Two-stage Stochastic Programming Research Articles

Consumer service level-oriented resilience optimisation for e-commerce supply chain considering hybrid strategies with blockchain adoption

ABSTRACT The booming development of customised e-commerce makes e-commerce supply chains experience a severe test of consumer service level (CSL). An efficient e-commerce supply chain resilience optimisation method addressing economic performance and service performance is proposed to improve customised services under disruption risks. The proposed method includes a hybrid strategy considering both a resistance strategy with blockchain (BCT) adoption and time-dependent recovery strategies simultaneously. A two-stage multi-period multi-product stochastic programming with BCT adoption is then proposed, which (1) implements BCT as the resistance strategy in the pre-disruption stage; (2) collaborates four recovery strategies in the post-disruption stage; (3) supports the e-tailer in making decisions and optimises his profit and order fulfillment time while taking product priorities into account. Using the actual data of Chinese e-commerce during the public health emergency in 2020, it is demonstrated that (1) the applicability of the model with BCT adoption in both supply chain resilience and CSL improvement; (2) the performance of the hybrid strategy in long-term disruption management; (3) the efficiency and robustness of the proposed solution approach for multi-objective high-dimensional stochastic programming problems. E-tailers can optimise decision-making under long-term disruptions by the proposed methodology in practices.

A disaggregated integer L-shaped method for stochastic vehicle routing problems with monotonic recourse

This paper proposes a new integer L-shaped method for solving two-stage stochastic integer programs whose first-stage solutions can decompose into disjoint components, each one having a monotonic recourse function. In a minimization problem, the monotonicity property stipulates that the recourse cost of a component must always be higher or equal to that of any of its subcomponents. The method exploits new types of optimality cuts and lower bounding functionals that are valid under this property. The stochastic vehicle routing problem is particularly well suited to be solved by this approach, as its solutions can be decomposed into a set of routes. We consider the variant with stochastic demands in which the recourse policy consists of performing a return trip to the depot whenever a vehicle does not have sufficient capacity to accommodate a newly realized customer demand. This work shows that this policy can lead to a non-monotonic recourse function, but that the monotonicity holds when the customer demands are modeled by several commonly used families of probability distributions. We also present new problem-specific lower bounds on the recourse that strengthen the lower bounding functionals and significantly speed up the solving process. Computational experiments on instances from the literature show that the new approach achieves state-of-the-art results.

Optimising microgrid energy management: Leveraging flexible storage systems and full integration of renewable energy sources

The significance of microgrid systems has grown considerably. This research proposes an innovative approach to manage uncertainty in microgrids by employing energy storage systems as the exclusive flexible resource. To address this challenge, a mathematical problem is formulated as a two-stage stochastic programming model, considering two uncertainties in the microgrid: wind and photovoltaic production. The microgrid system encompasses multiple components, including a diesel generator, a microturbine, wind and photovoltaic power generation, an energy storage system, and the microgrid’s demand. Notably, the microgrid exhibits two distinctive features: (i) the complete integration of wind and photovoltaic production, and (ii) the utilisation of an energy storage system as the sole flexible resource. The objective is to minimise the expected cost of the microgrid system while determining the optimal capacity of the energy storage system to meet the energy balance constraint. This constraint takes into account the varying scenarios of wind and photovoltaic production. The decisions are taking for a duration of 8760 h, a long-term evaluation. A case study is presented for actual data from Greece and the results show high volatility of the renewable energy sources implies higher energy storage system capacity as a sole flexible source for avoiding renewable curtailment.

Mixed-model sequencing with reinsertion of failed vehicles: a case study for automobile industry

In the automotive industry, some vehicles, failed vehicles, cannot be produced according to the planned schedule due to some reasons such as material shortage, paint failure, etc. These vehicles are pulled out of the sequence, potentially resulting in an increased work overload. On the other hand, the reinsertion of failed vehicles is executed dynamically as suitable positions occur. In case such positions do not occur enough, either the vehicles waiting for reinsertion accumulate or reinsertions are made to worse positions by sacrificing production efficiency. This study proposes a bi-objective two-stage stochastic program and formulation improvements for a mixed-model sequencing problem with stochastic product failures and integrated reinsertion process. Moreover, an evolutionary optimisation algorithm, a two-stage local search algorithm, and a hybrid approach are developed. Numerical experiments over a case study show that while the hybrid algorithm better explores the Pareto front representation, the local search algorithm provides more reliable solutions regarding work overload objective. Finally, the results of the dynamic reinsertion simulations show that we can decrease the work overload by 20% while significantly decreasing the waiting time of the failed vehicles by considering vehicle failures and integrating the reinsertion process into the mixed-model sequencing problem.

