Multi-objective Two-stage Stochastic Programming Research Articles

A multi-objective two-stage stochastic programming approach for wind farms and demand side provider decisions using the weighted metric method

The rapid expansion of wind-based renewable energy sources highlights the critical need to enhance reserve capacity within power systems. This research aims to develop an effective model for harnessing the combined potential of flexible demand-side resources, acting as demand-side providers (DSPs), to address the inherent unpredictability of wind power. The DSP framework comprises three demand response aggregators: the capacity market program (CAP), energy storage systems (ESSs), and plug-in electric vehicle parking lots (PEVs PLs). To achieve this, a multi-objective two-stage stochastic programming model is introduced. It not only minimizes operational costs for system operators but also maximizes returns for demand-side resource owners supervised by DSPs in the energy and reserve market environment. The problem is tackled using a mixed-integer linear programming (MILP) approach employing a weighted metric methodology. Within this framework, the Weibull probability distribution function (WPDF) is utilized to model wind power uncertainty, while the truncated Gaussian distribution function (GDF) encapsulates the uncertainty regarding electric vehicle owner behavior. The results demonstrate that employing this method for simultaneous programming of energy and reserve not only reduces system operator costs but also enhances profits for resource owners.

Air traffic management in parallel-point merge systems under wind uncertainties

The point merge system (PMS) was developed by EUROCONTROL in 2006 to enable controllers to implement systematic sequences and replace the traditional vector technique in Terminal Maneuver Area (TMA) even during peak hours. A conventional PMS includes two arc-shaped route segments referred to as sequencing legs and a single merge point to merge arrivals. When necessary, arrivals can be delayed along these legs using path stretching before they are directed to the merge point. After merging, aircraft join the final approach. The parallel-point merge system (P-PMS), however, has a more complex route structure which consists of two oppositely located PMS and a set of common points located between the merge points and final paths of runways. This system increases the capacity of the airspace and provides the advantages of the single PMS. However, especially in simulation studies where P-PMS has been tested, the emphasis on wind sensitivity came to the fore as one of the main issues affecting the use of the system. The use of P-PMS was stated as feasible but more difficult due to the loss of symmetry, especially when the wind is perpendicular to the sequencing legs. Controllers may need help in providing safe and efficient sequences in this type of structure. In this study, a multi-objective two-stage stochastic programming model is developed for P-PMS to obtain robust aircraft sequences, schedules, and runway assignments considering the uncertainties of both wind direction and speed. The model was implemented on the existing layout of Istanbul Airport having a P-PMS serving five parallel runways using the real traffic and wind data. A scenario-based approach was adopted to represent the uncertainties of the model. Also, to meet the demands of the various stakeholders in the air traffic system, the minimization of total fuel consumption, total flight time, and total delays were considered as single and multi-objectives. As a result, it was found that the stochastic approach provides better solutions in terms of objective values and emission outputs compared to deterministic and FCFS approaches under wind uncertainties. The differences between these solutions are quantified through numerical experiments.

Multi-objective two-stage stochastic programming for adaptive interdisciplinary pain management with piecewise linear network transition models

Pain is a major health problem for many people, and pain management is currently innovating because of the opioid crisis in the United States. Existing models optimizing personal adaptive treatment strategies for chronic pain management have only considered one pain outcome. However, most of the pain management centers consider multiple pain outcome measures to identify pain intensity. Consequently, this research uses five pain outcomes. Transition models are represented by piecewise linear networks (PLN). A multi-objective mixed integer linear program (MILP) is developed to optimize treatment strategies for patients based upon on these transition models. A convex quadratic program (QP) is developed to determine weights for multiple levels of multiple pain outcomes that are consistent with surveys submitted by pain management experts. Results show that the MILP that considers multiple pain outcomes yields treatment recommendations with better expected outcomes compared to observed data and to solutions from an optimization model with a single pain outcome objective.

