Mathematics

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23 pages, 651 KiB

Open AccessArticle

Allocating Benefits Due to Shared Resources Using Shapley Value and Nucleolus in Dynamic Network Data Envelopment Analysis

by Lívia Torres and Francisco S. Ramos

Mathematics 2024, 12(5), 698; https://doi.org/10.3390/math12050698 - 28 Feb 2024

Viewed by 564

Abstract

Shared resources are common among supply chain partners and also occur in multiple linked stages of an internal network. The sharing of these resources impacts the organization’s profits. This study is focused on the potential benefits of resource sharing on a three-stage network [...] Read more.

Shared resources are common among supply chain partners and also occur in multiple linked stages of an internal network. The sharing of these resources impacts the organization’s profits. This study is focused on the potential benefits of resource sharing on a three-stage network system and on the profit improvement allocation. Previous treatments concentrate on defining optimal proportions to allocate resources and disregard the impacts of allocations to promote cooperation and are limited to static evaluations. Data Envelopment Analysis performs the decision-making units (DMUs) efficiency measurement. Methodological advances have resulted in models that analyze their internal structure and temporal impacts on efficiency. We propose an integrated cooperative game and dynamic network DEA that considers known quantities of resources used in each stage and the time effects to optimize the system’s profit. Each DMU stage is a player, and we investigate performance before and after resource sharing. Using Shapley value and Nucleolus, it is possible to allocate the benefits obtained based on the marginal contributions of each stage, providing incentives to motivate and maintain cooperation. A numerical example is used to illustrate the method. The results confirm the identification of inefficient DMUs and that sharing resources allows for profit increase for all of them. Full article

(This article belongs to the Special Issue Application of Optimal Control and Game Theory to the Problem of Resource Management)

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22 pages, 484 KiB

Open AccessArticle

Robust Statistic Estimation in Constrained Optimal Control Problems of Pollution Accumulation (Part II: Markovian Switchings)

by Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos, Carmen Geraldi Higuera-Chan and Francisco Alejandro Alaffita-Hernández

Mathematics 2023, 11(4), 1045; https://doi.org/10.3390/math11041045 - 18 Feb 2023

Cited by 1 | Viewed by 1011

Abstract

This piece is a follow-up of the research started by the authors on the constrained optimal control problem applied to pollution accumulation. We consider a dynamic system governed by a diffusion process with multiple modes that depends on an unknown parameter. We will [...] Read more.

This piece is a follow-up of the research started by the authors on the constrained optimal control problem applied to pollution accumulation. We consider a dynamic system governed by a diffusion process with multiple modes that depends on an unknown parameter. We will study the components of the model and their restrictions and propose a scheme to solve the problem in which it is possible to determine (adaptive) policies that maximize a suitable discounted reward criterion using standard dynamic programming techniques in combination with discrete estimation methods for the unknown parameter. Finally, we develop a numerical example to illustrate our results with a particular case of the method of minimum least square error approximation. Full article

(This article belongs to the Special Issue Application of Optimal Control and Game Theory to the Problem of Resource Management)

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19 pages, 622 KiB

Open AccessArticle

Robust Statistic Estimation of Constrained Optimal Control Problems of Pollution Accumulation (Part I)

by Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos, Carmen Geraldi Higuera-Chan and Francisco Alejandro Alaffita-Hernández

Mathematics 2023, 11(4), 923; https://doi.org/10.3390/math11040923 - 11 Feb 2023

Cited by 2 | Viewed by 1335

Abstract

In this paper, we study a constrained optimal control on pollution accumulation where the dynamic system was governed by a diffusion process that depends on unknown parameters, which need to be estimated. As the true values are unknown, we intended to determine (adaptive) [...] Read more.

In this paper, we study a constrained optimal control on pollution accumulation where the dynamic system was governed by a diffusion process that depends on unknown parameters, which need to be estimated. As the true values are unknown, we intended to determine (adaptive) policies that maximize a discounted reward criterion with constraints, that is, we used Lagrange multipliers to find optimal (adaptive) policies for the unconstrained version of the optimal control problem. In the present context, the dynamic system evolves as a diffusion process, and the cost function is to be minimized by another function (typically a constant), which plays the role of a constraint in the control model. We offer solutions to this problem using standard dynamic programming tools under the constrained discounted payoff criterion on an infinite horizon and the so-called principle of estimation and control. We used maximum likelihood estimators by means of a minimum least square error approximation in a pollution accumulation model to illustrate our results. One of the advantages of our approach compared to others is the intuition behind it: find optimal policies for an estimated version of the problem and let this estimation tend toward the real version of the problem. However, most risk analysts will not be as used to our methods as they are to, for instance, the model predictive control, MATLAB’s robust control toolbox, or the polynomial chaos expansion method, which have been used in the literature to address similar issues. Full article