Decomposition algorithms for the robust unidirectional quay crane scheduling problems

Our study focuses on a two-stage robust optimization model for the unidirectional quay crane scheduling problem with uncertain handling times at container terminals. We first implement two classical algorithms adopted in Li and Zhang (2021), the Benders decomposition algorithm and the column-and-constraint generation algorithm, to solve the robust model. Based on analytical and numerical comparisons of them, we design an exact hybrid algorithm to leverage capabilities of both algorithms by alternatively adding cuts of both kinds. Extensive experiments validate the effectiveness of this mechanism. Numerical experiments also reveal that the benefit and the cost of robustness fade away as the uncertainty budget increases and highlight the advantage of our robust approach to deal with uncertainty over a two-stage stochastic program under extreme situations.

Integrated stochastic optimisation of stope design and long-term production scheduling at an operating underground copper mine

ABSTRACT Conventional underground long-term mine planning is based on a deterministic stepwise framework, which is unable to effectively manage the synergies between the mine planning components or to manage the orebody risk in production schedules and forecasts. The present study enhances prior integrated optimisation approaches to a variation of the sublevel longhole open stoping mining method using backfilling, through a new two-stage stochastic integer programming formulation. A comparison to a sequential stochastic approach shows different stope layouts and extraction sequences for a copper mine with secondary elements being gold and uranium. The integrated approach shows lower horizontal development costs and 6% higher net present value.

Joint stochastic optimisation of stope layout, production scheduling and access network

The three main optimisation components of sublevel stoping methods are stope layout, production schedule (or stope sequencing) and access networks. The joint optimisation of these components could further add value to an underground mining project. This potential has not been considered in the literature due to computational difficulties, and the problem was solved sequentially. This paper proposes a new joint optimisation model to integrate these components. In addition, the proposed optimisation model incorporates stochastic simulations to capture uncertainty and variability associated with the grades of the related mineral deposits mined. The optimisation model is based on a two-stage stochastic integer programming (SIP) formulation that maximises the project's net present value (NPV) and minimises the planned dilution. Applying the proposed method at a small copper deposit shows that the SIP outperforms the results obtained from mixed integer programming. For a seven-year mine life, the SIP model generated ∼20% more NPV, demonstrating the importance of developing a joint optimisation formulation and accounting for grade uncertainty and variability.

Risk-averse two-stage stochastic programming for assembly line reconfiguration with dynamic lot sizes

In this paper, a comprehensive optimization problem is developed for a composite of an assembly line reconfiguration problem with multiple lines and a capacitated lot-sizing problem. Multiple products are considered, whose demand is uncertain and is dynamically forecasted. The production planner is assumed to be risk-averse, and decisions are made contingent upon the risk preference. To model the problem, a stochastic program with two stages is utilized. A solution approach is devised using a divide-and-conquer algorithm, which incorporates a set of valid inequalities. The effectiveness and efficiency of the proposed solution approach are assessed through a series of computational tests. Finally, a case study focusing on an engine production process is presented, leading to the derivation of several valuable insights.

A multi-period vaccines supply chain network design with capacity expansion and different replenishment cycles under uncertain demand

The World Health Organization (WHO) proposes guidelines for vaccine distribution networks that incorporate economic order quantity and risk pooling principles, specifically through implementing different replenishment cycles between different levels of the distribution network. The substantial impact of different replenishment cycles on inventory levels is demonstrated in the existing literature. However, current optimization network design research has paid limited attention to this aspect. This article aims to address this research gap by investigating a two-stage stochastic programming for a multi-period perishable vaccine network design problem, taking into account the different replenishment cycles characteristic and capacity expansion under uncertain demand conditions. Given the inherent difficulty of solving the proposed problem, a matheuristic approach based on Variable Neighborhood Search (VNS) is developed. To illustrate the practical application and analyze the impact of uncertain demand, we apply the proposed model and its solution method using the Indonesia dataset in the COVID-19 situation. The computational results indicate that lower replenishment cycles leads to higher inventory levels, total costs, and capacity requirements for the top-level distribution center. Optimum replenishment cycles and further managerial implications and future potential research are also presented.

