Risk Attitude Of Decision Maker Research Articles

Green Supplier Selection Based on Sequential Group Three-Way Decision Making

Environmental protection and sustainable development have become the consensus of all countries in the world. Enterprises must pay attention to the impact on the environment in their operations. Therefore, the selection of green suppliers has become a crucial issue for companies. Supplier selection is a dynamic and complex multi-attribute group decision-making process. The decision results have the tripartite characteristics of “accepted”, “rejected” and “pending further investigation”, and experts need to constantly negotiate in the decision-making process to achieve consensus. In view of the above characteristics, this study constructs a sequential group three-way decision making (TWDM) method to support green supplier selection. Firstly, we review the existing literature on the evaluation criteria and selection methods of green suppliers. In this process, we construct an evaluation attribution system including the following four aspects: product formation, service level, development capability, and green level. Secondly, combining with the sequential and group characteristics of decision making, we propose a multiple-attribute sequential group TWDM method based on a multi-level granularity structure. The weight of each decision maker is determined by his influence weight and interval-valued intuitionistic fuzzy entropy weight. The attribute weight is determined by entropy weight and subjective weight. By using the VIsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method improved by grey relation analysis (GRA), we calculate conditional probabilities. Since the cumulative prospect theory can reflect the risk attitude of decision makers, we apply it to calculate the decision thresholds. Finally, we apply a case of Q automobile manufacturer to verify the effectiveness, applicability and feasibility of the method. The innovation of this study is to construct social networks at each granularity level and introduce an expert information exchange model to promote group consensus. The sequential group TWDM method provides a new reference and idea for the selection of green suppliers.

Integrated evaluation of the performance of phosphogypsum recycling technologies in China

The Chinese government is implementing policies, such as the “Guidance on comprehensive utilization of bulk solid waste for the 14th Five-Year Plan period”, to stimulate phosphogypsum (PG) reduction and recycling. Thus, the comprehensive evaluation of PG recycling technologies for sustainable development is crucial. This study proposes a novel multi-criteria decision analysis (MCDA) method that considers the criteria of resources, environment, economy, and society and risk attitudes of decision-makers and integrates game theory (GT) and utility theory for criteria weighting and ranking to assess industrial-scale PG recycling technologies in China. The results demonstrate that GT provides more reasonable criteria weights than individual weighting methods. PG-based lightweight plaster is the top performer in the resource and environmental dimensions owing to its exceptional resource and energy efficiency. PG utilized for dry-mix mortar and organic fertilizer production exhibited the best utility performance of 0.74 and 0.73, respectively. Measures, such as subsidies and product publicity, should be implemented to promote these technologies. However, technologies with poor performance, such as PG used for the co-production of sulfuric acid and fertilizer or cement, may require optimization or substitution for the sustainable recycling of PG. The proposed MCDA method is robust and can serve as a reliable decision-making tool for other waste-recycling technologies. However, caution must be exercised when determining risk attitude using the MCDA method as it may vary with the number of technologies and affect the final rankings.

Picture fuzzy prospect theory-MAIRCA method considering optimal reference point for emergency decision-making

In the emergency decision-making process, decision-makers usually cannot give rational evaluations, and existing decision-making methods do not adequately consider the risk attitude of decision-makers either. To solve these problems, a combined method based on the prospect theory and the multi-attributive ideal-real comparative analysis (MAIRCA) method is put forward in the picture fuzzy environment. Firstly, the optimal aggregation (OA) model is proposed to obtain the ideal evaluations with the least disagreement among decision-makers. Regarding the evaluations as reference systems, the OA-based prospect theory is put forward, which could calculate the prospect matrix more reasonably. Secondly, considering the prospect matrix and alternative preference, the improved MAIRCA method is proposed, which overcomes the defects of theory and has the better ranking ability. Then, the OA-based prospect theory-MAIRCA method is further put forward to effectively complete the decision-making process with risk attitudes. Finally, an illustrative example of earthquake emergency assessment and a series of comparative experiments are presented. The analyses of results show that the proposed method has great guiding significance in the field of emergency decision-making management.

