Two-stage Stochastic Mixed-integer Programming Research Articles

Self-consumption through power-to-heat and storage for enhanced PV integration in decentralised energy systems

Many countries have adopted schemes to promote investments into renewable energy sources resulting, among others, in a high penetration of solar PV energy. The system integration of the increasing amount of variable electricity generation is therefore a highly important task. This paper focuses on a residential quarter with PV systems and explores how heat pumps and thermal and electrical storages can help to integrate the PV generation through self-consumption. However, self-consumption and PV integration are not only affected by technologies but also by pricing mechanisms. This paper therefore analyses the impact of different tariffs on the investment and operation decisions in a residential quarter and its interaction with the external grid. The considered tariffs include a standard fixed per-kilowatt-hour price, a dynamic pricing scheme and a capacity pricing scheme. To account for the interdependent uncertainties of energy supply, demand and electricity prices, we use a module-based framework including a Markov process and a two-stage stochastic mixed-integer program. Analysing a residential quarter in Southern Germany as a case study, we find that the integration of a PV system is economically advantageous for all considered tariffs. The self-consumption rate varies between 58 and 75%. The largest PV system is built when dynamic prices are applied. However, the peak load from the external grid increases by a factor of two under this tariff without any incentive for reduction. In contrast, capacity pricing results in a reduction of the peak load by up to 35%.

Tight Second Stage Formulations in Two-Stage Stochastic Mixed Integer Programs

We study two-stage stochastic mixed integer programs (TSS-MIPs) with integer variables in the second stage. We show that under suitable conditions, the second stage MIPs can be convexified by adding parametric cuts a priori. As special cases, we extend the results of Miller and Wolsey [Math. Program., 98 (2003), pp. 73--88] to TSS-MIPs. Furthermore, we consider second stage programs that are generalizations of the well-known mixing (and continuous mixing) set, or certain piecewise-linear convex objective integer programs. These results allow us to relax the integrality restrictions on the second stage integer variables without effecting the integrality of the optimal solution of the TSS-MIP. We also use four variants of the two-stage stochastic capacitated lot-sizing problems as test problems for computational experiments and present tight second stage formulations for these problems. Our computational results show that adding parametric inequalities that a priori convexify the second stage formulation si...

Value of pumped hydro storage in a hybrid energy generation and allocation system

Transition from fossil fuels to renewable sources is inevitable. In this direction, variation and intermittency of renewables can be integrated into the grid by means of hybrid systems that operate asa combination of alternative resources, energy storage and long distance transmission lines. In this study, we propose a two-stage stochastic mixed-integer programming model for sizing an integrated hybrid energy system, in which intermittent solar generation in demand points is supported by pumped hydro storage (PHES) systems and diesel is used asan expensive back-up source. PHES systems work asa combination of pumped storage and conventional hydropower stations since there is also natural streamflow coming to the upper reservoirs that shows significant seasonal and inter-annual variability and uncertainty. With several case studies from India, we examine the role of high hydropower potential in the Himalaya Mountains to support solar energy generation in the form of pumped hydro or conventional hydro system while meeting the demand at various scales. We show that pumped hydro storage can keep the diesel contribution to meet the demand less than 10%, whereas this number can go up to more than 50% for conventional systems where the streamflow potential is limited compared to the demand. We also examine the role of pumped hydro systems in both isolated and connected systems (through inter-regional transmission lines) and show that the benefit of pumped hydro is more significant in isolated systems and resource-sharing in connected systems can substitute for energy storage. In addition, with the help of the proposed model, we show that the upper reservoir size of a pumped hydro system could be lower than the reservoir size of a conventional hydropower system depending on the demand scale and streamflow availability. This means that, most of the current conventional hydropower stations could be converted to pumped hydropower stations without needing to modify the upper reservoir, leading to a significantly reduced diesel contribution and lower system unit cost.