Resilience-oriented planning for active distribution systems: A probabilistic approach considering regional weather profiles

Recent experiences with increasing climatic variability highlighted the importance of enhanced power distribution system resilience. Conventional methods of power distribution system planning usually focus on persistentinvestments associated with predicted faulty events, aiming to enhance the overall system reliability. Improving distribution system resilience in response to extreme weather events involves reduction of impact of the events having varying probability of occurrence. The probability of an extreme event varies significantly by location due to a combination of geographical, climatological, and meteorological factors. Thus, instead of requiring persistent investments, the solutions for resilience-oriented system upgrades should be guided by the potential effects and the probability of occurrence of extreme weather events in a specific area. To achieve this objective, the present paper proposes a two-stage stochastic mixed-integer linear programming (MILP) model based on regional weather profiles for the long-term resilience planning of distribution systems against extreme weather events. In the planning stage, investment decisions are made regarding overhead distribution line hardening and the distributed generation (DG) installation. During restoration process in the operational stage, dynamic microgrid formation is carried out for advanced distribution grid operations to minimize the cost corresponding to loss of load. The inclusion of regional weather profiles into planning objectives provide added flexibility to the network operators for analysing the trade-off between investment cost and load loss cost in resilient distribution system planning strategies.

Resource allocation and route planning under the collection price uncertainty for the biomass supply chain

The escalating severity of global warming has drawn worldwide attention to ecological problems. Nevertheless, with the introduction of environmental policies, biomass resources have been effectively developed as a renewable energy source. This paper investigates an advanced biomass supply chain design wherein biomass resources are initially converted into bio-oil through widely distributed fast pyrolysis facilities, and are subsequently transported to a centralised biorefinery for further refining into biofuels. This novel biomass supply chain addressed three key issues: (1) The number of fast pyrolysis facilities, (2) The allocation of resources, and (3) The routes of resources transport. In respect of these problems, a two-stage stochastic mixed integer programming model is established to minimise the total cost of the biomass supply chain considering the uncertainty collection price of fast pyrolysis facilities. A hybrid simulated annealing algorithm which incorporates the sample average approximation method is proposed to solve the stochastic model and is effectiveness for large-scale examples. Finally, a sensitivity analysis is performed using the proposed algorithm and the results show that the proposed stochastic model outperforms the deterministic model under uncertain collection price. The model allows optimising the biomass supply chain economic performances and minimise financial risk on investment by determining the fast pyrolysis facility locations, reasonable resource allocation and optimal transport routes under uncertain collection price.

Two-stage stochastic programming for multi-objective optimization of sustainable utility systems integrating with combined heat and power units

Utility systems provide thermal energy and power for industrial processes while emitting significant amounts of carbon dioxide. This study investigates the integration of Combined Heat and Power (CHP) units and renewable energy sources into utility systems in the context of diverse energy demands and low-carbon operations. A two-stage stochastic programming model (TSSP) is proposed for sustainable utility systems. The main objective of the model is to minimize the annual total cost and the other two objectives are to maximize renewable energy penetration rate and minimize grid net interaction level. The model tackles renewable energy sources’ uncertainty and evaluates environmental metrics by computing carbon emission costs. In TSSP, the first stage involves new equipment capacity planning. The second stage involves considering uncertainties related to wind speed and solar radiation. Monte Carlo simulations and probability density functions were used to generate a variety of scenarios for the optimization of operational scheduling decisions. Finally, a case study was conducted to validate the effectiveness of the proposed model. Compared to traditional deterministic optimization in utility systems, the two-stage stochastic programming of sustainable utility systems can reduce carbon emissions by 5.6 percent annually and overall costs by 8,907,733 CNY. It provides decision-makers with a range of reference options by balancing multiple objectives, including economic factors, renewable energy penetration rate, and grid interaction level.

