Piecewise Linear Utility Function Research Articles

From local utility to neural networks

We introduce and analyze two preference-based notions of local linearity in the spirit of Machina (1982). We show how the weaker among the two extends Machina’s local utility analysis, and that the stronger among the two characterizes continuous finite piecewise linear (CFPL) utility functions. We introduce a representation of the decision maker’s preference called the neural-network utility representation that is equivalent to the CFPL representation, in which the decision maker evaluates an alternative through a neural network.

Optimal Mechanism in a Dynamic Stochastic Knapsack Environment

This study introduces an optimal mechanism in a dynamic stochastic knapsack environment. The model features a single seller who has a fixed quantity of a perfectly divisible item. Impatient buyers with a piece-wise linear utility function arrive randomly and they report the two-dimensional private information: marginal value and demanded quantity. We derive a revenue-maximizing dynamic mechanism in a finite discrete time framework that satisfies incentive compatibility, individual rationality, and feasibility conditions. This is achieved by characterizing buyers' utility and utilizing the Bellman equation. Moreover, we establish the essential penalty scheme for incentive compatibility, as well as the allocation and payment policies. Lastly, we propose algorithms to approximate the optimal policy, based on the Monte Carlo simulation-based regression method and reinforcement learning.

A 0–1 linear relaxation and stochastic approximation method for real-time pricing in a smart grid with non-convex constraints

In this article, real-time pricing (RTP) for a smart grid with complicated non-convex and/or non-smooth constraints is explored. First, piecewise linear functions are adopted to approximate any kinds of utility function and RTP is formulated into bilevel non-convex programming. Then, a 0–1 relaxation method is used to linearize sparse constraints and piecewise linear utility functions, to obtain mixed integer linear programming. Finally, a distributed simultaneous perturbation stochastic approximation (DSPSA) method is designed to solve the bilevel non-convex programming. The RTP strategy is generalizable because sparse constraints and non-differentiable and/or non-concave utility functions are widespread in practice, and the piecewise linear functions approximation method is applicable to all kinds of utility function, not just to non-smooth and/or non-concave utility functions. In addition, the 0–1 linear relaxation and stochastic approximation method make the bilevel non-convex programming easily solvable by DSPSA, which converges rapidly while maintaining the high accuracy of solutions.

Optimal insurance design under mean-variance preference with narrow framing

In this paper, we study an optimal insurance design problem under mean-variance criterion by considering the local gain-loss utility of the net payoff of insurance, namely, narrow framing. We extend the existing results in the literature to the case where the decision maker has mean-variance preference with a constraint on the expected utility of the net payoff of insurance, where the premium is determined by the mean-variance premium principle. We first show the existence and uniqueness of the optimal solution to the main problem studied in the paper. We find that the optimal indemnity function involves a deductible provided that the safety loading imposed on the “mean part” of the premium principle is strictly positive. Our main result shows that narrow framing indeed reduces the demand for insurance. The explicit optimal indemnity functions are derived under two special local gain-loss utility functions – the quadratic utility function and the piecewise linear utility function. As a spin-off result, the Bowley solution is also derived for a Stackelberg game between the decision maker and the insurer under the quadratic local gain-loss utility function. Several numerical examples are presented to further analyze the effects of narrow framing on the optimal indemnity function as well as the interests of both parties.

Risk Aversion and a Calculus for Finitely Generated Piecewise Linear Functions: A Calculus that Economists Ought to Develop?

In this note we propose a calculus for piece-wise linear functions, in order to obtain derivatives and second derivatives at points where the function is not differentiable. Such derivatives can be used to calculate coefficients of risk aversion at initial wealth for piece-wise linear utility functions for gains, which display loss aversion-and hence non differentiability at zero gains.

An Optimization Approach Considering User Utility for the PV-Storage Charging Station Planning Process

Based on the comprehensive utilization of energy storage, photovoltaic power generation, and intelligent charging piles, photovoltaic (PV)-storage charging stations can provide green energy for electric vehicles (EVs), which can significantly improve the green level of the transportation industry. However, there are many challenges in the PV-storage charging station planning process, making it theoretically and practically significant to study approaches to planning. This paper promotes a bi-level optimization planning approach for PV-storage charging stations. First, taking PV-storage charging stations and EV users as the upper- and lower-level problems, respectively, during the planning process, a bi-level optimization model for PV-storage charging stations considering user utility is established for capacity allocation and user behavior-based electricity pricing. Second, the model is converted into a single-level mixed-integer linear programming model using the piecewise linear utility function, Karush–Kuhn–Tucker (KKT) conditions, and linearization methods. Finally, to verify the validity of the proposed model and the solution algorithm, a commercial solver is used to solve the optimization model and obtain the planning scheme. The results show that the proposed bi-level optimization model can provide a more economical and reasonable planning scheme than the single-level model, and can reduce the investment cost by 8.84%, operation and maintenance cost by 13.23%, and increase net revenue by 5.11%.

