Number Of Decision Alternatives Research Articles

Methods for Minimizing the Savage Function with Various Constraints

Introduction. When making decisions under uncertainty, Savage's criterion is sometimes used, or the criterion of minimizing regrets [1]. Usually, in the literature, this decision situation is described in matrix language. In other words, both the number of decision alternatives and the number of states of nature are finite. Of particular interest are situations where the admissible domain of decision variants is a convex set, and the regrets with respect to each state of nature are expressed by convex functions. In this paper, we propose numerical methods for minimizing Savage's regret function, constructed based on the subgradient projection method with automatic step size adjustment [2, 3]. The convergence of these methods is demonstrated. Goal. In the article, the Savage function is defined as a function that expresses the maximum regret value, assumed to be a convex function with respect to the decision factors. This function measures the effectiveness of each decision relative to the set of states of nature. It is important to note that computing the values of these functions is complex because of the need to know the optimal solution for each state of nature. This difficulty is successfully overcome in the process of solving the problem of minimizing functions on convex sets, thanks to parallel solutions of m "internal" algorithms based on the number of states of nature, and one external algorithm, aimed at minimizing the Savage function. Each of the m+1 algorithms represent modifications of the subgradient projection method with a programmable way of adjusting the step size. Depending on the complexity of the constraints and the required precision, three theorems have been proven, confirming the convergence of the investigated methods. Results obtained. Constructive numerical algorithms have been developed for determining optimal decision alternatives under uncertainty, when the number of states of nature is finite, the admissible domain of control factors is convex and compact, and the Savage regret function serves as a decision criterion. The convergence of the corresponding algorithms to the set of optimal solutions has been proven, without knowing the exact values of the Savage function. Instead, estimates obtained from parallel runs of algorithms were used, aimed at determining optimal solutions for each state of nature. Conclusions. Uncertainty poses significant difficulties in designing and making decisions. Any decision made under uncertainty represents a certain risk or a certain regret. In cases where the number of states of nature is finite, the decision domain is convex, the target function with respect to each state of nature is convex, and the Savage regret function is adopted as the decision criterion, the decision-making problem can be successfully solved using numerical algorithms based on the generalized gradient method. The implementation of the algorithm is relatively simple, and the fields of application can be very diverse. Keywords: uncertainty, decisions, Savage function, optimization.

Uncertainty analysis for effluent trading planning using a Bayesian estimation-based simulation-optimization modeling approach

In this study, a Bayesian estimation-based simulation-optimization modeling approach (BESMA) is developed for identifying effluent trading strategies. BESMA incorporates nutrient fate modeling with soil and water assessment tool (SWAT), Bayesian estimation, and probabilistic–possibilistic interval programming with fuzzy random coefficients (PPI-FRC) within a general framework. Based on the water quality protocols provided by SWAT, posterior distributions of parameters can be analyzed through Bayesian estimation; stochastic characteristic of nutrient loading can be investigated which provides the inputs for the decision making. PPI-FRC can address multiple uncertainties in the form of intervals with fuzzy random boundaries and the associated system risk through incorporating the concept of possibility and necessity measures. The possibility and necessity measures are suitable for optimistic and pessimistic decision making, respectively. BESMA is applied to a real case of effluent trading planning in the Xiangxihe watershed, China. A number of decision alternatives can be obtained under different trading ratios and treatment rates. The results can not only facilitate identification of optimal effluent-trading schemes, but also gain insight into the effects of trading ratio and treatment rate on decision making. The results also reveal that decision maker's preference towards risk would affect decision alternatives on trading scheme as well as system benefit. Compared with the conventional optimization methods, it is proved that BESMA is advantageous in (i) dealing with multiple uncertainties associated with randomness and fuzziness in effluent-trading planning within a multi-source, multi-reach and multi-period context; (ii) reflecting uncertainties existing in nutrient transport behaviors to improve the accuracy in water quality prediction; and (iii) supporting pessimistic and optimistic decision making for effluent trading as well as promoting diversity of decision alternatives.

CVaR-based factorial stochastic optimization of water resources systems with correlated uncertainties

In this paper, a conditional value-at-risk based factorial stochastic programming approach is proposed to address random uncertainties and their interactions in a systematic manner. Random variables can be addressed through a risk-averse method within the two-stage stochastic programming framework. Interactions between random variables are examined through conducting a multi-level factorial analysis. The proposed approach is applied to a case study of water resources management to demonstrate its validity and applicability. A number of decision alternatives are obtained under different risk coefficients, which are useful for decision-makers to make sound water management plan and to perform an in-depth analysis of trade-offs between economic objectives and associated risks. Results obtained from the factorial experiment uncover the multi-level interactions between uncertain parameters and their contributions to the variability of net benefits. The performance of the proposed approach is compared with a factorial two-stage stochastic programming method.