Problem-based scenario generation by decomposing output distributions

Scenario generation is required for most applications of stochastic programming to evaluate the expected effect of decisions made under uncertainty. We propose a novel and effective problem-based scenario generation method for two-stage stochastic programming that is agnostic to the specific stochastic program and kind of distribution. Our contribution lies in studying how an output distribution may change across decisions and exploit this for scenario generation. From a collection of output distributions, we find a few components that largely compose these, and such components are used directly for scenario generation. Computationally, the procedure relies on evaluating the recourse function over a large discrete distribution across a set of candidate decisions, while the scenario set itself is found using standard and efficient linear algebra algorithms that scale well. The method’s effectiveness is demonstrated on four case study problems from typical applications of stochastic programming to show it is more effective than its distribution-based alternatives. Due to its generality, the method is especially well suited to address scenario generation for distributions that are particularly challenging.

Optimizing electricity peak shaving through stochastic programming and probabilistic time series forecasting

This paper proposes a predict-then-optimize framework to optimally schedule the charging and discharging activities of battery energy storage systems (BESS). BESS are used to eliminate peak electricity consumption loads, which pose a significant risk of microgrid system failure and increase electricity costs to end users. A major challenge in BESS lies in determining the battery usage schedule, which must be determined before actual energy consumption materializes. To address this challenge, we propose a two-phase approach. In the first phase, we develop a deep-learning-based probabilistic time-series forecasting model to predict future electricity consumption. In the second phase, the output of the prediction model is used to generate future load scenarios, which are incorporated into a two-stage stochastic programming model that determines the optimal battery usage schedule. Extensive computational experiments on real-world datasets demonstrate the effectiveness of the proposed framework in shaving peak loads and minimizing energy costs. Specifically, our proposed framework reduces daily energy peaks by up to 26%, with the potential for greater improvement as the forecast of future energy loads improves. To the best of our knowledge, this work is the first to investigate the integration of probabilistic forecasting and stochastic optimization to enhance the effectiveness of BESS in managing peak energy loads, leading to more energy-efficient buildings.

Application of Stochastic Programming in Agricultural and Newsvendor Problems and It's Application in Real Life

The study of making the best decision under risk management in a variety of areas of our lives is known as Stochastic Programming. We will go through two-stage Stochastic Linear Programming approaches for a variety of real-world choice issues, as well as how to solve them. We will achieve this by constructing stochastic linear programming models based on real-world situations like the well-known Farmer's situation and News Vendors problems. The influence of pricing, Stochastic Integer Linear Programming problem, second stage Stochastic Integer Linear Programming problem, first stage Stochastic Binary Linear Programming problem, risk aversion problem, and continuous function for random variables based on two-stage SLP with the aid of Farmer's problem will all be examined. We will address the Newsvendor’s problem with Deterministic Equivalent Stochastic Linear Programming, an extension of Deterministic Stochastic Linear Programming for risk aversion with a high number of decision variables and restrictions, utilizing the two-stage Stochastic Linear Programming approach once more. Hand calculation is a challenging way to acquire the solution to the problems. As a result, we will use the programming language AMPL to design computer solutions for tackling both farmer and newsvendor difficulties. We will also utilize MATLAB to create graphs for the farmer's problem's continuous function. Dhaka Univ. J. Sci. 72(1): 30-45, 2024 (January)

Two-stage stochastic program for environmental resettlement decision-making

Large and irregular movements of forcibly displaced people are an escalating global issue today. As a result, forced displacement has become an international problem. It requires planning for the relocation and integration of displaced people. However, forced displacement is not a one-time event. On the contrary, it is an ongoing, long-term process with dynamic parameters. We are interested in the long-term displacement problem, especially for climate-driven cases in which people will be forced to leave uninhabitable regions to escape slow-onset climate change impacts. Creating an optimization model that could cover all the requirements and considerations of such a problem would encounter many challenges. Nonetheless, planning in advance for the upcoming climate-change-driven displacement crisis is critical. Unfortunately, this topic has not been explored sufficiently, especially in the operations research literature. To fill this gap, we focus on developing a mathematical model for the long-term relocation planning of people displaced due to climate change. In this model, uncertainty in demand is represented with various demand scenarios, demand and capacity are managed dynamically, and integration outcomes and related costs are optimized. Ultimately, the purpose of this paper is to provide decision-makers with an initial resource for preparing for upcoming movements under uncertainty. To this end, we propose a two-stage stochastic model to determine optimal flow and location decisions. We run this model under three different demand scenarios with and without considering societal impacts measured by diversity indicators. The results show how profoundly the uncertainty in demand and differences in our focus change the future steps that should be taken.