Fragility-based lot-sizing in veterinary pharmaceutical plants under demand uncertainty

We study a production lot-sizing problem inspired by a veterinary pharmaceutical plant in which demands are uncertain. First, we develop a deterministic capacitated lot-sizing model for the production of animal pesticides, performed in three machine-specific stages. Second, we propose a traditional robust optimisation formulation following the popular budget-of-uncertainty approach. Third, we derive a novel fragility-based approach that circumvents well-known issues with traditional robust optimisation approaches, such as the estimation of budgets of uncertainty, the over-conservatism of robust solutions and the sensitivity of solutions to the decision maker's risk attitude. The fragility-based approach is grounded in the idea of minimising violations, over the full uncertainty support, from a user-specified cost target. It avoids the estimation of budgets of uncertainty and produces less conservative solutions via explicit modelling of constraint violation. We demonstrate the effectiveness of our approach on instances built upon real data provided by our industrial partner, a major player in the Brazilian veterinary pharmaceutical sector. The results show that our fragility-based approach reduces average total costs across all instances and maintains greater model stability under different target estimations. It also preserves cost savings when bottlenecks are introduced in production and when inventory costs and capacities are varied.

A clustering-based method for large-scale group decision making in the hesitant fuzzy set environment

The hesitant fuzzy set (HFS) has received wide attention because it can neatly describe the fuzziness and hesitancy of decision information with several possible values. Hence, this paper proposes a clustering-based method for resolving large-scale group decision making problems with HFSs. Considering the different risk attitudes of decision makers (DMs), a new score function of hesitant fuzzy elements (HFEs) is defined by introducing weight factors of all possible values. According to the scores of attribute values, a neighborhood-based approach is put forward to detect and process abnormal evaluations that may influence decision results. Afterwards, a two-stage clustering approach along with three algorithms is presented to cluster DMs into sub-groups. The first stage clusters DMs with finite possible cluster numbers based on the supporters of each DM and the separations between DMs. In the second stage, according to intra and inter-differences among DMs, an index is introduced to evaluate clustering results and select the best cluster number from the finite ones given in the first stage. Finally, by fusing the information densities and cluster sizes, cluster weights are determined and ranking values of alternatives are generated by aggregating evaluation information of the alternatives. At length, a case study is provided to illustrate the applications of the proposed method. Furthermore, the experiments and comparative analyses are conducted to show the effectiveness and superiority of the proposed method.

The influence of context effects on exit choice behavior during building evacuation combining virtual reality and discrete choice modeling

Exit choice is critical to saving lives during a building evacuation. Numerous studies have examined how environmental factors, social interactions, and individuals’ internal factors affect exit choice behavior. However, due to their limited ability to perceive and combine these multi-attributes, people do not always make strategic and rational decisions but instead rely on preferences. The choice context can facilitate or hinder the evaluation of attributes, and thus preference reversals may occur in the same choice set but in different contexts. Researchers have long known that the context effect – i.e. preference reversals depending on the availability of other options – plays an important role in decision-making behavior. However, no research on decision-making behavior has been conducted that considers the context effect on exit choice during building evacuation. To fill this gap, we conducted a series of virtual reality (VR) experiments to examine the influence of context effect on exit choice. Before conducting the experiments, the validity of exit choice behavior in the VR-based method was analysed by comparing controlled experiments in both the real world and virtual environment. The results from our VR experiments provide novel and clear empirical evidence that contextual preference reversals involving three major effects, the compromise effect, the similarity effect, and the attraction effect, play an important role in exit choice during evacuation. Additionally, we compared three models for estimating discrete choice – expected utility theory (EUT), cumulative prospect theory (CPT), and accentuation of differences (AOD) – which primarily focus on the option attributes, the risk attitude of decision-makers, and the context of options, respectively. The outstanding performance of AOD, which is based on a psychophysical law and takes into account the impact of the context of options, highlights the psychological perspective of evacuees towards various option attributes and the importance of context effects in exit choice.

