Value Of The Stochastic Solution Research Articles

Optimizing Fleet Size in Point-to-Point Shared Demand Responsive Transportation Service: A Network Decomposition Approach

With increasing urbanization and the demand for efficient, flexible transportation solutions, demand-responsive transportation services (DTRS) has emerged as a viable alternative to traditional public transit. However, determining the optimal fleet size to balance the investment and operational revenue remains a significant challenge for service providers. In this article, we address the optimization of fleet size in point-to-point shared demand DRTS, which widely operates within many cities. To capture the uncertain passenger demands in the future when planning the fleet size currently, we model this problem with a framework of two-stage stochastic programming with recourse. Fleet sizing decisions are made in the first stage before the uncertain demands are revealed. After the uncertainty is revealed, the second stage involves making additional decisions to maximize operational revenue. The objective is to optimize the total revenue of the first-stage decisions and the expected revenue of the recourse actions. To solve this practical problem, we resort to the Model Predictive Control method (MPC) and propose a network decomposition approach that first converts the transportation network to a nodal tree structure and then develops a Nodal Tree Recourse with Dependent Arc Capacities (NTRDAC) algorithm to obtain the exact value of the expected recourse functions. In the experiments, NTRDAC is able to produce results within seconds for transportation networks with over 30 nodes. In contrast, a commercial solver is only capable of solving networks with up to five nodes. The stability tests show that NTRDAC remains robust as the problem size varies. Lastly, the value of the stochastic solution (VSS) was evaluated, and the results indicate that it consistently outperforms the expected value solutions. Numerical experiments show that the performance of the NTRDAC algorithm is quite encouraging and fit for large-scale practical problems.

Co-Evolutionary Algorithm for Two-Stage Hybrid Flow Shop Scheduling Problem with Suspension Shifts

Demand fluctuates in actual production. When manufacturers face demand under their maximum capacity, suspension shifts are crucial for cost reduction and on-time delivery. In this case, suspension shifts are needed to minimize idle time and prevent inventory buildup. Thus, it is essential to integrate suspension shifts with scheduling under an uncertain production environment. This paper addresses the two-stage hybrid flow shop scheduling problem (THFSP) with suspension shifts under uncertain processing times, aiming to minimize the weighted sum of earliness and tardiness. We develop a stochastic integer programming model and validate it using the Gurobi solver. Additionally, we propose a dual-space co-evolutionary biased random key genetic algorithm (DCE-BRKGA) with parallel evolution of solutions and scenarios. Considering decision-makers’ risk preferences, we use both average and pessimistic criteria for fitness evaluation, generating two types of solutions and scenario populations. Testing with 28 datasets, we use the value of the stochastic solution (VSS) and the expected value of perfect information (EVPI) to quantify benefits. Compared to the average scenario, the VSS shows that the proposed algorithm achieves additional value gains of 0.9% to 69.9%. Furthermore, the EVPI indicates that after eliminating uncertainty, the algorithm yields potential improvements of 2.4% to 20.3%. These findings indicate that DCE-BRKGA effectively supports varying decision-making risk preferences, providing robust solutions even without known processing time distributions.

Optimization of crude oil operations scheduling by applying a two-stage stochastic programming approach with risk management

This paper focuses on the problem of crude oil operations scheduling carried out in a system composed of a refinery and a marine terminal, considering uncertainty in the arrival date of the ships that supply the crudes. To tackle this problem, we develop a two-stage stochastic mixed-integer nonlinear programming (MINLP) model based on continuous-time representation. Furthermore, we extend the proposed model to include risk management by considering the Conditional Value-at-Risk (CVaR) measure as the objective function, and we analyze the solutions obtained for different risk levels. Finally, to evaluate the solution obtained, we calculate the Expected Value of Perfect Information (EVPI) and the Value of the Stochastic Solution (VSS) to assess whether two-stage stochastic programming model offers any advantage over simpler deterministic approaches.

Optimization under uncertainty of a hybrid waste tire and natural gas feedstock flexible polygeneration system using a decomposition algorithm

Market uncertainties motivate the development of flexible polygeneration systems that are able to adjust operating conditions to favor production of the most profitable product portfolio. However, this operational flexibility comes at the cost of higher capital expenditure. A scenario-based two-stage stochastic nonconvex Mixed-Integer Nonlinear Programming (MINLP) approach lends itself naturally to optimizing these trade-offs. This work studies the optimal design and operation under uncertainty of a hybrid feedstock flexible polygeneration system producing electricity, methanol, dimethyl ether, olefins or liquefied (synthetic) natural gas. A recently developed C++ based software framework (named GOSSIP) is used for modeling the optimization problem as well as its efficient solution using the Nonconvex Generalized Benders Decomposition (NGBD) algorithm. Two different cases are studied: The first uses estimates of the means and variances of the uncertain parameters from historical data, whereas the second assesses the impact of increased uncertain parameter volatility. The value of implementing flexible designs characterized by the value of the stochastic solution (VSS) is in the range of 260–405 M$ for a scale of approximately 893 MW of thermal input. Increased price volatility around the same mean results in higher expected net present value and VSS as operational flexibility allows for asymmetric exploitation of price peaks.