Planning renewable energy in electric power system for sustainable development under uncertainty – A case study of Beijing

An interval type-2 fuzzy fractional programming (IT2FFP) method is developed for planning the renewable energy in electric power system for supporting sustainable development under uncertainty. IT2FFP can tackle output/input ratio problems where complex uncertainties are expressed as type-2 fuzzy intervals (T2FI) with uncertain membership functions. The IT2FFP method is then applied to planning Beijing electric power system, where issues of renewable energy utilization, electricity supply security, and pollutant/greenhouse gas (GHG) emissions mitigation are incorporated within the modeling formulation. The obtained results suggest that the coal-fired power would continue to decrease and the share of renewable energy in gross electricity supply would maintain an increasing trend. Results also reveal that imported electricity plays a significant role in the city’s energy supply. A number of decision alternatives are also analyzed based on the interval solutions as well as the projected applicable conditions, which represent multiple options with sustainable and economic considerations. The optimal alternative that can give rise to the desirable sustainable option under the maximization of the share of renewable power generation has been suggested. The findings can help decision makers identify desired alternatives for managing such a mixed energy system in association with sustainable development. Compared with the conventional optimization methods that optimize single criterion, it is proved that IT2FFP is advantageous in balancing conflicting objectives and reflecting complicated relationships among multiple system factors as well as in tackling various subjective judgments of decision makers with different interests and preferences.

Risk-based factorial probabilistic inference for optimization of flood control systems with correlated uncertainties

In this paper, a risk-based factorial probabilistic inference method is proposed to address the stochastic objective function and constraints as well as their interactions in a systematic manner. To tackle random uncertainties, decision makers’ risk preferences are taken into account in the decision process. Statistical significance for each of the linear, nonlinear, and interaction effects of risk parameters is uncovered through conducting a multi-factorial analysis. The proposed methodology is applied to a case study of flood control to demonstrate its validity and applicability. A number of decision alternatives are obtained under various combinations of risk levels associated with the objective function and chance constraints, facilitating an in-depth analysis of trade-offs between economic outcomes and associated risks. Dynamic complexities are addressed through a two-stage decision process as well as through capacity expansion planning for flood diversion within a multi-region, multi-flood-level, and multi-option context. Findings from the factorial experiment reveal the multi-level interactions between risk parameters and quantify their contributions to the variability of the total system cost. The proposed method is compared against the fractile criterion optimization model and the chance-constrained programming technique, respectively.

An Inexact Probabilistic–Possibilistic Optimization Framework for Flood Management in a Hybrid Uncertain Environment

Flooding is one of the leading causes of loss due to natural catastrophes, and at least one third of all losses due to natural forces can be attributed to flooding. Flood management systems involve a variety of complexities, such as multiple uncertainties, dynamic variations, and policy implications. This paper presents an inexact probabilistic–possibilistic programming with fuzzy random coefficients (IPP-FRC) model for flood management in a hybrid uncertain environment. IPP-FRC is capable not only of tackling multiple uncertainties in the form of intervals with fuzzy random boundaries but of addressing the dynamic complexity through capacity expansion planning within a multi-region, multi-flood-level, and multi-option context. The possibility and necessity measures used in IPP-FRC are suitable for risk-seeking and risk-averse decision making, respectively. A case study is used to demonstrate the applicability of the proposed methodology for facilitating flood management. The results indicate that the inexact degrees of possibility and necessity would decrease with increased probabilities of occurrence, implying a potential tradeoff between fulfillment of objectives and associated risks. A number of decision alternatives can be obtained under different policy scenarios. They are helpful for decision makers to formulate the appropriate flood management policy according to practical situations. The performance of IPP-FRC is analyzed and compared with a possibility-based fractile model.