Three-way decision for probabilistic linguistic conflict analysis via compounded risk preference

Three-way decision, an essential granular computing research tool, provides an efficient solution to complex and uncertain problems. Behavioral decision theory can analyze the risk preferences of decision-makers effectively. Scholars have conducted preliminary exploration on the fusion of these two theories, but it is still challenging to describe the different types of risk preferences of decision-makers. This paper combines prospect theory with regret theory and studies the compound risk preference modeling of three-way decision to address this issue. Because three attitudes of conflicts coincide with three-way decision, many scholars have conducted multi-dimensional research on three-way conflict analysis and accomplished remarkable results. However, few relevant studies consider psychological factors and risk attitudes of decision-makers, and it is more appropriate to describe agents' attitudes on issues using linguistic terms. This paper applies the proposed three-way decision model based on compounded risk preference and probabilistic linguistic term sets to the conflict analysis problem. We utilize examples to explain the decision-making process of the proposed model and three-way conflict analysis method with the influence of the compounded risk preference under the action of reference point and regret avoidance coefficient. The illustrative example illustrates that the proposed three-way decision model can effectively solve the software development conflict analysis problem for different decision-makers and the comparative analysis shows the advantages of the proposed model and method compared with the two existing methods. Finally, we verify the performance of the three-way decision model based on compounded risk preference by UCI data sets in parameter experiments. The changes of the reference point from 10 to 0 and regret avoidance coefficient in 0, 0.15 and 0.3 respectively demonstrate the trend rule of the model's thresholds and delay-decision rate index.

A GIS-Based Probabilistic Spatial Multicriteria Roof Water Inrush Risk Evaluation Method Considering Decision Makers’ Risk-Coping Attitude

A combination of geographic information system (GIS) and spatial multicriteria decision making (MCDA) in mine water inrush risk evaluation is widely used, but the randomness in the process of index weight determination and the risk-coping attitude of decision makers are not considered in the decision making process. Therefore, this paper proposes a probability-based roof water inrush risk evaluation method (GIS-MCDA) by combining the Monte Carlo analytic hierarchy process (MAHP) and ordered weighted averaging (OWA) operator. This method uses MAHP to determine the weight of the evaluation indicators, reducing the randomness of the analytic hierarchy process (AHP) to determine the weight of the evaluation indicators using the OWA operator to quantify the five risk-coping attitudes of decision makers and incorporate the risk attitude of decision makers into the evaluation process. Taking the Liangshuijing Coal Mine in northern Shaanxi as an example, the application of the GIS-MCDA method showed that the method makes the risk results of roof water inrush more objective and comprehensive and reduces or avoids the risk of decision making due to human subjective tendency change.

A study on offshore wind farm site selection based on CRITIC and CPT-TOPSIS: a case study of China

PurposeThis study aims to improve the offshore wind farm (OWF) site selection evaluation index system and establishes a decision-making model for OWF site selection. It is expected to provide helpful references for the progress of offshore wind power.Design/methodology/approachFirstly, this paper establishes an evaluation criteria system for OWF site selection, considering six criteria (wind resource, environment, economic, technical, social and risk) and related subcriteria. Then, the Criteria Importance Though Intercrieria Correlation (CRITIC) method is introduced to figure out the weights of evaluation indexes. In addition, the cumulative prospect theory and technique for order preference by similarity to an ideal solution (CPT-TOPSIS) method are employed to construct the OWF site selection decision-making model. Finally, taking the OWF site selection in China as an example, the effectiveness and robustness of the framework are verified by sensitivity analysis and comparative analysis.FindingsThis study establishes the OWF site selection evaluation system and constructs a decision-making model under the spherical fuzzy environment. A case of China is employed to verify the effectiveness and feasibility of the model.Originality/valueIn this paper, a new decision-making model is proposed for the first time, considering the ambiguity and uncertainty of information and the risk attitudes of decision-makers (DMs) in the decision-making process.

Dual-Source Procurement Strategies of Emergency Materials Considering Risk Attitudes

The uncertainty of emergencies makes the emergency procurement face many risks, so the risk management is particularly important of the emergency procurement. The risk attitude of decision makers will significantly affect the decision-making of risk management. In this paper, the risk management problem with different risk attitudes of emergency procurement consisting of dual-source suppliers and the single government is studied, and a government-led Stackelberg game is used to analyze the risks of each link to establish an emergency procurement model under the option contract, and the optimal decision-making is obtained. The effects of reserve period, risk avoidance coefficient, and probability of emergency on optimal decision-making are analyzed with different risk attitude. Moreover, we investigate the coordination of the government-led supply chain coordination under the risk aversion and risk-neutral conditions of emergency supply chain participants. The results show that the model can control the risk while reducing the cost of government procurement and ensuring the revenue of suppliers. Finally, the influence of each parameter on the optimization decision is verified by a numerical example.