Bi-objective stochastic closed-loop supply chain network design under uncertain quantity and quality of returns

In this paper, we seek to optimize closed loop supply chain (CLSC) network design by determining the location of distribution and collection facilities and the assignment of forward and reverse flows that maximize the system profit and minimize CO2 emissions. To account for the difficulty of predicting and controlling recoverable items in practice, we incorporate into the proposed bi-objective model three types of uncertainties: return quantity, return quality and remanufacturing costs. We use the ε-constraint method to solve the resulting bi-objective scenario-based two-stage stochastic program and generate a set of Pareto-optimal solutions. Our numerical experiments show that both network configuration and performance levels are sensitive to variations of the quality and quantity of returns, in particular when the penalty for not processing returns is high. They also reveal the importance of stochastic modelling since the value of the stochastic solution can be significant in many cases. However, by increasing the size of the problem, the runtime of the ε-constrained stochastic model can be dramatically increased and memory shortage issues may occur. Therefore, we investigate the computational performance of a metaheuristic method based on a non-dominated sorting genetic algorithm (NSGAII) and a linear programming (LP) relaxation. Our method is capable of approximating the Pareto frontier on instances where the ε-constraint method cannot provide any feasible solution within the same time limit.

Police manpower supply planning for regular duties with stochastic demands

ABSTRACT We develop modeling strategies to alleviate the shortage of police manpower in a stochastic environment, based on flexible manpower supply planning using off-duty police officers and mutual support between police stations, along with flexible working hours and shifts. A stochastic independent assignment model (SIAM) and a stochastic mutual support model (SMSM) are constructed for manpower supply planning for regular policing duties with stochastic demands. These models are designed using mathematical programming techniques to minimize total manpower supply hours, subject to practical constraints. The SIAM is solved using CPLEX, but the SMSM is too complicated for this approach, so we develop our own heuristic algorithm to solve it efficiently. Additionally, we use the expected value of perfect information (EVPI) and the value of the stochastic solution (VSS) to evaluate the stochastic models. Numerical test results indicate that both models perform well, but the SMSM produces better manpower plans for police than the SIAM.

A STOCHASTIC OPTIMIZATION MODEL FOR THE IRREGULAR KNAPSACK PROBLEM WITH UNCERTAINTY IN THE PLATE DEFECTS

The present research deals with the two-dimensional knapsack problem by considering the cutting of irregular items from a rectangular plate with defects. While the defects are only known at the time of cutting (in the future), we need first to select which items to produce from cutting the plate. The final items cannot have any defects and the goal is to maximize the profit from cutting the plate and producing the items. We propose a two-stage stochastic optimization model that makes use of a discrete set of scenarios with the realization of the plate defects. The first-stage decisions involve selecting items for cutting and possible production. The second-stage decisions consider the positioning of items in the plate given the scenarios with defects, and then the cancellation and non-production of some selected items, if any. We also extend this model to include a measure of risk, aiming at robust solutions. We perform computational tests on instances adapted from the literature that consider three types of defects, eight scenarios, and four cases for determining each scenario’s probability. The tests evaluate the impact of uncertainties on the problem by calculating the expected value of perfect information and the value of the stochastic solution. The results indicate a percentage reduction in the profit of up to 27.7%, on average, when considering a fully risk-averse decision-maker.