Topology design of remote patient monitoring system concerning qualitative and quantitative issues

Piecewise linear functions have been used to model and solve non-linear problems in science, social science, and technology fields since the 1950′s. Multi-choice goal programming (MCGP) has been widely used to solve multiple objective decision-making (MODM) problems in which each goal mapping with multiple aspiration levels is allowed, expanding the original feasible region to obtain better solutions in the MODM problems. This paper integrates the efficient S-shaped penalty method, arbitrary piecewise linear utility functions, trapezoidal utility functions, and MCGP to solve a topology design problem in a remote patient monitoring system (RPMS) by providing universal senior citizen coverage in which quantitative (e.g., patient satisfaction-related cost) and qualitative (e.g., satisfaction) issues are considered simultaneously. In addition, some novel utility functions such as the force utility function, indicator utility function, and arbitrary utility function are proposed to improve the usefulness of MCGP in the field of management science. These proposed utility functions can be easily used to model qualitative issues in real-world problems. A topology design problem for an RPMS is demonstrated to justify the feasibility, usefulness, and compatibility of the proposed methods. Further, sensitivity analysis and managerial implications are provided using an RPMS in Taiwan.

Real-Time Pricing of Smart Grid Based on Piece-Wise Linear Functions

Abstract In a power grid system, utility is a measure of the satisfaction of users’ electricity consumption; cost is a monetary value of electricity generated by the supplier. The utility and cost functions represent the satisfaction of different users and the supplier. Quadratic utility, logarithmic utility, and quadratic cost functions are widely used in social welfare maximization models of real-time pricing. These functions are not universal; they have to be discussed in detail for individual models. To overcome this problem, a piece-wise linear utility function and a piece-wise linear cost function with general properties are proposed in this paper. By smoothing the piece-wise linear utility and cost functions, a social welfare maximization model can be transformed into a differentiable convex optimization problem. A dual optimization method is used to solve the smoothed model. Through mathematical deduction and numerical simulations, the rationality of the model and the validity of the algorithm are verified as long as the elastic and cost coefficients take appropriate values. Thus, different user types and the supplier can be determined by selecting different elastic and cost coefficients.

Discrete-time mean-CVaR portfolio selection and time-consistency induced term structure of the CVaR

We investigate a discrete-time mean-risk portfolio selection problem, where risk is measured by the conditional value-at-risk (CVaR). A substantial challenge is the combination of a time-inconsistent objective with an incomplete and dynamic model for the financial market. We are able to solve this problem analytically by embedding the original, time-inconsistent problem into a family of time-consistent expected utility maximization problems with a piecewise linear utility function. The optimal investment strategy is a fully adaptive feedback policy and the cumulated amount invested in the risky assets is of a characteristic V-shaped pattern as a function of the current wealth. For the incomplete, discrete-time market considered herein, the mean-CVaR efficient frontier is a straight line in the mean-CVaR plane and thus economically meaningful. This contrasts the complete, continuous-time setting where the mean-CVaR efficient frontier is degenerate or does not exist at all. We further solve an inverse investment problem, where we investigate how mean-CVaR preferences need to adapt in order for the pre-committed optimal strategy to remain optimal at any point in time. Our result shows that a pre-committed mean-CVaR investor behaves like a naive mean-CVaR investor with a time-increasing confidence level for the CVaR, who revises the investment decision at every point in time. Finally, an empirical application of our results suggests that risk measured by the CVaR might help to understand the long-standing equity premium puzzle.

Discrete-Time Mean-CVaR Portfolio Selection and Time-Consistency Induced Term Structure of the CVaR

We investigate a discrete-time mean-risk portfolio selection problem, where risk is measured by the conditional value-at-risk (CVaR). By embedding this time-inconsistent problem into a family of expected utility maximization problems with a piecewise linear utility function, we solve the problem analytically. In contrast to the case of a complete, continuous-time market, the mean-CVaR efficient frontier in this generally incomplete, discrete-time setting is a straight line in the mean-CVaR plane and there is in particular a constant trade-off between risk and return. The cumulated amount invested in the risky assets under the optimal strategy is of a V -shaped pattern as a function of the current wealth. We further solve an inverse investment problem, where we investigate how mean-CVaR preferences need to adapt such that the pre-commited optimal strategy remains optimal at any point in time. Our result shows that, although conceptually distinct, a pre-commited mean-CVaR investor behaves like a naive mean-CVaR investor with a time-increasing confidence level for the CVaR, who revises her investment decision at every point in time. Finally, an empirical application of our results suggests that risk measured by the CVaR might help to understand the long-standing equity premium puzzle.