(This article belongs to the Special Issue Application of Optimal Control and Game Theory to the Problem of Resource Management)

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11 pages, 321 KiB

Open AccessFeature PaperArticle

Estimation of Initial Stock in Pollution Control Problem

by Shimai Su and Anna Tur

Mathematics 2022, 10(19), 3457; https://doi.org/10.3390/math10193457 - 22 Sep 2022

Cited by 1 | Viewed by 1053

Abstract

A two-player differential game of pollution control with uncertain initial disturbance stock is considered. In pace with contemporary policy in the resource extraction industry, we initiate our research based on a resource extraction differential model with a rehabilitation process in which the firms [...] Read more.

A two-player differential game of pollution control with uncertain initial disturbance stock is considered. In pace with contemporary policy in the resource extraction industry, we initiate our research based on a resource extraction differential model with a rehabilitation process in which the firms are required to compensate the local to rehabilitate the polluted and dilapidated areas. Given the reality that the initial pollution stock plays a critical role in the production, and we cannot rigorously determine its actual value, a simulation of the estimation of the initial stock is alternatively investigated through the Pontryagin maximum principle (PMP). The later analytical results by normalized value of information (NVI) indicate the precious influence brought to the final payoff under various estimations of the initial stock both in the cooperative and non-cooperative cases. With such guidance, the player is capable of making a much more judicious decision when it comes to the determination of the initial stock. Furthermore, a numerical example is additionally presented for better comprehension. Full article

(This article belongs to the Special Issue Application of Optimal Control and Game Theory to the Problem of Resource Management)

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10 pages, 317 KiB

Open AccessArticle

On the Estimation of the Initial Stock in the Problem of Resource Extraction

by Anna Tur, Ekaterina Gromova and Dmitry Gromov

Mathematics 2021, 9(23), 3099; https://doi.org/10.3390/math9233099 - 01 Dec 2021

Cited by 3 | Viewed by 1631

Abstract

We consider a differential game of non-renewable resource extraction, in which the players do not know the precise value of the resource stock and, thus, have to make an estimate. We define the value of information about the initial stock and give recommendations [...] Read more.

We consider a differential game of non-renewable resource extraction, in which the players do not know the precise value of the resource stock and, thus, have to make an estimate. We define the value of information about the initial stock and give recommendations for the choice of the estimate depending on the parameters of the problem. Further, we consider the situation where the players only know the bounds for the stock of the resource and solve the problem of computing the optimal estimate, such that it minimizes the players’ losses in the worst-case scenario. The analysis allows us to give a simple rule for the choice of the optimal estimate of the resource stock. Full article

(This article belongs to the Special Issue Application of Optimal Control and Game Theory to the Problem of Resource Management)

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10 pages, 281 KiB

Open AccessArticle

The Cost of Work Discrimination: A Market Capture Differential Game Model

by Mario Alberto García-Meza

Mathematics 2021, 9(19), 2419; https://doi.org/10.3390/math9192419 - 28 Sep 2021

Cited by 2 | Viewed by 1760

Abstract

Discrimination in the workplace can be a source of opportunity costs for firms that desire to enter a new market. In this article, we model how an exogenous technological change introduces both new potential workers in the labor market and, as the economy [...] Read more.

Discrimination in the workplace can be a source of opportunity costs for firms that desire to enter a new market. In this article, we model how an exogenous technological change introduces both new potential workers in the labor market and, as the economy grows, a new set of consumers for firms to conquer. We then use a differential model to show how the payoff of a firm that discriminates in the labor market loses potential market from this same demographic. Our results imply that anti-discriminatory policies within the firm constitute an important element for the solution of this problem, and that there might be incentives for the firms to apply such policies. Full article

(This article belongs to the Special Issue Application of Optimal Control and Game Theory to the Problem of Resource Management)

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24 pages, 907 KiB

Open AccessFeature PaperArticle

A Resource Extraction Model with Technology Adoption under Time Inconsistent Preferences

by Carles Mañó-Cabello, Jesús Marín-Solano and Jorge Navas

Mathematics 2021, 9(18), 2205; https://doi.org/10.3390/math9182205 - 08 Sep 2021

Viewed by 2168

Abstract

A two-stage non-standard optimal control problem with time inconsistent preferences is studied. In an infinite horizon setting, a time consistent (sophisticated) decision maker chooses the time of switching between two consecutive regimes. The second regime corresponds to the implementation of a new technology, [...] Read more.