The efficient and stable planning for interrupted supply chain with dual‐sourcing strategy: a robust optimization approach considering decision maker's risk attitude

• Proposing an efficient and robust dual-sourcing model for disrupted supply chain • Considering various risk attitudes and interruption uncertain situations • Processing supply interruption uncertainty with robust optimization method • Applying the real case study to analyze validity and stability of model • Providing useful managerial implications for interrupted supply chain sourcing In the face of the complex market environment and the in-depth development of globalization, ensuring the supply chain's stability and cost leadership by developing a differentiated dual-source procurement strategy is extremely important. The present study contemplates the supply chain as the research object. It also considers disruption uncertainty and decision-makers’ risk attitudes. To minimize costs and shortages, we build a dual-objective robust optimization model for a dual-sourcing strategy under interruption risk. The authors aim to explore efficient and robust planning under supply disruption uncertainty in dual-sourcing strategies by different risk attitudes. The research employs the augmented ε-constraint method to solve the proposed model by transforming the model into a single-objective mixed-integer programming model. Additionally, our study uses enterprise examples to prove the feasibility and effectiveness of the model. Moreover, it conducts risk attitude analysis, dual objective function trade-off analysis, and superiority analysis of procurement strategies. Finally, we put forward some management insights to provide a new decision-making idea for real supply chain enterprises in the case of supply interruption risk.

Optimal shipping quantity, product pricing strategies and carbon emissions of online retailers under a new logistics mode

A new logistics mode has emerged in online retailing, which ships products before customers place orders online to reduce the waiting time for customers to receive them. The existing research on this logistics mode mainly focuses on the shipping quantity (i.e., the quantity of products to be shipped in advance from the order fulfillment center to the distribution center nearby customers before the order is placed online), while the product pricing problem and carbon emissions have not been studied comprehensively. Moreover, the risk attitudes of decision-makers have been ignored in the studies related to this new logistics mode. This paper builds an extended newsvendor model to explore the optimal joint decision problem of shipping quantity and product pricing under this new logistics mode and analyzes the carbon emissions of this new logistics mode compared with the traditional logistics mode. We consider decision-makers to be risk averse and take Conditional Value-at-Risk (CVaR) as the risk measure criterion. For both multiplicative and additive demand models, we obtain the closed-form solution of the optimal strategy. The differences between the optimal strategies of risk-averse and risk-neutral decision-makers and between the optimal strategies of this logistics mode and the traditional one have also been explored. Based on the optimal shipping quantity and product pricing strategies, we analyze the carbon emission variation of this logistics mode compared with the traditional one and provide sufficient conditions for reducing carbon emissions. Numerical experiments verify the theoretical conclusions and analyze the effect of some parameters on the optimal strategy, profit and carbon emission variation. The results show that online retailers should reduce product price in this new logistics mode compared with the traditional one. The risk-aversion degree slightly affects the optimal price but significantly affects the shipping quantity. Moreover, implementing this new logistics mode contributes to reducing carbon emissions compared with the traditional logistics mode in most cases. This study helps support online retailers in developing shipping quantity and product pricing strategies under this new logistics mode and helps the Environmental Protection Association to understand the carbon emissions under this new logistics mode so that relevant strategies can be developed.

A three-way decision approach with S-shaped utility function under Pythagorean fuzzy information

Three-way decision, as opposed to conventional two-way decision-making, can more effectively reduce decision-making risks by adding a postponed decision-making. With respect to the three-way decision problem consisting of Pythagorean fuzzy information, considering the risk attitudes of decision makers(DMs), we exploit Pythagorean fuzzy sets (PFNs), S-shaped utility and three-way decision to establish a new Pythagorean three-way decision model. The model realizes the effective integration of Pythagorean fuzzy theory and three-way decision, and enriches three-way decision based on risk preference. Firstly, we define a new PFN score function and construct a new condition probability by using Pythagorean fuzzy similarity, which not only reflects the difference in distance but conforms to human cognitive habits. Subsequently, considering the impact of the changes in risk preferences on the measurement of decision risks, we define an action utility function and construct the risk measurement matrix function based on S-shaped utility function, calculate the thresholds by employing optimization formulation. According to key steps and algorithms, the Pythagorean fuzzy-three-way decision rules based on S-shaped utility function can be obtained. Finally, we utilize an example of selecting film projects to validate the interpretability and applicability of our proposed model.