A risk-based method for planning of bus–subway corridor evacuation under hybrid uncertainties

Emergencies involved in a bus–subway corridor system are associated with many processes and factors with social and economic implications. These processes and factors and their interactions are related to a variety of uncertainties. In this study, an interval chance-constrained integer programming (EICI) method is developed in response to such challenges for bus–subway corridor based evacuation planning. The method couples a chance-constrained programming with an interval integer programming model framework. It can thus deal with interval uncertainties that cannot be quantified with specified probability distribution functions. Meanwhile, it can also reflect stochastic features of traffic flow capacity, and thereby help examine the related violation risk of constraint. The EICI method is applied to a subway incident based evacuation case study. It is solved through an interactive algorithm that does not lead to more complicated intermediate submodels and has a relatively low computational requirement. A number of decision alternatives could be directly generated based on results from the EICI method. It is indicated that the solutions cannot only help decision makers identify desired population evacuation and vehicle dispatch schemes under hybrid uncertainties, but also provide bases for in-depth analyses of tradeoffs among evacuation plans, total evacuation time, and constraint-violation risks.

Coupled planning of water resources and agricultural landuse based on an inexact-stochastic programming model

Water resources are fundamental for support of regional development. Effective planning can facilitate sustainable management of water resources to balance socioeconomic development and water conservation. In this research, coupled planning of water resources and agricultural land use was undertaken through the development of an inexact-stochastic programming approach. Such an inexact modeling approach was the integration of interval linear programming and chance-constraint programming methods. It was employed to successfully tackle uncertainty in the form of interval numbers and probabilistic distributions existing in water resource systems. Then it was applied to a typical regional water resource system for demonstrating its applicability and validity through generating efficient system solutions. Based on the process of modeling formulation and result analysis, the developed model could be used for helping identify optimal water resource utilization patterns and the corresponding agricultural land-use schemes in three sub-regions. Furthermore, a number of decision alternatives were generated under multiple water-supply conditions, which could help decision makers identify desired management policies.

An integrated approach for water resources decision making under interactive and compound uncertainties

In recent years, the issue of water allocation among competing users has been of great concern for many countries due to increasing water demand from population growth and economic development. In water management systems, the inherent uncertainties and their potential interactions pose a significant challenge for water managers to identify optimal water-allocation schemes in a complex and uncertain environment. This paper thus proposes a methodology that incorporates optimization techniques and statistical experimental designs within a general framework to address the issues of uncertainty and risk as well as their correlations in a systematic manner. A water resources management problem is used to demonstrate the applicability of the proposed methodology. The results indicate that interval solutions can be generated for the objective function and decision variables, and a number of decision alternatives can be obtained under different policy scenarios. The solutions with different risk levels of constraint violation can help quantify the relationship between the economic objective and the system risk, which is meaningful for supporting risk management. The experimental data obtained from the Taguchi's orthogonal array design are useful for identifying the significant factors affecting the means of total net benefits. Then the findings from the mixed-level factorial experiment can help reveal the latent interactions between those significant factors at different levels and their effects on the modeling response.

Interval Binary Programming Model for Noise Control Within an Urban Environment

This paper introduces an interval binary programming (IBP) method to the selection of control measures for noise reduction under uncertainty, by incorporating the concepts of interval numbers and interval mathematical programming into a binary programming optimization framework. As an extension of the binary programming method, IBP can explicitly address complexities and uncertainties in a noise control system. Parameters in the IBP model can be expressed as intervals, and uncertainties are effectively incorporated within the model solution process. The modelling approach is applied to a representative control measure selection problem for noise reduction in an urban environment. Results of the application indicate that useful solutions for noise control practices can be generated. A number of decision alternatives have been obtained and analyzed under different acceptable noise levels for two communities, and they reflect complex tradeoffs between environmental and economic considerations.

Interactive Fuzzy Boundary Interval Programming for Air Quality Management Under Uncertainty

This paper presents an interactive fuzzy boundary interval programming (IFBIP) approach for addressing dual uncertainties that exist in the objective function and the left- and right-hand sides of constraints. IFBIP permits an interactive participation of the decision maker (DM) in all steps of the decision process through expressing his/her preferences in linguistic terms. A regional air quality management problem is studied to illustrate the proposed approach. The results indicate that a number of decision alternatives can be obtained under different feasibility degrees of constraints. They can help quantify the relationship between the total operating cost and the constraint violation risk, which is meaningful not only for supporting an in-depth analysis of the tradeoff between the economic objective and the system safety but also for the DM to identify a desired compromise between two conflicting concerns: the satisfaction degree of the goal and the feasibility degree of constraints.