Multi-Objective Two-Stage Stochastic Programming Model for a Proposed Casualty Transportation System in Large-Scale Disasters: A Case Study

Disaster management is a process that includes mitigation, preparedness, response and recovery stages. Operational strategies covering all stages must be developed in order to alleviate the negative effects of the disasters. In this study, we aimed at minimizing the number of casualties that could not be transported to the hospitals after the disaster, the number of additional ambulances required in the response stage, and the total transportation time. Besides, we assumed that a data-driven decision support tool is employed to track casualties and up-to-date hospital capacities, so as to direct the ambulances to the available hospitals. For this purpose, a multi-objective two-stage stochastic programming model was developed. The model was applied to a district in Istanbul city of Turkey, for a major earthquake. Accordingly, the model was developed with a holistic perspective with multiple objectives, periods and locations. The developed multi-objective stochastic programming model was solved using an improved version of the augmented ε-constraint (AUGMECON2) method. Hence, the Pareto optimal solutions set has been obtained and compared with the best solution achieved according to the objective of total transportation time, to see the effect of the ambulance direction decisions based on hospital capacity availability. All of the decisions examined in these comparisons were evaluated in terms of effectiveness and equity. Finally, managerial implication strategies were presented to contribute decision-makers according to the results obtained. Results showed that without implementing a data-driven decision support tool, equity in casualty transportation cannot be achieved among the demand points.

Designing profitable and responsive supply chains under uncertainty

In this paper, we develop a multi-objective two-stage stochastic programming model, which takes into account the selection of warehouse and retailer sites and the decision about production levels, inventory levels, and shipping quantities among the entities of the supply chain network. The first objective function is to maximise the chain’s total profit over multiple periods, and the second objective function is to minimise the total travel times for unsatisfied customers, whose demands must be met by retailers which have been established in other markets, to maximise the chain’s responsiveness. Demands, selling prices and productions times at manufacturing sites are all considered as uncertain parameters. The two objective functions are in conflict with each other, and we use ϵ-constraint method to generate a set of Pareto optimal solutions for the proposed multi-objective problem. We then generalise the case and assume the uncertain parameters are continuously distributed random variables and use a simulation approach called sample average approximation (SAA) scheme to compute near optimal solutions to the stochastic model with potentially infinite number of scenarios. A computational study involving hypothetical networks of different sizes and a real supply chain network are presented to highlight the efficiency of the proposed solution methodology.

Pre-positioning and dynamic operations planning in pre- and post-disaster phases with lateral transhipment under uncertainty and disruption

Humanitarian organizations pre-position relief items in pre-disaster and distribute them to the affected areas in post-disaster. Improper planning of emergency operations causes more deaths in post-disaster. In this paper, the problem of relief item pre-positioning and multi-period distribution planning is addressed considering lateral transhipment among distribution centres to improve the efficiency of the humanitarian relief chain. The proposed model not only improves cost efficiency of the relief chain, but also enhances the equity and fairness. Therefore, a multi-objective two-stage stochastic programming, which involves imprecise parameters, is developed to address the problem. Moreover, TH method is utilized to solve the proposed multi-objective programming and efficient possibilistic programming is adopted to deal with the imprecise input parameters. Applicability of the proposed model and the effectiveness of the solutions are examined through a numerical analysis. Finally, sensitivity analyses are conducted on key input parameters to extract managerial insights.

Accounting for uncertainty in eco-efficient agri-food supply chains: A case study for mushroom production planning

Due to the increasing awareness of climate change, depletion of natural resources, and increasing world population, companies in the agri-food sector need to redesign their existing supply chains and take into account both the economic and environmental impact of their operations. In practice not all the required information is available in advance due to various sources of uncertainty in agri-food supply chains. In this research a multi-objective two-stage stochastic programming model is proposed to analyse and evaluate the economic and environmental impacts to account for uncertainty in agri-food supply chains. A mushroom supply chain in the Netherlands is presented as an illustrative case study. Optimal production planning decisions calculated with a two-stage stochastic programming model are compared with the results of an equivalent deterministic model. The results of the optimizations show that accounting for stochasticity in important model parameters can reduce the difference between expected and realized economic performance by approximately 4% on average. Moreover, this paper demonstrates that including stochastic model parameters can reduce the environmental impact without compromising the current economic performance. Given the assumptions in the setup of the case study and the available information, it is concluded that applying a 2-stage stochastic programming approach for production planning decisions can lead to improved economic and environmental performance in an agri-food supply chain. New findings in real-life case studies are needed to get profound insights and understanding on the impact of uncertainty on production planning decisions in sustainable agri-food supply chains.