Vehicle Routing with Stochastic Supply of Crowd Vehicles and Time Windows

The growth of e-commerce has increased demand for last-mile deliveries, increasing the level of congestion in the existing transportation infrastructure in urban areas. Crowdsourcing deliveries can provide the additional capacity needed to meet the growing demand in a cost-effective way. We introduce a setting where a crowd-shipping platform sells heterogeneous products of different sizes from a central depot. Items sold vary from groceries to electronics. Some items must be delivered within a time window, whereas others need a customer signature. Furthermore, customer presence is not guaranteed, and some deliveries may need to be returned to the depot. Delivery requests are fulfilled by a fleet of professional drivers and a pool of crowd drivers. We present a crowd-shipping platform that standardizes crowd drivers’ capacities and compensates them to return undelivered packages back to the depot. We formulate a two-stage stochastic model, and we propose a branch and price algorithm to solve the problem exactly and a column generation heuristic to solve larger problems quickly. We further develop an analytical method to calculate upper bounds on the supply of vehicles and an innovative cohesive pricing problem to generate columns for the pool of crowd drivers. Computational experiments are carried out on modified Solomon instances with a pool of 100 crowd vehicles. The branch and price algorithm is able to solve instances of up to 100 customers. We show that the value of the stochastic solution can be as high as 18% when compared with the solution obtained from a deterministic simplification of the model. Significant cost reductions of up to 28% are achieved by implementing crowd drivers with low compensations or higher capacities. Finally, we evaluate what happens when crowd drivers are given the autonomy to select routes based on rational and irrational behavior. There is no cost increase when crowd drivers are rational and select routes that have a higher compensation first. However, when crowd drivers are irrational and select routes randomly, the cost can increase up to 4.2% for some instances.

Artificial lift infrastructure planning of shale gas horizontal wells considering endogenous and exogenous uncertainties

AbstractArtificial lift methods (ALMs) lift the accumulated fluids from horizontal shale‐gas‐producing wells and help sustain well performance. An artificial lift infrastructure plan includes the selection of ALMs and their operating schedule. This paper presents two discrete‐time large‐scale nonconvex mixed‐integer nonlinear programming models to solve the artificial lift infrastructure planning problem. Two equivalent mixed‐integer linear programming models are formulated using the special structure of the nonlinear terms. A set of valid inequalities is defined to tighten the models and shorten solution times to two orders of magnitude, considering well production limitations. We incorporate endogenous uncertainty in ALM‐dependent production rates and exogenous uncertainty in shale gas prices into the models. For a hypothetical case study under only endogenous uncertainties, the value of the stochastic solution is 5%. For the same case study, the exogenous uncertainty in gas prices does not change the optimum solution.

A multi-stage stochastic programming approach to epidemic resource allocation with equity considerations.

Existing compartmental models in epidemiology are limited in terms of optimizing the resource allocation to control an epidemic outbreak under disease growth uncertainty. In this study, we address this core limitation by presenting a multi-stage stochastic programming compartmental model, which integrates the uncertain disease progression and resource allocation to control an infectious disease outbreak. The proposed multi-stage stochastic program involves various disease growth scenarios and optimizes the distribution of treatment centers and resources while minimizing the total expected number of new infections and funerals. We define two new equity metrics, namely infection and capacity equity, and explicitly consider equity for allocating treatment funds and facilities over multiple time stages. We also study the multi-stage value of the stochastic solution (VSS), which demonstrates the superiority of the proposed stochastic programming model over its deterministic counterpart. We apply the proposed formulation to control the Ebola Virus Disease (EVD) in Guinea, Sierra Leone, and Liberia of West Africa to determine the optimal and fair resource-allocation strategies. Our model balances the proportion of infections over all regions, even without including the infection equity or prevalence equity constraints. Model results also show that allocating treatment resources proportional to population is sub-optimal, and enforcing such a resource allocation policy might adversely impact the total number of infections and deaths, and thus resulting in a high cost that we have to pay for the fairness. Our multi-stage stochastic epidemic-logistics model is practical and can be adapted to control other infectious diseases in meta-populations and dynamically evolving situations.

A stochastic approach for the single-machine scheduling problem to minimize total expected cost with client-dependent tardiness costs

In the injection moulding process, a set-up containing positioning moulds, a cleaning raw material hopper and machine settings is required. The set-up is uncertain owing to random factors, such as the crew’s skill levels and unexpected breakdowns. The general version of this problem is known as the single-machine scheduling problem with stochastic sequence-dependent set-up times. In this specialized form of the problem, the objective function is defined so as to minimize the total expected cost, which includes set-up, tardiness and earliness costs. Moreover, the effect of the client-dependent tardiness cost on the optimal machine schedule under uncertainty is explored. As a solution approach, a two-stage stochastic programming method is used. For the solution of large-size problems, the harmony search algorithm is proposed, and its performance is compared with the exact solution approach. Finally, the value of the stochastic solution is assessed by comparing the results with the deterministic approach.

