Marginal Value Functions Research Articles

A scenario-based MCDA framework for wastewater infrastructure planning under uncertainty

Wastewater infrastructure management is increasingly important because of urbanization, environmental pollutants, aging infrastructures, and climate change. We propose a scenario-based multi-criteria decision analysis (MCDA) framework to compare different infrastructure alternatives in terms of their sustainability. These range from the current centralized system to semi- and fully decentralized options. Various sources of uncertainty are considered, including external socio-economic uncertainty captured by future scenarios, uncertainty in predicting outcomes of alternatives, and incomplete preferences of stakeholders. Stochastic Multi-criteria Acceptability Analysis (SMAA) with Monte Carlo simulation is performed, and rank acceptability indices help identify robust alternatives. We propose step-wise local sensitivity analysis, which is useful for practitioners to effectively elicit preferences and identify major sources of uncertainty. The approach is demonstrated in a Swiss case study where ten stakeholders are involved throughout. Their preferences are quantitatively elicited by combining an online questionnaire with face-to-face interviews. The trade-off questions reveal a high concern about environmental and an unexpectedly low importance of economic criteria. This results in a surprisingly good ranking of high-tech decentralized wastewater alternatives using urine source separation for most stakeholders in all scenarios. Combining scenario planning and MCDA proves useful, as the performance of wastewater infrastructure systems is indeed sensitive to socio-economic boundary conditions and the other sources of uncertainty. The proposed sensitivity analysis suggests that a simplified elicitation procedure is sufficient in many cases. Elicitation of more information such as detailed marginal value functions should only follow if the sensitivity analysis finds this necessary. Moreover, the uncertainty of rankings can be considerably reduced by better predictions of the outcomes of alternatives. Although the results are case based, the proposed decision framework is generalizable to other decision contexts.

Interdependent Value Multi-unit Auctions for Initial Allocation of Emission Permits

Auctions have been shown to be able to tackle the efficient allocation of initial emission permits, but most of existing works only consider the problem with private values. They can not deal with the information externalities that each polluter's value depends on his own private information as well as the information of other polluters. In this paper, we model the allocation problem of initial emission permits as a multi-unit auction with interdependent values and propose a mechanism. First, we give the assumptions, i.e., each polluter is allowed to submit a series of marginal value functions which are declining in the number of units allocated, each polluter's signal to marginal values is independent private information and his marginal values are interdependent. Second, we establish a designing model for efficient multi-unit auction mechanism with interdependent values according to above assumptions. Third, we get the auction mechanism by solving this model and give the detail steps to show how to implement the auction mechanism in practice. Finally, we give an application example to show the operability of the auction mechanism proposed by this paper. The conclusion can be drawn that our auction mechanism can prevent polluters from cheating in the auction, achieve the social welfare optimization, and provide the environmental management departments with the theoretical basis and reference methods to realize the fairness and effectiveness of initial allocation of emission permits.

Robust extensions of the MUSA method based on additional properties and preferences

The MUSA method is a collective preference disaggregation approach following the main principles of ordinal regression analysis under constraints using linear programming techniques. The method has been developed in order to measure and analyse customer satisfaction and it is used for the assessment of a set of marginal satisfaction functions in such a way that the global satisfaction criterion becomes as consistent as possible with customer's judgments. The main objective of the method is to assess collective global and marginal value functions by aggregating individual judgments. This study evaluates different extensions of the MUSA method with the introduction of additional constraints in the basic linear programming formulation of the method. These constraints concern special properties for the assessed average indices and additional customer preferences about the importance of the criteria and they may be modelled as multiobjective linear programming problems. The main aim of the study is to show how the introduction of these additional constraints and information may improve the stability of the estimated results. An illustrative example is presented in order to show the applicability of this approach.

Modeling assignment-based pairwise comparisons within integrated framework for value-driven multiple criteria sorting

We introduce a new preference disaggregation modeling formulations for multiple criteria sorting with a set of additive value functions. The preference information supplied by the Decision Maker (DM) is composed of: (1) possibly imprecise assignment examples, (2) desired class cardinalities, and (3) assignment-based pairwise comparisons. The latter have the form of imprecise statements referring to the desired assignments for pairs of alternatives, but without specifying any concrete class. Additionally, we account for preferences concerning the shape of the marginal value functions and desired comprehensive values of alternatives assigned to a given class or class range. The exploitation of all value functions compatible with these preferences results in three types of results: (1) necessary and possible assignments, (2) extreme class cardinalities, and (3) necessary and possible assignment-based preference relations. These outputs correspond to different types of admitted preference information. By exhibiting different outcomes, we encourage the DM in various ways to enrich her/his preference information interactively. The applicability of the framework is demonstrated on data involving the classification of cities into liveability classes.