Optimal Day-Ahead Power Procurement With Renewable Energy and Demand Response

This study proposes the demand-side power procurement problem to optimally reduce consumer's energy cost. The motivation stems from pressing issues on an increase of energy cost in an industrial section. From an energy consumer's perspective, there exists an opportunity to reduce energy cost by adjusting purchase and consumption of energy in response to time-varying electricity price while utilizing renewable energy, which is called demand response. In this case, energy storage can be used to mitigate fluctuation of intermittent renewable supply and volatile electricity price. Although it is anticipated to serve a significant amount of energy consumption from renewable energy and to avoid peak electricity price, variability and uncertainty in power demand, renewable supply, and electricity price make it challenging to determine an optimal power procurement. The main objective of this study is to suggest a decision-making methodology that enables energy consumers to optimally determine power procurement against time varying and stochastic electricity price and renewable supply. Specifically, this study formulates an optimal day-ahead power procurement as a two-stage stochastic mixed-integer program and proposes a solution approach based on Benders decomposition. The proposed methodology can be successfully applied to energy-intensive industries, such as data centers.

Sustainable network design for multi-purpose pellet processing depots under biomass supply uncertainty

This study develops a two-stage stochastic mixed-integer programming model to manage multi-purpose pellet processing depots under feedstock supply uncertainty. The proposed optimization model would help to minimize cost and mitigate emissions from the supply chain network. We consider three alternative Biomass Processing and Densification Depot (BPDD) technologies; namely, conventional pellet processing, high moisture pellet processing, and ammonia fiber expansion. These three technologies pre-process/pre-treat and densify different types of biomass into more highly densified intermediate products for different markets in order to improve movability and overall supply network performance in terms of costs and emissions. A hybrid decomposition algorithm was developed that combines Sample Average Approximation with an enhanced Progressive Hedging (PH) algorithm to solve this challenging NP-hard problem. Some heuristics such as Rolling Horizon (RH) heuristic, variable fixing technique were later applied to further enhance the PH algorithm. Mississippi and Alabama were selected as a testing ground and ArcGIS was employed to visualize and validate the modeling results. The results of the analysis reveal promising insights that could lead to recommendations to help decision makers achieve a more cost-effective environmentally-friendly supply chain network.

A two-stage stochastic mixed-integer programming approach to the competition of biofuel and food production

The multi-attribute biomass and food production (BFP) problem facing farmers and co-operatives is further complicated by uncertainties in crop yield and prices. In this paper, we present a two-stage stochastic mixed-integer programming (MIP) model that maximizes the economic and environmental benefits of food and biofuel production. The uncertain parameters of yield amount and price level are calculated using real data. Economic aspects include revenue obtained from biomass and food crop sales as well as costs related to seeding, production, harvesting, and transportation operations at the farm level. Environmental effects include greenhouse gas (GHG) emissions, carbon sequestration, soil erosion, and nitrogen leakage to water. The first-stage variables define binary decisions for allocating various land types to food and energy crops, while the second-stage variables are operational decisions related to harvesting, budget allocation, and amounts of different yield types. We present a decomposition algorithm, which is enhanced with specialized Benders cuts for solving this stochastic MIP problem. The computational efficiency of the proposed model and approach is demonstrated by applying it to a real case study involving switchgrass and corn production in the state of Kansas. We measure the solution quality and speed of the decomposition method over stochastic and deterministic models. Results indicate the significant benefit of using the stochastic yield-level information in an optimization model. The proposed stochastic MIP model provides important strategies and insights into decision making for biofuel and food production under uncertainty.

A Scalable Bounding Method for Multistage Stochastic Programs

Many dynamic decision problems involving uncertainty can be appropriately modeled as multistage stochastic programs. However, most practical instances are so large and/or complex that it is impossible to solve them on a single computer, especially due to memory limitations. Extending the work of [B. Sandikci, N. Kong, and A. J. Schaefer, Math. Program., 138 (2013), pp. 253--272] on two-stage stochastic mixed-integer programs, this paper considers general multistage stochastic programs and develops a bounding method based on scenario decomposition. This method is broadly applicable, as it does not assume any problem structure including convexity. Moreover, it naturally fits into a distributed computing environment. Computational experiments with large-scale instances (with up to 100 million scenarios, about 1.5 billion decision variables---85% binary---and 800 million constraints) demonstrate that the proposed method scales nicely with problem size and has immense potential to obtain high-quality solutions...