Incorporating the status quo effect into the decision making process: The case of municipal companies merger

Decision-making for cooperation between municipal entities involves elected members serving public interest who argue in both quantitative (e.g. financial performance), and qualitative (e.g. social acceptance) grounds. Policy support should consider conflicting objectives in a context of uncertainty and imprecision as well as in presence of organizational lock-ins that tend to favor incumbent arrangements. In search of appropriate algorithm to support decision making in such cases we opted for the PROMETHEE outranking method integrating novel findings of behavioral theory. Thus its preference functions are specified so that they reflect bounded rationality that shapes risk attitudes of decision makers within the uncertain environment of public management. Moreover alternative weighting procedures have been applied for group decisions. The prospect theory multi-criteria methodology, termed PT-PROMETHEE is implemented relating proposed options to observed behavior providing useful insights especially regarding status quo options also allowing for sensitivity analysis. Thus acceptance of MCDM support by the stakeholders is enhanced leading to solid compromise institutional arrangements.

A Wasserstein metric-based distributionally robust optimization approach for reliable-economic equilibrium operation of hydro-wind-solar energy systems

Hydro-wind-solar integrated operation is a promising way to balance the growing amount of variable renewable energy (RE) and enhance energy utilization efficiency. This study focuses on the short-term reliable-economic equilibrium operation of the hydro-wind-solar energy systems. A Distributionally Robust Optimization based on Wasserstein metric, which considers multiple power supply risks (output shortage, power curtailment and spilled water) and the economy (the water consumption and the stored energy) of hydropower operation simultaneously is proposed to address the above issue. The model is transformed into a mixed-integer linear programming framework using a reformation approach based on ϵ-constraint method, strong duality theory, and linearization technology. Case studies are conducted for a hydro-wind-solar energy system in Southwest China's Beipan River basin. Results indicate that: (1) with more available data, more information about RE uncertainties is involved in the decision-making process to hedge against RE variability's interference. (2) Decisions obtained from the proposed model overcome the issues of stochastic optimization's over-optimism and robust optimization's over-conservativeness. (3) Reliable-economic trade-off mechanisms are derived by imposing a decision-maker's risk attitude, the risks in dry and wet season typical cases decline from 10483.05 MW to 1606.39 MW and from 8394.15 MW to 118.08 MW by adjusting the economy.

Dual-Source Procurement Strategy of Cross-Border E-Commerce Supply Chain considering Members’ Risk Attitude

The risk attitude of decision-makers will significantly affect the decision-making of enterprise risk management. Specifically, high risk represents the potential premise of high return for risk preference decision-makers, and for risk-averse decision-makers, the increase of risk degree will stimulate decision-makers’ aversion to uncertainty and turn to seek safer business strategies. Although there are many pieces of literature on the risk preference of decision-makers, they usually only assume the risk attitude of one party and rarely consider the risk attitude of suppliers and retailers in the scenario of cross-border e-commerce at the same time. Therefore, under the background of supply disruption, for the cross-border e-commerce supply chain composed of cross-border suppliers, overseas suppliers, overseas retailers, and consumers, combined with the risk attitude preference of enterprise subjects, this paper constructs the mean-variance model dominated by overseas retailers and reversely solves the risk-aversion attitude of a single cross-border supplier. When a single overseas retailer maintains a risk-aversion attitude and both cross-border suppliers and overseas retailers hold a risk-neutral or risk-aversion attitude, the pricing of products in different channels is analyzed. Finally, an example is given to analyze the impact of supply disruption probability, risk-aversion coefficient, channel distribution coefficient, and other parameters on purchase price, market demand, target profit, and utility. It is of great practical significance for improving the stability of cross-border e-commerce supply chain system and reducing revenue loss to study how different degrees of risk-aversion attitudes of cross-border suppliers and overseas retailers affect enterprise procurement pricing strategy, target profit, and utility in case of supply disruption.