Development of an Interval-Based Evacuation Management Model in Response to Nuclear-Power Plant Accident

Nuclear-power-plant emergency management is very important, which can be used to avoid nuclear accident and radiation leak accident and the emergency action swiftly taken beyond the normal work procedure to mitigate the accident consequences. Evacuation management for large crowds after accidents involves a number of processes and factors with socio-economic and environmental implications. These processes and factors, as well as their interactions, are associated with a variety of uncertainties. In this study, an interval-based evacuation management (IBEM) model is developed in response to such challenges, based on interval-parameter linear programming (ILP) technique that can tackle uncertainties presented as interval values. The IBEM model is applied to a case study and then solved through an interactive algorithm that does not lead to more complicated intermediate submodels and has a relatively low computational requirement. Two scenarios are analyzed based on different policies of total capital considerations. A number of decision alternatives could be directly generated based on results from the IBEM model, which provide bases for in-depth analyses of tradeoffs among evacuation population, system cost, and constraint-violation risk.

Single or Multiple Sourcing: A Method for Determining the Optimal Size of the Supply Base

Consider a situation where a buyer has to procure an item from outside suppliers and is faced with the decision whether to procure the item from a single supplier or from multiple suppliers. Supply risk has become, in the recent years, a key consideration for a manager while taking such decisions and often mitigation of such risks is done either by building up inventory or having multiple suppliers. In this paper, we address the problem of determining the optimal number of suppliers to be engaged in order to minimize the effects of supply risks on the focal firm. In the literature, the events leading to supply risks have been classified into three categories viz. super events, semi-super events and unique events. However, none have considered all the three types of events together into a single model in order to determine the optimal size of the supply base. In the present work, we have considered all three types of events and calculated the probability of complete supply disruption. We formulate the problem as a cost minimization problem so as to find the optimal size of the supplier base that minimizes the effects of such supply disruptions. The mathematical formulation of the problem is combinatorial in nature. When a decision tree is used, a moderate size problem results in an unmanageable number of decision alternatives. In this paper, we propose mathematical theorems and rules that helps avoid considering many non-optimal decision alternatives for evaluation. The proposed solution procedure is very simple and reduces the number of decision alternatives to be evaluated significantly, thus saving time and effort in solving the problem.

The Usage of Influence Diagram for Decision Making in Textiles

Naturally, managers are frequently faced with decision making processes. The optimal solution of the decision making process rises to the for which in itself depends on the number of decision alternatives and state of the worlds which in themselves may affect the decision alternatives. The scenarios or possibilities faced in decision problem are interlaced to a graphical technique commonly refered to as decision tree to which managers confer to be guided. In this study, the influence diagram, which is similar to a decision tree but also provides room for the decision maker to include extra information about the relationships between variables, is reviewed. An example decision problem is used to show the usage of the influence diagram within the context of Textiles industry.

A two-phase fuzzy decision-making method based on multigranular linguistic assessment

In general, the decision-makers or experts should evaluate and select the best alternative from the huge number of decision alternatives. Under this situation, it is difficult for decision-maker to make the right comparison between alternatives and choose the best one in a short time. The aim of this paper is to present a two-phase fuzzy decision-making method to deal with the kind of decision problems. In this paper, experts’ opinions can be expressed by different types of 2-tuple linguistic variables. Experts’ opinions are including the performance of each alternative with respect to each criterion and the weight of each criterion. The proposed method was divided into two phases to reach the final decision. First, some unsuitable alternatives will be eliminated to reduce the number of decision alternatives. And then, the ranking order of the rest alternatives will be determined by a systematic method. Therefore, linguistic ELECTRE are used at the first phase to eliminate some unsuitable alternatives in accordance with pair-wise comparison among all alternatives. At the second phase, the linguistic TOPSIS are used to determine the total ranking order of the rest alternatives by closeness coefficient of each alternative. Finally, an example is shown to highlight the procedure of the proposed method and the conclusion is discussed at the end of this paper. Key words: Two-phase fuzzy MCDM, 2-tuple linguistic variable, ELECTRE, TOPSIS.

Identifying Optimal Water Resources Allocation Strategies through an Interactive Multi-Stage Stochastic Fuzzy Programming Approach

In this study, an interactive multi-stage stochastic fuzzy programming (IMSFP) approach has been developed through incorporating an interactive fuzzy resolution (IFR) method within an inexact multi-stage stochastic programming framework. IMSFP can deal with dual uncertainties expressed as fuzzy boundary intervals that exist in the objective function and the left- and right-hand sides of constraints. Moreover, IMSFP is capable of reflecting dynamics of uncertainties and the related decision processes through constructing a set of representative scenarios within a multi-stage context. A management problem in terms of water resources allocation has been studied to illustrate applicability of the proposed approach. The results indicate that a set of solutions under different feasibility degrees (i.e., risk of constraint violation) has been generated for planning the water resources allocation. They can not only help quantify the relationship between the objective-function value and the risk of violating the constraints, but also enable decision makers (DMs) to identify, in an interactive way, a desired compromise between two factors in conflict: satisfaction degree of the goal and feasibility degree of constraints. Besides, a number of decision alternatives have been generated under different policies for water resources management, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic penalties when the promised water-allocation targets are violated, and thus help DMs to identify desired water-allocation schemes under uncertainty.