Optimal sizing of hybrid renewable energy systems using quasi-optimal control

This paper addresses the critical challenge of sizing hybrid renewable energy systems over their lifespan while accounting for uncertainties in energy sources and load demands. By leveraging stochastic programming, we introduce a novel modeling technique that ensures robust optimization without compromising numerical tractability. Unlike conventional methods, which determine system parameters for the entire project duration upfront, our approach enables annual adjustments to the renewable components’ sizing. This dynamic strategy alleviates early-stage under-utilization, aligning system capacity with evolving demand patterns. Central to our approach is the reformulation of the two-stage stochastic program as a quasi-optimal control. This reduces the number of optimization variables significantly, simplifying the solution process. Moreover, thousands of constraints are replaced with a system of differential equations, enhancing computational efficiency. By minimizing capital costs and dynamic operating expenses, we achieve an optimal system size. Through a real-world application in a rural area of South Africa, we demonstrate the effectiveness of our approach. The results showed a remarkable improvement in problem-solving efficiency, reduced under-utilization, and heightened adaptability to unpredictable weather conditions. This innovative framework represents a substantial leap forward in hybrid renewable energy systems design, offering a more agile and efficient solution for sustainable energy generation.

A stochastic programming model for designing bus bridging services under metro disruptions

ABSTRACT With the growing reliance on urban metro networks, any accidental disruption can lead to rapid degradation and significant economic losses. Bus bridging services are common and efficient ways to minimize such adverse impacts. In this study, we investigate the problem of designing bus bridging services in response to unexpected metro disruptions, and propose a routing strategy with multiple bridging routes. In particular, to respond to uncertain factors such as passenger arrivals and bus travel times in the disruption environment, we develop a two-stage stochastic programming model for the collaborative optimization of bus bridging routes, schedules, and passenger assignments. To solve the computational challenges arising with the proposed model, a tailored tabu search algorithm is developed. Finally, several sets of numerical experiments are conducted and experimental results reveal that our proposed routing strategy can effectively improve the service level for the affected passengers during metro disruptions.

Stochastic Modelling Frameworks for Dragon Fruit Supply Chains in Vietnam under Uncertain Factors

Managing uncertainties and risks is always a difficult but fascinating task in fresh fruit supply chains, especially when dealing with the strategy for the production and conveyance of fresh fruit in Vietnam. Following the COVID-19 outbreak, the confluence of economic recession and persistent adverse weather conditions has exacerbated challenges faced by dragon fruit cultivators. This research investigates a two-stage stochastic programming (TSSP) approach which is developed and served as a valuable tool for analyzing uncertainties, optimizing operations, and managing risks in the fresh fruit industry, ultimately contributing to the sustainability and resilience of supply chains in the agricultural sector. A prototype is provided to illustrate the complex and dynamic nature of dragon fruit cultivation and consumption in Vietnam. Data on the selling prices of dragon fruit were collected from several sources between 2013 and 2022 in Binh Thuan Province, Vietnam. The results were obtained from the model by using three different approaches in order of their versatility and efficacy: (1) Scenario tree generation; (2) Sample average approximation; (3) Chance-constrained programming.

Hydrogen refueling station location optimization under uncertainty

Hydrogen refueling stations (HRS) are essential for the growth of the hydrogen energy and fuel cell vehicle industry, making the strategic decision of their location and scale critical. This study advances the HRS location optimization problem by integrating network planning and hydrogen scheduling optimization, a novel approach that addresses both where and how big HRSs should be, alongside managing supply relations and hydrogen transportation scheduling. Uniquely, it incorporates uncertainty factors in hydrogen demand, using a two-stage multi-period stochastic programming model designed to minimize both construction and operational costs under variable demand scenarios, while also ensuring customer satisfaction. Our main contributions include the development of this comprehensive model and the introduction of an efficient metaheuristic algorithm to solve it, significantly enhancing computational efficiency. The effectiveness of our approach is demonstrated through numerical experiments, and a sensitivity analysis provides crucial insights, underlining the practical implications for HRS infrastructure development. The findings not only offer a methodological advancement in HRS location selection but also deliver tangible results that can guide industry stakeholders in strategic decision-making, marking a significant step forward in the sustainable development of hydrogen energy infrastructure.