A two-stage non-standard optimal control problem with time inconsistent preferences is studied. In an infinite horizon setting, a time consistent (sophisticated) decision maker chooses the time of switching between two consecutive regimes. The second regime corresponds to the implementation of a new technology, and a cost must be paid at the switching time. Although the problem is formulated for a general discount function, special attention is devoted to models with nonconstant discounting and heterogeneous discounting. The problem is solved by transforming it into a problem in a finite horizon and free terminal time. The corresponding dynamic programming equations are presented, and conditions for the derivation of the switching time by decision makers with different degrees of sophistication are studied. A resource extraction model with technology adoption is solved in detail. Effects of the adoption of different discount functions are illustrated numerically. Full article

(This article belongs to the Special Issue Application of Optimal Control and Game Theory to the Problem of Resource Management)

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14 pages, 336 KiB

Open AccessArticle

Time-Consistency of an Imputation in a Cooperative Hybrid Differential Game

by Ekaterina Gromova, Anastasiia Zaremba and Shimai Su

Mathematics 2021, 9(15), 1830; https://doi.org/10.3390/math9151830 - 03 Aug 2021

Cited by 2 | Viewed by 1350

Abstract

This work is aimed at studying the problem of maintaining the sustainability of a cooperative solution in an n-person hybrid differential game. Specifically, we consider a differential game whose payoff function is discounted with a discounting function that changes its structure with [...] Read more.

This work is aimed at studying the problem of maintaining the sustainability of a cooperative solution in an n-person hybrid differential game. Specifically, we consider a differential game whose payoff function is discounted with a discounting function that changes its structure with time. We solve the problem of time-inconsistency of the cooperative solution using a so-called imputation distribution procedure, which was adjusted for this general class of differential games. The obtained results are illustrated with a specific example of a differential game with random duration and a hybrid cumulative distribution function (CDF). We completely solved the presented example to demonstrate the application of the developed scheme in detail. All results were obtained in analytical form and illustrated by numerical simulations. Full article

(This article belongs to the Special Issue Application of Optimal Control and Game Theory to the Problem of Resource Management)

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29 pages, 482 KiB

Open AccessArticle

A Constrained Markovian Diffusion Model for Controlling the Pollution Accumulation

by Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos and Javier Garrido-Meléndez

Mathematics 2021, 9(13), 1466; https://doi.org/10.3390/math9131466 - 22 Jun 2021

Cited by 4 | Viewed by 1460

Abstract

This work presents a study of a finite-time horizon stochastic control problem with restrictions on both the reward and the cost functions. To this end, it uses standard dynamic programming techniques, and an extension of the classic Lagrange multipliers approach. The coefficients considered [...] Read more.

This work presents a study of a finite-time horizon stochastic control problem with restrictions on both the reward and the cost functions. To this end, it uses standard dynamic programming techniques, and an extension of the classic Lagrange multipliers approach. The coefficients considered here are supposed to be unbounded, and the obtained strategies are of non-stationary closed-loop type. The driving thread of the paper is a sequence of examples on a pollution accumulation model, which is used for the purpose of showing three algorithms for the purpose of replicating the results. There, the reader can find a result on the interchangeability of limits in a Dirichlet problem. Full article

(This article belongs to the Special Issue Application of Optimal Control and Game Theory to the Problem of Resource Management)

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15 pages, 498 KiB

Open AccessArticle

Greenness as a Differentiating Strategy

by Nahid Masoudi

Mathematics 2021, 9(11), 1300; https://doi.org/10.3390/math9111300 - 06 Jun 2021

Viewed by 1668

Abstract

In a vertical differentiation model, we study a market where consumers, depending on their level of environmental consciousness, value the greenness of the product they consume and are distributed according to a Kumaraswamy distribution. Three scenarios are studied: only one firm takes some [...] Read more.

In a vertical differentiation model, we study a market where consumers, depending on their level of environmental consciousness, value the greenness of the product they consume and are distributed according to a Kumaraswamy distribution. Three scenarios are studied: only one firm takes some green measures and firms compete upon prices; only one firm takes some green measures, and this firm acts as the leader of the price competition; and finally, both firms choose their level of greenness and compete upon their location and price. The results suggest that as consumers become more environmentally conscious, the marginal consumer and the greener firm’s location move to the right. In contrast, the less green firm’s response is non-monotonic. In fact, when the two firms choose their location along with their prices, the latter firm chooses to produce a less green product in response to more environmentally conscious consumers. In the extreme case where all consumers are fully environmentally conscious, the latter firm produces a brown product and sells it at a price equal to its marginal cost. In this case, the greener firm’s price and location choices make the consumers indifferent between the two products. These results could explain why despite all the improvements in the consumers’ environmental consciousness, brown (in its general term) products are still widely produced and consumed, even by environmentally conscious consumers. Full article

(This article belongs to the Special Issue Application of Optimal Control and Game Theory to the Problem of Resource Management)

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