Fuzzy C-Means-Based Scenario Bundling for Stochastic Service Network Design

Stochastic service network designs with uncertain demand represented by a set of scenarios can be modelled as a large-scale two-stage stochastic mixed-integer program (SMIP). The progressive hedging algorithm (PHA) is a decomposition method for solving the resulting SMIP. The computational performance of the PHA can be greatly enhanced by decomposing according to scenario bundles instead of individual scenarios. At the heart of bundle-based decomposition is the method for grouping the scenarios into bundles. In this paper, we present a fuzzy c-means-based scenario bundling method to address this problem. Rather than full membership of a bundle, which is typically the case in existing scenario bundling strategies such as k-means, a scenario has partial membership in each of the bundles and can be assigned to more than one bundle in our method. Since the multiple bundle membership of a scenario induces overlap between the bundles, we empirically investigate whether and how the amount of overlap controlled by a fuzzy exponent would affect the performance of the PHA. Experimental results for a less-than-truckload transportation network optimization problem show that the number of iterations required by the PHA to achieve convergence reduces dramatically with large fuzzy exponents, whereas the computation time increases significantly. Experimental studies were conducted to find out a good fuzzy exponent to strike a trade-off between the solution quality and the computational time.

A stochastic location and allocation model for critical items to response large-scale emergencies: A case of Turkey

This paper aims to decide on the number of facilities and their locations, procurement for pre and post-disaster, and allocation to mitigate the effects of large-scale emergencies. A two-stage stochastic mixed integer programming model is proposed that combines facility location- prepositioning, decisions on pre-stocking levels for emergency supplies, and allocation of located distribution centers (DCs) to affected locations and distribution of those supplies to several demand locations after large-scale emergencies with uncertainty in demand. Also, the use of the model is demonstrated through a case study for prepositioning of supplies in probable large-scale emergencies in the eastern and southeastern Anatolian sides of Turkey. The results provide a framework for relief organizations to determine the location and number of DCs in different settings, by using the proposed model considering the main parameters, as; capacity of facilities, probability of being affected for each demand points, severity of events, maximum distance between a demand point and distribution center.

Closed-loop supply chain network design under uncertain quality status: Case of durable products

This paper proposes a two-stage stochastic mixed-integer programming model for a closed-loop supply chain network design problem in the context of modular structured products in which the reverse network entails several types of recovery options. It accounts for uncertainty in the quality status of the return stream, modeled as binary scenarios for each component in the reverse bill of material corresponding to such products. To deal with the intractable number of scenarios in the proposed model, a scenario reduction scheme is adapted to the problem of interest to preserve the most pertinent scenarios based on a modified Euclidean distance measure. The reduced stochastic large-scale optimization problem is then solved via a L-shaped algorithm enhanced with surrogate constraints and Pareto-optimal cuts. Numerical results indicate that the scenario reduction algorithm provides good quality solutions to the stochastic problem in a reasonable amount of time through applying the enhanced L-shaped method.

Large-scale stochastic power grid islanding operations by line switching and controlled load shedding

Recently, there has been an increasing concern regarding the security and reliability of power systems due to the onerous consequences of cascading failures. Among many emergency control operations, controlled power grid islanding is a last resort yet powerful method to prevent large-scale blackouts. Islanding operations split the power grid into self-sufficient operational subnetworks and avoid cascading failures by isolating the failed elements of the power system into a non-operational island. In this paper, we consider a two-stage stochastic mixed-integer program to seek the optimal islanding operations under severe contingency states. Line switching and controlled load shedding are the main tools for the islanding operations and load shedding is considered as a measurement to gauge system’s inability to respond to disruption. The number of possible extreme contingencies grows exponentially as the size of the grid increases, and this results in a large-scale mixed-integer program, which is a computationally challenging problem to solve. We present an efficient decomposition method to solve this problem for large-scale power systems.