Tackling uncertainty in multi-criteria decision analysis – An application to water supply infrastructure planning

We present a novel approach for practically tackling uncertainty in preference elicitation and predictive modeling to support complex multi-criteria decisions based on multi-attribute utility theory (MAUT). A simplified two-step elicitation procedure consisting of an online survey and face-to-face interviews is followed by an extensive uncertainty analysis. This covers uncertainty of the preference components (marginal value and utility functions, hierarchical aggregation functions, aggregation parameters) and the attribute predictions. Context uncertainties about future socio-economic developments are captured by combining MAUT with scenario planning. We perform a global sensitivity analysis (GSA) to assess the contribution of single uncertain preference parameters to the uncertainty of the ranking of alternatives. This is exemplified for sustainable water infrastructure planning in a case study in Switzerland. We compare 11 water supply alternatives ranging from conventional water supply systems to novel technologies and management schemes regarding 44 objectives. Their performance is assessed for four future scenarios and 10 stakeholders from different backgrounds and decision-making levels. Despite uncertainty in the ranking of alternatives, potential best and worst solutions could be identified. We demonstrate that a priori assumptions such as linear value functions or additive aggregation can result in misleading recommendations, unless thoroughly checked during preference elicitation and modeling. We suggest GSA to focus elicitation on most sensitive preference parameters. Our GSA results indicate that output uncertainty can be considerably reduced by additional elicitation of few parameters, e.g. the overall risk attitude and aggregation functions at higher-level nodes. Here, rough value function elicitation was sufficient, thereby substantially reducing elicitation time.

A synergy of multicriteria techniques to assess additive value models

The assessment of additive value functions in Multicriteria Decision Aid (MCDA) has to face issues of legitimacy and technical difficulties when real decision makers are involved. This paper presents a synergy of three complementary techniques to assess additive models on the whole criteria space. The synergy includes a revised MACBETH technique, the standard MAUT trade-off analysis and UTA-based methods for the assessment of both the marginal value functions and the weighting factors. The paper uses a set of original robustness measures and rules associated with revised MACBETH and UTA in order to manage multiple linear programming solutions and to extract robust conclusions from them. Finally, to illustrate the methods’ synergy, an application example is presented, dealing with the planning of metro extension lines.

The Value of Urban Green Space in Britain: A Methodological Framework for Spatially Referenced Benefit Transfer

A meta-analysis of studies valuing urban greenspace in the UK is undertaken to yield spatially sensitive marginal value functions. A geographical information system (GIS) is used to apply these functions to spatial data detailing the location of such greenspace resources in five British cities. Changes in monetary values are computed for the six future scenarios used in the UK National Ecosystem Assessment for the period 2010–2060. Different degrees of substitutability between urban greenspaces are considered. These findings are then extrapolated to all major British cities to obtain per household and aggregate valuation estimates for each scenario both with and without distributional weights. While subject to a number of shortcomings in both data availability and methodology, this represents the first systematic and comprehensive attempt to value marginal changes in urban greenspace while accounting for spatial heterogeneity.

The Breast Ultrasound Lexicon: Breast Imaging Reporting and Data System (BI-RADS)

Decision models based on value function can help learn the preferences of decision makers using historical data. However, inconsistency of the preferences learned from data with the real preferences of decision makers may occur. Suppose decision data or scenarios in the target domain are insufficient to guarantee the consistency of learning preferences, and there exists one similar source domain with sufficient effective data. Then, knowledge from the source domain can help learn preferences in the target domain. Following this idea, this paper proposes a cross-domain decision making method using a parameter transfer strategy with homogeneous and heterogeneous criteria. A decision model based on the value function is constructed using a generalized additive model to ensure the interpretability of the model. The marginal value functions are adjusted using Sturm’s theorem to keep the monotonicity of the learned marginal value functions on the criteria. Then, the parameter transfer strategy is adopted to transfer the obtained value functions to help learn the real preferences of the decision maker for the decision problem where preferences in historical data may not be consistent. The effectiveness of the proposed method is validated by solving the problem of diagnosing breast lesions in a hospital.

Selection of a representative value function in robust multiple criteria ranking and choice

We introduce the concept of a representative value function in robust ordinal regression applied to multiple criteria ranking and choice problems. The proposed method can be seen as a new interactive UTA-like procedure, which extends the UTA GMS and GRIP methods. The preference information supplied by the decision maker (DM) is composed of a partial preorder and intensities of preference on a subset of reference alternatives. Robust ordinal regression builds a set of general additive value functions which are compatible with the preference information, and returns two binary preference relations: necessary and possible. They identify recommendations which are compatible with all or at least one compatible value function, respectively. In this paper, we propose a general framework for selection of a representative value function from among the set of compatibles ones. There are a few targets which build on results of robust ordinal regression, and could be attained by a representative value function. In general, according to the interactively elicited preferences of the DM, the representative value function may emphasize the advantage of some alternatives over the others when all compatible value functions acknowledge this advantage, or reduce the ambiguity in the advantage of some alternatives over the others when some compatible value functions acknowledge an advantage and other ones acknowledge a disadvantage. The basic procedure is refined by few extensions. They enable emphasizing the advantage of alternatives that could be considered as potential best options, accounting for intensities of preference, or obtaining a desired type of the marginal value functions.