The Impact of Uncertainty and Time Structure on Optimal Flexibility Scheduling in Active Distribution Networks

The authors focus on a model for system operators that uses centralized scheduling of multiple flexibility assets and services to minimize the cost of managing problems with grid congestion, voltages, and losses. The model schedules flexibility assets using stochastic optimization for AC optimal power flow in an active distribution network. The novelty of the contribution lies in its focus on how the dynamic capabilities of the flexibility resources are defined with regard to how uncertainty is resolved in the model. The impact of uncertainty is studied by using well-known quality measures from stochastic programming, such as the value of the stochastic solution. Moreover, the authors introduce a new measure related to the impact of representing uncertainty and flexibility when considering reactive power. By changing the time attributes of flexibility assets, the authors show the impact of uncertainty and time structure on a scheduling problem. The uncertainties considered are price and load levels. The findings reveal that the quality of the scheduling of each flexibility resource depends on using a stochastic model with a rigorous consideration of time and uncertainty.

A Multi-Stage Stochastic Programming Approach to Epidemic Resource Allocation with Equity Considerations

Existing compartmental models in epidemiology are limited in terms of optimizing the resource allocation to control an epidemic outbreak under disease growth uncertainty. In this study, we address this core limitation by presenting a multi-stage stochastic programming compartmental model, which integrates the uncertain disease progression and resource allocation to control an infectious disease outbreak. The proposed multi-stage stochastic program involves various disease growth scenarios and optimizes the distribution of treatment centers and resources while minimizing the total expected number of new infections and funerals. We define two new equity metrics, namely infection and capacity equity, and explicitly consider equity for allocating treatment funds and facilities over multiple time stages. We also study the multi-stage value of the stochastic solution (VSS), which demonstrates the superiority of the proposed stochastic programming model over its deterministic counterpart. We apply the proposed formulation to control the Ebola Virus Disease (EVD) in Guinea, Sierra Leone, and Liberia of West Africa to determine the optimal and fair resource-allocation strategies. Our model balances the proportion of infections over all regions, even without including the infection equity or prevalence equity constraints. Model results also show that allocating treatment resources proportional to population is sub-optimal, and enforcing such a resource allocation policy might adversely impact the total number of infections and deaths, and thus resulting in a high cost that we have to pay for the fairness. Our multi-stage stochastic epidemic-logistics model is practical and can be adapted to control other infectious diseases in meta-populations and dynamically evolving situations.

Closed-loop supply chain network design under demand, return and quality uncertainty

We consider the problem of designing a closed-loop supply chain (CLSC) network in the presence of uncertainty in demand quantities, return rates, and quality of the returned products. We formulate the problem as a two-stage stochastic mixed-integer program (SMIP) that maximizes the total expected profit. The first-stage decisions in our model are facility location and capacity decisions, and the second-stage decisions are the forward/reverse flows on the network and hence the production/recovery quantities defined by the flow amounts. We solve the problem by using the L-shaped method in iterative and branch-and-cut frameworks. To improve the computational efficiency, we consider various cut generation strategies. Besides testing the performance of the considered solution methods, we also use our numerical results to estimate the value of the stochastic solution (VSS), the expected value of perfect information (EVPI), and the benefit of utilizing a CLSC network. Our results indicate that the uncertainty in demand has the highest impact and the uncertainty in return rate has the lowest impact on VSS and EVPI values, and including reverse chain increases the expected profit significantly.

Short-Term Production Optimization Under Water-Cut Uncertainty

Summary Short-term production optimization is an essential activity in the oil/gasfield-development process because it allows for the maximization of field production by finding the optimal operational point. In the fields that use gas lift as an artificial-lift method, the gas-lift optimization is a short-term problem. This paper presents a stochastic approach to include uncertainties from production parameters in gas-lift optimization, called the uncertain-gas-lift-optimization problem (UGLOP). Uncertainties from production variables are originated from the measurement process and the intrinsic stochastic phenomena of the production activity. The production variables usually obtained from production tests play an important role in the optimization process because they are used to update reservoir and well models. To include the uncertainties, the strategy involves representing the well-test data using nonlinear regression [support-vector regression (SVR)] and using the Latin-hypercube-sampling (LHS) method. The optimization gives a stochastic solution for the operational point. In the solved problem, this operational point is composed of the individual wells’ gas-lift-injection rate, choke opening, and well/separator routing. The value of the stochastic solution is computed to evaluate the benefit of solving the stochastic problem over the deterministic. The developed methodology is applied to wells of a Brazilian field considering uncertainty in water-cut (WC) values. As a result, an up-to-4.5% gain in oil production is observed using this approach.