Loss-averse decision analysis in overbooking

ABSTRACTThe strategy of overbooking capacity is commonly practiced in business that accepts reservations and subsequently runs the risk of cancellations and no-shows. Traditional overbooking models are mainly based on the assumption that decision-makers are rationally loss-neutral. This paper seeks the optimal overbooking policies for managers with a loss-averse preference towards the loss of compensations when excessive customers actually show up. This study presents overbooking models in both single-period and multiperiod settings in which the loss-averse preference is described with a ¡°kinked¡± piecewise-linear utility function. The analysis demonstrates the optimal policy exhibiting a booking limit structure in both settings. The manager accepts reservation requests up to a booking limit if the number of initial reservations is less than that booking limit, and otherwise declines reservation requests. This study finds that loss-averse managers are cautious and prefer lower booking limits compared to loss-neutral managers. From numerical studies, we investigate the biases between predictions of loss-neutral and loss-averse models. This study also investigates how the optimal policy changes based on certain parameters, including the decision-maker's degree of loss-aversion.

DEA Efficiency of Energy Consumption in China’s Manufacturing Sectors with Environmental Regulation Policy Constraints

Because overall energy consumption intensity in China’s manufacturing industry is extremely high, the study of energy efficiency in that industry, with an analysis of the policy impacts of energy intensity reduction and other key factors, will no doubt improve energy utilization in the industry and stimulate sustainable development within it. This paper uses 2004–2014 panel data of 28 manufacturing industries and a piecewise linear utility function to construct a data envelopment analysis (DEA) model of energy consumption with environmental regulations constraints. We also examine the DEA evaluation of energy efficiency in manufacturing industries. We integrate environmental regulations as qualitative variables into the energy consumption evaluation model to research the coupling effects on energy consumption intensity of energy consumption structure, opening up, environmental regulations, technological progress, and competition within industries. The research shows that energy efficiency policy intensity is not the major effect on the development of low or moderate energy-consumption industries, whereas low-energy-efficiency policy is very favorable for the development of high energy-consumption industries.

A two-resource allocation algorithm with an application to large-scale zero-sum defensive games

This paper investigates efficient computation schemes for allocating two defensive resources to multiple sites to protect against possible attacks by an adversary. The availability of the two resources is constrained and the effectiveness of each may vary over the sites. The problem is formulated as a two-person zero-sum game with particular piecewise linear utility functions: the expected damage to a site that is attacked linearly decreases in the allocated resource amounts up to a point that a site is fully protected. The utility of the attacker, equivalently the defender's disutility, is the total expected damage over all sites. A fast algorithm is devised for computing the game's Nash equilibria; it is shown to be more efficient in practice than both general purpose linear programming solvers and a specialized method developed in the mid-1980s. To develop the algorithm, optimal solution properties are explored.

A Neurodynamic Approach for Real-Time Scheduling via Maximizing Piecewise Linear Utility.

In this paper, we study a set of real-time scheduling problems whose objectives can be expressed as piecewise linear utility functions. This model has very wide applications in scheduling-related problems, such as mixed criticality, response time minimization, and tardiness analysis. Approximation schemes and matrix vectorization techniques are applied to transform scheduling problems into linear constraint optimization with a piecewise linear and concave objective; thus, a neural network-based optimization method can be adopted to solve such scheduling problems efficiently. This neural network model has a parallel structure, and can also be implemented on circuits, on which the converging time can be significantly limited to meet real-time requirements. Examples are provided to illustrate how to solve the optimization problem and to form a schedule. An approximation ratio bound of 0.5 is further provided. Experimental studies on a large number of randomly generated sets suggest that our algorithm is optimal when the set is nonoverloaded, and outperforms existing typical scheduling strategies when there is overload. Moreover, the number of steps for finding an approximate solution remains at the same level when the size of the problem (number of jobs within a set) increases.

Assessing Trade‐offs among Multiple Objectives for Humanitarian Aid Delivery Using Expert Preferences

Humanitarian aid agencies deliver emergency supplies and services to people affected by disasters. Scholars and practitioners have developed modeling approaches to support aid delivery planning, but they have used objective functions with little validation as to the trade‐offs among the multiple goals of aid delivery. We develop a method to value the performance of aid delivery plans based on expert preferences over five key attributes: the amount of cargo delivered, the prioritization of aid by commodity type, the prioritization of aid by delivery location, the speed of delivery, and the operational cost. Through a conjoint analysis survey, we measure the preferences of 18 experienced humanitarian logisticians. The survey results quantify the importance of each attribute and enable the development of a piecewise linear utility function that can be used as an objective function in optimization models. The results show that the amount of cargo delivered is the most valued objective and cost the least important. In addition, experts prioritize more vulnerable communities and more critical commodities, but not to the exclusion of others. With these insights and the experts’ utility functions, better humanitarian objective functions can be developed to enable better aid delivery in emergency response.