A novel LOPCOW‐DOBI multi‐criteria sustainability performance assessment methodology: An application in developing country banking sector

Banks do not directly contribute to sustainable development, yet they indirectly play an undeniable role. The sustainability of the banking sector, which is the heart of finance, is crucial both for this sector and economy as well as for the realization of the 2030 sustainable development goals. Despite the growing interest of practitioners and scholars on this issue, previous works are rare for developing countries. With a novel multi-criteria framework developed, therefore, this paper assesses banks in terms of sustainability and clarifies their levels of corporate sustainability performances. For the first time, this work introduces an objective weighting method named LOgarithmic Percentage Change-driven Objective Weighting (LOPCOW) with the advantages of eliminating the gap due to the size of the data, generating more reasonable weightings, and considering both positive and negative data in the weighting process. Further, a new multi-criteria method named DOmbi Bonferroni (DOBI) is developed and utilized for deciding the ranking of the alternatives. The technique provides flexible decision-making, considering the risk attitude of decision-makers. The applicability of the proposed framework is demonstrated by a case study of Turkish banks. Banks are analyzed regarding the three pillars of sustainability and their seventeen drivers. The environmental pillar indicates the top significant impact on banks' sustainability. The “average return on shareholders' equity”, “electricity consumption”, “number of branches,” and “number of staff” are the top four key drivers for banks' sustainability. As a result of ranking the nine banks that release sustainability reports in Turkey concerning their sustainability performances, it is highlighted that Garanti BBVA outperforms the others. With a comprehensive sensitivity check, the robustness and applicability of the model are further tested. The developed decision support tool could allow different organizations to conduct sustainability assessments under the main pillars and help monitor their improvement over time. Some political and managerial implications are as well mentioned.

A Dempster-Shafer-theory-based entry screening mechanism for small and medium-sized enterprises under uncertainty

The evaluation of star-up small and medium-sized enterprises (SMEs) is an important part of ensuring the success of development, but it is not an easy task because of the multiple criteria to be considered and uncertainties caused by incomplete decision information, limited individual perceptions and differences in perceptions of experts. By integrating Dempster-Shafer theory into a probabilistic linguistic setting that takes into account the uncertainties in the evaluation process by combining linguistic terms with subjective probabilities, this study proposes a multi-criteria decision-making model for the management of SMEs. We incorporate different decision factors regarding (a) the risk attitude of decision makers, (b) the relative importance of criteria, (c) the unreliability and incompleteness of evaluation on alternatives in one preference model of multi-criteria decision making. The applicability of the proposed preference model is validated by a case study of the entry screening of SMEs for business incubators. Through comparative analysis and sensitive analysis, the reliability of the proposed model is verified, which can not only express uncertain evaluation information, but also avoid information loss and defuzzification in the aggregation process.

A PROBABILISTIC LINGUISTIC VIKOR METHOD TO SOLVE MCDM PROBLEMS WITH INCONSISTENT CRITERIA FOR DIFFERENT ALTERNATIVES

VIKOR is a well-defined multiple criteria decision-making (MCDM) method since it reflects different risk attitudes of decision-makers by measuring the overall performance of an alternative under all criteria as “group value” and the worst performance of the alternative under all criteria as “individual regret”, but it cannot deal with MCDM problems where the criteria of different alternatives are inconsistent (different) and the decision information is uncertain. To address these problems, we present a probabilistic linguistic VIKOR method by combining with probabilistic linguistic term sets which portrays uncertain information such as individual hesitancy and incomplete belief flexibly. In addition, we introduce the aspired and tolerable values of criteria as reference points to measure the closeness degrees of alternatives to the ideal solution. To compare group values and individual regrets of different alternatives, we develop a vector normalization method that considers the number of criteria. The robustness of the aggregation results of group values and individual regrets is improved based on the extended Borda rule, which takes into account both values and ranks of alternatives in the aggregation. A case study of personnel evaluation demonstrates the effectiveness of the proposed method for solving MCDM problems with inconsistent criteria and uncertain decision information.

Shortfall Risk Models When Information on Loss Function Is Incomplete

Utility-based shortfall risk measures effectively captures a decision maker's risk attitude on tail losses. In this paper, we consider a situation where the decision maker's risk attitude toward tail losses is ambiguous and introduce a robust version of shortfall risk, which mitigates the risk arising from such ambiguity. Specifically, we use some available partial information or subjective judgement to construct a set of plausible utility-based shortfall risk measures and define a so-called preference robust shortfall risk as through the worst risk that can be measured in this (ambiguity) set. We then apply the robust shortfall risk paradigm to optimal decision-making problems and demonstrate how the latter can be reformulated as tractable convex programs when the underlying exogenous uncertainty is discretely distributed.