Compliance Strategies under Permits for Emissions

We characterize the trade‐offs among firms' compliance strategies in a market‐based program where a regulator interested in controlling emissions from a given set of sources auctions off a fixed number of emissions permits. We model a three‐stage game in which firms invest in emissions abatement, participate in a share auction for permits, and produce output. We develop a methodology for a profit‐maximizing firm to derive its marginal value function for permits and translate this value function into an optimal bidding strategy in the auction. We analyze two end‐product market scenarios independent demands and Cournot competition. In both scenarios we find that changing the number of available permits influences abatement to a lesser extent in a dirty industry than in a cleaner one. In addition, abatement levels taper off with increasing industry dirtiness levels. In the presence of competition, firms in a relatively clean industry can, in fact, benefit from a reduction in the number of available permits. Our findings are robust to changes in certain modeling assumptions.

A foundation for the solution of consumption-saving behavior with a borrowing constraint and unbounded marginal utility

Models of precautionary saving or storage include cases where the marginal value of accumulated balances is unbounded, with an invariant distribution with infinite mean. Based on a uniform continuity argument, we show that a model of saving with bounded marginal value can be used to approximate the unbounded marginal value function, and the quantiles of its invariant distribution, arbitrarily accurately. These results offer a foundation for a strategy for numerical solution of marginal values in cases where they are unbounded, and for derivation of the quantiles of their invariant distributions.

On robustness of set-valued maps and marginal value functions

On robustness of set-valued maps and marginal value functions

Integration of ordinal and cardinal information in multi-criteria ranking with imperfect compensation

The method presented in this paper is a MCDA method, which outranks a certain number of choice options that are evaluated on a mixture of cardinal and ordinal judgement criteria. Characteristics of the method are a theoretically sound integration of cardinal and ordinal information and a decision-makers' preference structure that allows for less than perfect compensation between criteria. The method is based on a pairwise comparison of choice-options. It belongs to the family of decomposed scaling methods: the assessment of the relative importance of the criteria on one hand and the assessment of the scores (cardinal criteria) and rankings (ordinal criteria) on the other hand are performed separately. The assessment of the scores and rankings is realised by means of (marginal) value functions in such a way that the ordinal information is fully comparable with the cardinal information. Essentially, the ordinal information is translated into stochastical information without imposing any assumption on the probability distributions attached to the ordinal data. The relative importance of the judgement criteria is represented by a set of cardinal weights or by a ranking of weights. The final ranking of the choice options is based upon an overall value function that is some kind of weighed aggregation of the marginal value functions. In order to allow for less-than-perfect-compensation between criteria, the method uses a convex combination of two different CES-type value functions. The paper ends with an empirical application, on the planning of public transport systems in The Netherlands.

Supporting others and the evolution of influence

In this paper we study environments in which agents can transfer influence to others by supporting them. When planning whom to support, they should take into account the future effect of this, since the receiving agent might use this influence to support others in the future. We show that in the presence of a finite horizon there is an essentially unique optimal support behavior which can be characterized in terms of associated marginal value functions. The analysis of these marginal value functions allows us to derive qualitative properties of optimal support strategies under different specific environments and to explicitly compute the optimal support behavior in some numerical examples. We also investigate the case of an infinite horizon. Examples show that multiple equilibria may appear in this setting, some of which sustaining a degree of cooperation that would not be possible under a finite horizon.

Long cycles and neighborhood union in 1-tough graphs with large degree sums

The ideas of robust sets, robust functions and robustness of general set-valued maps were introduced by Chew and Zheng [7, 26], and further developed by Shi, Zheng, Zhuang [18, 19, 20], Phu, Hoffmann and Hichert [8, 9, 10, 17] to weaken up the semi-continuity requirements of certain global optimization algorithms. Robust analysis, along with measure theory, has well served as the basis for the integral global optimization method (IGOM) (Chew and Zheng [7]). Hence, we have attempted to extend the robust analysis of Zheng et al. to that of robustness of set-valued maps with given structures and marginal value functions. We are also of strong conviction that the results of our investigation could open a way to apply the IGOM for the numerical treatment of some class of parametric optimization problems, when global optima are required.

Solving the discrete multiple criteria problem using linear prospect theory

Prospect theory developed by Kahneman and Tversky is a popular model of choice in decision problems under uncertainty. Prospect theory has recently been extended to multiple criteria choice problems. In this paper, an interactive method for solving discrete multiple criteria decision problems, based on prospect theory type value functions, has been developed. Piecewise linear marginal value functions are assumed to approximate the S-shaped value functions of prospect theory. Therefore, the proposed procedure is valid only for convex preferences.

A Model for Managing Reservoir Water Releases

AbstractTwo surface reservoirs in California's Salinas Basin are used to increase recharge of valley aquifers. Under the assumptions of zero price elasticity of demand for groundwater, unregulated groundwater withdrawals, and the objective of economic efficiency, the management criterion is to maximize the present value of expected aquifer recharge where the value of a unit of recharge is the associated reduction in pumping costs. A cyclical Markov dynamic programming management model is developed; optimal strategies are obtained. Results indicate the optimal strategy is relatively stable within the realistic range of alternative groundwater demand growth rates. The marginal value function for reservoir capacity is obtained by parametric analysis.