient and vice versa.

A memetic algorithm for a relocation-routing problem in green production of gas considering uncertainties

• A fuzzy chance constraint programming model for relocation-routing problem is formulated. • The uncertainty of amount of generated sewage and travel time are considered in the model. • Memetic algorithm and Monte Carlo simulation are integrated to solve the models. • Several series of experiments are conducted to evaluate the proposed model and algorithm. This study introduces a relocation-routing problem with a fuzzy amount of sewage and stochastic travel time in natural gas production. In this study, a set of sewage treatment plants (STPs) and a certain number of gas-gathering stations (GGSs) are distributed on the field. With the increasing amount of production, however, the current STPs cannot satisfy the production level. Policymakers propose several location candidates to build new STPs and aim to minimize the total cost of running the newly built STPs and the original STPs. The practical attributes of the sewage return logistics, capacity of vehicles and STPs, uncertain amount of sewage, stochastic travel times, and other constraints are taken into account. The new problem proposed in this study is defined as Relocation-Routing Problem with Fuzzy Sewage and Stochastic Travel Time (RLRPFSSTT), which has never been investigated before. To minimize the total cost, including the construction cost of newly opened STPs and transportation cost between STPs and GGSs, this paper designs a memetic algorithm to optimize location and routing problems simultaneously. Benchmark-based experimental data is designed, and the computational results demonstrate the effectiveness of the proposed memetic algorithm. Sensitivity analysis and comparisons are also carried out to validate the advantage of considering uncertainties. The proposed model and algorithm are meant to further supplement and extend the location and routing models, as well as have great significance for the decision-makers of industrial logistics in oil fields and coal mines.

Synergetic management of water-energy-food nexus system and GHG emissions under multiple uncertainties: An inexact fractional fuzzy chance constraint programming method

Management of water-food-energy nexus (WEFN) is of great importance to achieve the Sustainable Development Goals. The development of WEFN management strategies is challenged by extensive uncertainties in different system components. Also, agricultural activities would contribute a large portion of the total GHG emissions in many countries, which are affecting the promised carbon neutrality in future. In this study, an inexact fractional fuzzy chance constraint programming method was developed towards planning the water-food-energy nexus system under consideration of both uncertainties and greenhouse gases (GHG) emission. An inexact fractional fuzzy chance constraint programming-based water-energy-food nexus (IFFCCP-WEFN) model has been established under consideration of various restrictions and GHG emissions. Solutions of the planting areas for different crops in different periods have been generated. These results imply that the corn cultivation would be prioritized to satisfy cereal demand due to its relatively lower GHG emission intensity. But the residual resources, after satisfying cereal demand, would tend to be allocated to vegetable planting. Comparison has been conducted among the IFFCCP-WEFN model and WEFN models based the inexact fuzzy chance constraint programming approach with and without GHG emissions. The results indicate that, the results from IFFCCP-WEFN model would achieve a highest unit benefit and lowest total GHG emissions. The total GHG emissions can be 11% less at most than GHG emissions from the resulting crop structures of the other two comparable models. Consequently, the developed IFFCCP-WEFN model can help decision-makers identify the desirable planting structure for crops with a priority of low GHG emission rate. The major contributions in this study include (i) the inexact fractional fuzzy chance constraint programming method to deal with interval and fuzzy parameters, reflect decision makers’ preferences and handle conflicts among contradictory objectives, (ii) the IFFCCP-WEFN model to achieve a maximized unit benefit with respect GHG emissions

Weighting Factors Optimization of Model Predictive Control Based on Fuzzy Thrust Constraints for Linear Induction Machine

The model predictive thrust control (MPTC) is emerging as a promising control technique for the linear induction machine (LIM) drive applications. However, the weighting factor is designed based on empirical procedures that are tedious and time consuming. In this work, a novel fuzzy weighting factor optimization method for LIM is developed to identify the suitable control action. Different from conventional fuzzy methods, the tuning process of weighting factors is regarded as a fuzzy chance constraint so as to constitute a new optimal problem without weighting factors. Hence, the optimal model can be solved by probability optimization algorithm and the corresponding equivalence class is also conducted. At last, simulation results are presented to validate the feasibility and effectiveness of this method.

Multi-objective optimization strategy of multi-sources power system operation based on fuzzy chance constraint programming and improved analytic hierarchy process

Nowadays, the valid development and full utilization of renewable energy sources such as wind power and solar energy are considered to be one of the most significant approaches to get rid of the energy dilemma and realize environmental governance. However, although there are more and more renewable power sources connected to power systems, the renewable energy has not been fully used yet. In this context, an optimization model for the regional power system with the pump storage power, wind, hydro, and thermal power is proposed to enhance the economy, environmental protection, energy-saving and stability of the power system. Besides, the fuzzy chance constraint programming and the improved analytic hierarchy process are used to deal with the uncertainty of the wind power and the load demand and the multi-objectives function, respectively. The case study not only verify the effectiveness of the proposed model but also shows that better comprehensive performance of the power system can be achieved through the proposed model when considering the effect of the pump storage power.

A novel two-phase approach for the bi-objective simultaneous delivery and pickup problem with fuzzy pickup demands

This paper is dedicated to providing a solution framework for the bi-objective vehicle routing problem with simultaneous delivery and pickup which aims to minimize the comprehensive cost as well as maximize the recycling revenue in each round of dispatching. Considering that the real weights of goods to be recycled from the customers are fixed but cannot be precisely given while making one-time routing scheme in the daily operations, the pickup demands are considered as fuzzy numbers, and accordingly a fuzzy chance-constraint programming model for obtaining a prior route solution from the perspective of risk is presented. Afterwards, by demonstrating the continuity and monotonicity of the objective functions, a two-phase approach based on the operational law of the inverse credibility distribution is introduced to solve the model, including translating it into an equivalent nonlinear programming model and resorting to the existing algorithms for obtaining optimal solutions afterwards. Subsequently, in order to validate the performance of the proposed approach and in consideration of the inherent complexity of the vehicle routing problem, a two-phase-based genetic algorithm and the conventional fuzzy simulation-based genetic algorithm are designed and compared by a clothes delivery and pickup problem. The computational results demonstrate that the proposed solution framework is competitive in effectiveness and efficiency, and the parameter analyses provide some suggestions for guidance.