A multi-stage stochastic programming approach for supply chain risk mitigation via product substitution

Trends like globalization, shorter product life-cycles, and cost reduction strategies in the global business environment have exposed many supply chains to various risks. Disruptions are one of the supply chain risks that can interrupt product flow, delay customer deliveries, and reduce supply chain revenues considerably. Prior planning for disruptions could greatly alleviate these consequences. A method to cope with disruptions is to use product substitution in the case of a product shortage. In this research, the supply chain of a livestock-drug distribution company in Iran, facing demand disruptions, has been chosen as a case study. For this purpose, a multi-stage stochastic integer programming model is proposed and solved using a customized progressive hedging algorithm. Moreover, the effect of uncertainty on the supply chain performance is measured using the value of the stochastic solution (VSS) and the expected value of perfect information (EVPI) metrics. Based on the different instances of the problem solved, the VSS metric shows that modeling and solving the proposed stochastic model could enhance the company profit by about 3.27 percent on average. In addition, the EVPI metric demonstrates that planning and investing in proactive demand management could enhance the profit up to 9.42 percent. Finally, analyses indicate that when dealing with increased demand uncertainty levels, the importance of using the proposed method increases as the profitability of the company decreases.

A stochastic approach for integrated production and distribution planning in dairy supply chains

This work addresses production and distribution planning for a real-world dairy supply chain. The planning model accounts for the production and distribution of Cheese, Yogurt, Powdered Milk and UHT milk products across a two-echelon Supply Chain. This task is undermined by the inherent variability of raw materials and finished products demand. The integrated production and distribution planning methodology introduced is based on a two-stage stochastic mixed integer linear programming formulation. In real-world settings the number of scenarios grows substantially; thus, a scenario reduction strategy based on clustering techniques is given. A decomposition and solving strategy is also introduced and applied to a real-world case study. This study showed that the value of the stochastic solution might rise to 21.1% above the deterministic solution. This indicates the importance of considering uncertainty for dairy production and distribution. Besides, different-size instances are tested to study the scalability of the solution approach.

NURSE SCHEDULING AND RESCHEDULING UNDER UNCERTAINTY

Nurse planning decisions play a critical role on hospital budgeting, quality of nursing services and job satisfaction of nurses. Nurse planning in a hospital is composed of four main phases: nurse budgeting, nurse scheduling (rostering), nurse rescheduling (staffing) and nurse assignment. In this study, the integrated problem of nurse scheduling and rescheduling under demand uncertainty is considered. The problem is formulated and solved as a two-stage stochastic integer program. The value of the stochastic solution is estimated using realistic problem instances over a monthly planning horizon generated based on the data provided by a private healthcare provider in Ankara, Turkey. The computational results show that the value of the stochastic solution can be more than 9% in some problem instances, and hence healthcare providers can benefit from using models that capture uncertainty rather than deterministic models.

Contract-based utilization of plug-in electric vehicle batteries for day-ahead optimal operation of a smart micro-grid

Abstract The development of smart micro-grids makes the use of new and green technologies like plug-in electric vehicles more suitable. The plug-in electric vehicles can play a role in storing energy due to their network connection capability. The smart micro-grid can use this stored energy to reduce its operating cost. In this paper, a new mechanism based on the contractual agreements between the owners of plug-in electric vehicles and the smart micro-grid is proposed to provide the system's energy during the operating day. The proposed model enables plug-in electric vehicles parked in official parking to be integrated into the operation of smart micro-grid and earn revenue. The operation optimization is formulated as a two-stage stochastic mixed-integer linear problem and is implemented on energy management of a typical smart micro-grid as a case study. The simulation results confirm the effectiveness of the proposed design for both smart micro-grid and plug-in electric vehicles in the case of expected cost reduction. Also, the performance of the stochastic model for uncertainty handling is shown by the value of the stochastic solution.

Sample average approximation and stability tests applied to energy system design

This paper uses confidence intervals from sample average approximation (SAA) and stability tests to evaluate the quality of the solution of a long-term energy system model with stochastic wind power production. Using poorly designed scenarios can give stochastic model results that depend on the scenario representation rather than the actual underlying uncertainty. Nevertheless, there is little focus on the quality of the solutions of stochastic energy models in the applied literature. Our results demonstrate how too small a sample size can give a poor energy system design and misrepresent the value of the stochastic solution (VSS). We demonstrate how to evaluate the number of scenarios needed to ensure in-sample and out-of-sample stability. We also show how replication and testing of many candidate solutions using SAA iterations can provide a solution with a satisfactory confidence interval, including when the samples contain fewer scenarios than required for stability. An important observation, though, is that if SAA repeatedly solves the model with a sample size that satisfies in-sample and out-of-sample stability, the confidence interval is narrow, and the solutions are of high quality in terms of providing a tight bound for the optimal solution.