Allocating multiple defensive resources in a zero-sum game setting

This paper investigates the problem of allocating multiple defensive resources to protect multiple sites against possible attacks by an adversary. The effectiveness of the re- sources in reducing potential damage to the sites is assumed to vary across the resources and across the sites and their availability is constrained. The problem is formulated as a two-person zero-sum game with piecewise linear utility functions and polyhedral action sets. Linearization of the utility functions is applied in order to reduce the computation of the game's Nash equilibria to the solution of a pair of linear programs (LPs). The reduction facilitates revelation of structure of Nash equilibrium allocations, in particular, of mono- tonicity properties of these allocations with respect to the amounts of available resources. Finally, allocation problems in non-competitive settings are examined (i.e., situations where the attacker chooses its targets independently of actions taken by the defender) and the struc- ture of solutions in such settings is compared to that of Nash equilibria.

Offering Strategies and Simulation of Multi-Item Iterative Auctions of Energy Contracts

The objective of this work is threefold. We firstly present an optimization model for a price-taker hydrothermal generation company (Genco) to devise bidding strategies in multi-item iterative auctions of long-term contracts. The bidding model calculates a bidder's best-response function, which takes into account the key issues on the auctioned contracts, such as its time horizon, the risk factors that affect the future contract outcomes, interdependence between auctioned products, and the agents' risk profile. The risk profile of the Gencos are represented as piecewise linear utility functions and a practical specification approach is proposed. We finally present a simulator of a uniform-pricing and iterative multi-item contract auction, where the set of auction rules is implemented and the optimization model developed is used to determine the Gencos's best responses at each auction round. The simulator enables auctioneers and generators to estimate the auction's competitive equilibrium price and to study the auction dynamics. A real multiproduct descending clock auction is simulated for the Brazilian power system under the proposed bidding scheme.

Nonlinear Effects in Brand Choice Models: Comparing Heterogeneous Latent Class To Homogeneous Nonlinear Models

We investigate whether the latent class multinomial logit choice model with segmentspecific linear utility functions implies effects that are similar to those of parametric homogeneous nonlinear models given that this latent class model performs at least as well. The two nonlinear models have higher-order polynomial (i.e. quadratic and cubic) and piecewise linear utility functions, respectively. Piecewise linear functions are represented by linear splines and can reproduce threshold, saturation and asymmetric effects. We evaluate models and their variants using a tenfold cross-validation. As criterion we use the geometric mean of choice probabilities across all purchases for the brand actually chosen. We measure the similarity of effects between two models by the absolute differences of choice probabilities implied by these models for varying values of a predictor. Logits of choice probabilities provide a more detailed insight into the effects implied by models. For the data set we analyze, the latent class model with linear utility is clearly superior to the two homogeneous nonlinear models. Overall, the effects implied by the latent class models are similar to those of the two parametric nonlinear models.

Stochastic Shortest Path Problems with Piecewise-Linear Concave Utility Functions

This paper considers a stochastic shortest path problem where the arc lengths are independent random variables following a normal distribution. In this problem, the optimal path is one that maximizes the expected utility, with the utility function being piecewise-linear and concave. Such a utility function can be used to approximate nonlinear utility functions that capture risk averse behaviour for a wide class of problems. The principal contribution of this paper is the development of exact algorithms to solve large problem instances. Two algorithms are developed and incorporated in labelling procedures. Computational testing is done to evaluate the performance of the algorithms. Overall, both algorithms are very effective in solving large problems quickly. The relative performance of the two algorithms is found to depend on the “curvature” of the piecewise linear utility function.

A multicriterion approach to groundwater management

A new approach to groundwater quantity (hydraulic) management, based on a multicriterion decision aid methodology, is presented. The method couples a hydrodynamic groundwater flow simulation model with the decision aid one. It takes into account multiple criteria used by the decision maker (DM) in evaluating a management scenario as well as the hydrodynamic behavior of the groundwater system. The finite element and embedding methods are used to integrate the groundwater flow model into a multiobjective linear programming (LP) problem. Constraints on head, pumping rates, hydraulic gradient and velocity vector may be included in the management model. The piecewise linear utility function is assessed for modeling the DM's preferences. The best compromise solution is determined from the continuous Pareto set by solving a piecewise LP problem. User friendly software was developed to realize this methodology which is able to treat real scale problems. An illustrative example of an unconfined aquifer management is presented. Nonlinearities resulting from functional dependence of aquifer parameters on hydraulic head are handled iteratively.