The Ancestral Benders’ cutting plane algorithm with multi-term disjunctions for mixed-integer recourse decisions in stochastic programming

This paper focuses on solving two-stage stochastic mixed integer programs (SMIPs) with general mixed integer decision variables in both stages. We develop a decomposition algorithm in which the first-stage approximation is solved by a branch-and-bound algorithm with its nodes inheriting Benders' cuts that are valid for their ancestor nodes. In addition, we develop two closely related convexification schemes which use multi-term disjunctive cuts to obtain approximations of the second-stage mixed-integer programs. We prove that the proposed methods are finitely convergent. One of the main advantages of our decomposition scheme is that we use a Benders-based branch-and-cut approach in which linear programming approximations are strengthened sequentially. Moreover as in many decomposition schemes, these subproblems can be solved in parallel. We also illustrate these algorithms using several variants of an SMIP example from the literature, as well as a new set of test problems, which we refer to as Stochastic Server Location and Sizing. Finally, we present our computational experience with previously known examples as well as the new collection of SMIP instances. Our experiments reveal that our algorithm is able to produce provably optimal solutions (within an hour of CPU time) even in instances for which a highly reliable commercial MIP solver is unable to provide an optimal solution within an hour of CPU time.

Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs

We present a method for computing lower bounds in the progressive hedging algorithm (PHA) for two-stage and multi-stage stochastic mixed-integer programs. Computing lower bounds in the PHA allows one to assess the quality of the solutions generated by the algorithm contemporaneously. The lower bounds can be computed in any iteration of the algorithm by using dual prices that are calculated during execution of the standard PHA. We report computational results on stochastic unit commitment and stochastic server location problem instances, and explore the relationship between key PHA parameters and the quality of the resulting lower bounds.

Supplier selection in the processed food industry under uncertainty

This paper addresses an integrated framework for deciding about the supplier selection in the processed food industry under uncertainty. The relevance of including tactical production and distribution planning in this procurement decision is assessed. The contribution of this paper is three-fold. Firstly, we propose a new two-stage stochastic mixed-integer programming model for the supplier selection in the process food industry that maximizes profit and minimizes risk of low customer service. Secondly, we reiterate the importance of considering main complexities of food supply chain management such as: perishability of both raw materials and final products; uncertainty at both downstream and upstream parameters; and age dependent demand. Thirdly, we develop a solution method based on a multi-cut Benders decomposition and generalized disjunctive programming. Results indicate that sourcing and branding actions vary significantly between using an integrated and a decoupled approach. The proposed multi-cut Benders decomposition algorithm improved the solutions of the larger instances of this problem when compared with a classical Benders decomposition algorithm and with the solution of the monolithic model.

Design of a hydrogen supply chain with uncertainty

The use of fuels with low environmental impact has been recently highlighted in the media. In this context, the use of hydrogen as a fuel has been considered an alternative with significant potential to integrate a more sustainable energy matrix. However, there is still no appropriate infrastructure available for its commercialization. This study proposes a methodology for designing a hydrogen supply chain while considering the inherent uncertainty associated with the demand for this fuel in the future. To represent the problem and evaluate investment alternatives for the logistic infrastructure, an optimization model is proposed based on two-stage stochastic mixed-integer programming. To obtain solutions from the proposed model, the sample average approximation (SAA) method is used to obtain statistically certified solutions from a reduced number of scenarios. The proposed methodology was applied to the design of Great Britain's liquid hydrogen supply chain using real data. The proposed framework was able to provide solutions with optimality gaps estimated to be below 1% within an acceptable computational time, demonstrating the adequacy of the developed methodology.

A scenario planning approach for propositioning rescue centers for urban waterlog disasters

An urban waterlog disaster can produce severe results, such as residents’ property loss, environmental damages and pollution, and even casualties. This paper presents a system specification for urban waterlog disasters according to the analysis of urban waterlog disaster risks. Then, a two-stage stochastic mixed-integer programming model is formulated. The model minimizes the total logistics cost, and risk-induced penalties. Moreover, a deterministic counterpart of the stochastic model is proposed to study the expected value of perfect information. The multi-attribute utility theory is used to build assessment functions that assess the utility of the rescue system and the degree contributed to disaster relief for each rescue center. Finally, a real example of rescue logistics is examined for the urban waterlog disasters in Pudong District of Shanghai, China. Using the proposed model, two main results can be obtained. First, the expected value of perfect information experiment reveals that an additional ¥45,005 logistics cost and an additional ¥2417 risk-induced penalties can be incurred due to the presence of uncertainty. Second, as the weight of risk-induced penalty increases from 0.1 to 0.9, the logistics cost is increased by 41.21%, which thus contributes to a decrease of risk-induced penalty by 97.44%. Some managerial implications are discussed based on the numerical studies.