Designing Profitable and Responsive Supply Chains under Uncertainty

In this paper, we develop a multi-objective two-stage stochastic programming model, which takes into account the selection of warehouse and retailer sites and the decision about production levels, inventory levels, and shipping quantities among the entities of the supply chain network. The first objective function is to maximize the chain’s total profit over multiple periods, and the second objective function is to minimize the total travel times for unsatisfied customers, whose demands must be met by retailers which are established in other markets, to maximize the chain’s responsiveness. Demands, selling prices and productions times at manufacturing sites are all considered as uncertain parameters. The two objective functions are in conflict with each other, and we use ε-constraint method to generate a set of Pareto optimal solutions for the proposed multi-objective two-stage stochastic programming problem.

Integrated IMO-TSP and AHP Method for Regional Water Allocation under Uncertainty

AbstractThis paper developed an interval multiobjective two-stage stochastic programming (IMO-TSP) model. The model integrated interval two-stage stochastic programming (ITSP), interval linear mult...

An integrated supply chain model for new products with imprecise production and supply under scenario dependent fuzzy random demand

In the present day business scenario, instant changes in market demand, different source of materials and manufacturing technologies force many companies to change their supply chain planning in order to tackle the real-world uncertainty. The purpose of this paper is to develop a multi-objective two-stage stochastic programming supply chain model that incorporates imprecise production rate and supplier capacity under scenario dependent fuzzy random demand associated with new product supply chains. The objectives are to maximise the supply chain profit, achieve desired service level and minimise financial risk. The proposed model allows simultaneous determination of optimum supply chain design, procurement and production quantities across the different plants, and trade-offs between inventory and transportation modes for both inbound and outbound logistics. Analogous to chance constraints, we have used the possibility measure to quantify the demand uncertainties and the model is solved using fuzzy linear programming approach. An illustration is presented to demonstrate the effectiveness of the proposed model. Sensitivity analysis is performed for maximisation of the supply chain profit with respect to different confidence level of service, risk and possibility measure. It is found that when one considers the service level and risk as robustness measure the variability in profit reduces.

Multiobjective Two-Stage Stochastic Programming Problems with Interval Discrete Random Variables

Most of the real-life decision-making problems have more than one conflicting and incommensurable objective functions. In this paper, we present a multiobjective two-stage stochastic linear programming problem considering some parameters of the linear constraints as interval type discrete random variables with known probability distribution. Randomness of the discrete intervals are considered for the model parameters. Further, the concepts of best optimum and worst optimum solution are analyzed in two-stage stochastic programming. To solve the stated problem, first we remove the randomness of the problem and formulate an equivalent deterministic linear programming model with multiobjective interval coefficients. Then the deterministic multiobjective model is solved using weighting method, where we apply the solution procedure of interval linear programming technique. We obtain the upper and lower bound of the objective function as the best and the worst value, respectively. It highlights the possible risk involved in the decision-making tool. A numerical example is presented to demonstrate the proposed solution procedure.

Stochastic Programs with Recourse

Abstract : So far the study of stochastic programs with recourse has been limited to the case (called by G. Dantzig programming under uncertainty) when only the right-hand sides or resources of the problem are random. In this paper the authors extend the theory to the general case when essentially all the parameters involved are random. This generalization immediately raises the problem of attributing a precise meaning to the stochastic constraints. They examine a probability formulation (satisfying the constraints almost surely) and a possibility formulation (satisfying the constraints for all values of the random parameters in the support of their joint distribution) and show them equivalent under a rather weak but curious W-condition. Finally, they prove that without restriction the equivalent deterministic form of a stochastic program with recourse is a convex program for which we obtain some additional properties when some of the parameters of the original problem are constant. The applications of the theoretical results of this paper to certain classes of stochastic programs which have arisen from practical problems will be presented in a separate paper: 'Stochastic Programs with Recourse: Special Forms.' (Author)