A two-stage programming approach for water resources management under randomness and fuzziness

In this study, a fuzzy stochastic two-stage programming (FSTP) approach is developed for water resources management under uncertainty. The concept of fuzzy random variable expressed as parameters’ uncertainties with both stochastic and fuzzy characteristics was used in the method. FSTP has advantages in uncertainty reflection and policy analysis. FSTP integrates the fuzzy robust programming, chance-constrained programming and two-stage stochastic programming (TSP) within a general optimization framework. FSTP can incorporate pre-regulated water resources management policies directly into its optimization process. Thus, various policy scenarios with different economic penalties (when the promised amounts are not delivered) can be analyzed. FSTP is applied to a water resources management system with three users. The results indicate that reasonable solutions were generated, thus a number of decision alternatives can be generated under different levels of stream flows, α-cut levels and different levels of constraint-violation probability. The developed FSTP was also compared with TSP to exhibit its advantages in dealing with multiple forms of uncertainties.

Decision-Making Groups Attenuate the Discussion Bias in Favor of Shared Information: A Meta-Analysis

Groups often focus their discussions on information that all members know at the outset. To test how robust the sampling advantage for shared information is, a meta-analysis was conducted. The analysis integrated findings from 20 publications (45 independent effects), in which information sharedness was manipulated. Groups discussed more shared than unshared information overall. However, the observed sampling advantage was smaller than expected. Groups attenuated the discussion bias in particular when they had to choose among a small number of decision alternatives and when they had less than 30 minutes discussion time. Moreover, groups performing a hidden-profile task tended to display a smaller discussion bias than groups performing tasks with equally attractive alternatives.

A Superiority-Inferiority-Based Inexact Fuzzy Stochastic Programming Approach for Solid Waste Management Under Uncertainty

A superiority–inferiority-based inexact fuzzy stochastic programming (SI-IFSP) model was developed for planning municipal solid waste management systems under uncertainty. The SI-IFSP approach represents a new attempt to tackle multiple uncertainties in objective function coefficients which are beyond the capabilities of existing inexact programming methods. Through introducing the concept of fuzzy random boundary interval, SI-IFSP is capable of reflecting multiple uncertainties (i.e., interval values, fuzzy sets, probability distributions, and their combinations) in both the objective function and constraints, leading to enhanced system robustness. The developed SI-IFSP method was applied to a case study of long-term municipal solid waste management. Useful solutions were generated. A number of decision alternatives could be generated based on projected applicable conditions, reflecting the compromise between system optimality and reliability as well as the tradeoffs between economic and environmental objectives. Moreover, the consequences of system violations could be quantified through introducing a set of economic penalties, reflecting the relationships between system costs and constraint violation risks. The results suggest that the proposed SI-IFSP method can explicitly address complexities in municipal solid waste management systems and is applicable to practical waste management problems.

An inexact fuzzy-chance-constrained two-stage mixed-integer linear programming approach for flood diversion planning under multiple uncertainties

In this study, an inexact fuzzy-chance-constrained two-stage mixed-integer linear programming (IFCTIP) approach is developed for flood diversion planning under multiple uncertainties. A concept of the distribution with fuzzy boundary interval probability is defined to address multiple uncertainties expressed as integration of intervals, fuzzy sets and probability distributions. IFCTIP integrates the inexact programming, two-stage stochastic programming, integer programming and fuzzy-stochastic programming within a general optimization framework. IFCTIP incorporates the pre-regulated water-diversion policies directly into its optimization process to analyze various policy scenarios; each scenario has different economic penalty when the promised targets are violated. More importantly, it can facilitate dynamic programming for decisions of capacity-expansion planning under fuzzy-stochastic conditions. IFCTIP is applied to a flood management system. Solutions from IFCTIP provide desired flood diversion plans with a minimized system cost and a maximized safety level. The results indicate that reasonable solutions are generated for objective function values and decision variables, thus a number of decision alternatives can be generated under different levels of flood flows.