Online appointment sequencing and scheduling

We formulate and solve a new stochastic integer programming model for dynamic sequencing and scheduling of appointments to a single stochastic server. We assume that service durations and the number of customers to be served on a particular day are uncertain. Customers are sequenced and scheduled dynamically (online) one at a time as they request appointments. We present a two-stage stochastic mixed integer program that uses a novel set of non-anticipativity constraints to capture the dynamic multi-stage nature of appointment requests as well as the sequencing of customers. We describe several ways to improve the computational efficiency of decomposition methods to solve our model. We also present some theoretical findings based on small problems to help motivate decision rules for larger problems. Our numerical experiments provide insights into optimal sequencing and scheduling decisions and the performance of the solution methods we propose.

A two-stage stochastic mixed-integer programming approach to physical distribution network design

In many industries, distribution activities are realised in a dynamic environment including uncertainties. Besides, adding transportation mode alternatives, inventory-stocking opportunities in wholesalers, unmet demand permission in distribution centres, etc. increase the difficulty of problem modelling and solving for large-scale networks. In this study, the problem of physical distribution network (DN) design with profit maximisation objective function is modelled to tackle with realistic cases. Two-stage stochastic mixed-integer programming method is used to handle the uncertainties and to consider the probable scenarios. The first-stage decisions of the proposed model are related with the selection of facility location in strategic level, and the second-stage decisions are related with the transported and stocked products or unmet demand quantities. Here, a multi-product, two-echelon, multi-mode and multi-period network model is applied to a hypothetically created problem, inspired from the physical DN of home appliance companies. Various scenarios including stochastic demand and price data with different realisation probabilities are used in the model. The motivation of this study is the lack of reaching a global optimum result using transportation modes as stochastic parameters, considering their own lead times and capacities. Finally, various results are obtained for different cases and analysed in detail.

A two-stage stochastic mixed-integer program modelling and hybrid solution approach to portfolio selection problems

In this paper, we investigate a multi-period portfolio selection problem with a comprehensive set of real-world trading constraints as well as market random uncertainty in terms of asset prices. We formulate the problem into a two-stage stochastic mixed-integer program (SMIP) with recourse. The set of constraints is modelled as mixed-integer program, while a set of decision variables to rebalance the portfolio in multiple periods is explicitly introduced as the recourse variables in the second stage of stochastic program. Although the combination of stochastic program and mixed-integer program leads to computational challenges in finding solutions to the problem, the proposed SMIP model provides an insightful and flexible description of the problem. The model also enables the investors to make decisions subject to real-world trading constraints and market uncertainty.To deal with the computational difficulty of the proposed model, a simplification and hybrid solution method is applied in the paper. The simplification method aims to eliminate the difficult constraints in the model, resulting into easier sub-problems compared to the original one. The hybrid method is developed to integrate local search with Branch-and-Bound (B&B) to solve the problem heuristically. We present computational results of the hybrid approach to analyse the performance of the proposed method. The results illustrate that the hybrid method can generate good solutions in a reasonable amount of computational time. We also compare the obtained portfolio values against an index value to illustrate the performance and strengths of the proposed SMIP model. Implications of the model and future work are also discussed.

Optimal booking and scheduling in outpatient procedure centers

Patient appointment booking, sequencing, and scheduling decisions are challenging for outpatient procedure centers due to uncertainty in procedure times and patient attendance. We extend a previously developed appointment scheduling model to formulate a model based on a two-stage stochastic mixed integer program for optimizing booking and appointment times in the presence of uncertainty. The objective is to maximize expected profit. Analytical insights are reported for special cases and experimental results show that they provide useful rules of thumb for more general problems. Three solution methods are described which take advantage of the underlying structure of the stochastic program, and a series of experiments are performed to determine the best method. A case study based on an endoscopy suite at a large medical center is used to draw a number of useful managerial insights for procedure center managers.