A stochastic model for the daily coordination of pumped storage hydro plants and wind power plants

We propose a stochastic model for the daily operation scheduling of a generation system including pumped storage hydro plants and wind power plants, where the uncertainty is represented by the hourly wind power production. In order to assess the value of the stochastic modeling, we discuss two case studies: in the former the scenario tree is built so as to include both low and high wind power production scenarios, in the latter the scenario tree is built on historical wind speed data covering a time span of one and a half year. The Value of the Stochastic Solution, computed by a modified new procedure, shows that in scenarios with low wind power production the stochastic solution allows the producer to obtain a profit which is greater than the one associated to the deterministic solution. In-sample stability of the optimal function values for increasing number of scenarios is reported.     

A stochastic approach to a case study for product recovery network design

Uncertainty is one of the characteristics of product recovery networks. In particular the strategic design of their logistic infrastructure has to take uncertain information into account. In this paper we present a stochastic programming based approach by which a deterministic location model for product recovery network design may be extended to explicitly account for the uncertainties. Such a stochastic model seeks a solution which is appropriately balanced between some alternative scenarios identified by field experts. We apply the stochastic models to a representative real case study on recycling sand from demolition waste in The Netherlands. The interpretation of the results is meant to give more insight into decision-making under uncertainty for reverse logistics.     

Indifference pricing and hedging in a multiple-priors model with trading constraints

This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth, and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein (2002). The price is determined by two optimal stochastic control problems (mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations. By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates. The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.     

Stochastic modelling of reservoir operations

The nature of hydrologic parameters in reservoir management models is uncertain. In mathematical programming models the uncertainties are dealt with either indirectly (sensitivity analysis of a deterministic model) or directly by applying a chance-constrained type of formulation or some of the stochastic programming techniques (LP and DP based models). Various approaches are reviewed in the paper. Moran's theory of storage is an alternative stochastic modelling approach to mathematical programming techniques. The basis of the approach and its application is presented. Reliability programming is a stochastic technique based on the chance-constrained approach, where the reliabilities of the chance constraints are considered as extra decision variables in the model. The problem of random event treatment in the reservoir management model formulation using reliability programming is addressed in this paper.     

Two-stage workforce planning under demand fluctuations and uncertainty

We address a multi-category workforce planning problem for functional areas located at different service centres, each having office-space and recruitment capacity constraints, and facing fluctuating and uncertain workforce demand. A deterministic model is initially developed to deal with workforce fluctuations based on an expected demand profile over the horizon. To hedge against the demand uncertainty, we also propose a two-stage stochastic program, in which the first stage makes personnel recruiting and allocation decisions, while the second stage reassigns workforce demand among all units. A Benders’ decomposition-based algorithm is designed to solve this two-stage stochastic mixed-integer program. Computational results based on some practical numerical experiments are presented to provide insights on applying the deterministic versus the stochastic programming approach, and to demonstrate the efficacy of the proposed algorithm as compared with directly solving the model using its deterministic equivalent.     

A stochastic programming model for scheduling call centers with global Service Level Agreements

We consider the issue of call center scheduling in an environment where arrivals rates are highly variable, aggregate volumes are uncertain, and the call center is subject to a global service level constraint. This paper is motivated by work with a provider of outsourced technical support services where call volumes exhibit significant variability and uncertainty. The outsourcing contract specifies a Service Level Agreement that must be satisfied over an extended period of a week or month. We formulate the problem as a mixed-integer stochastic program. Our model has two distinctive features. Firstly, we combine the server sizing and staff scheduling steps into a single optimization program. Secondly, we explicitly recognize the uncertainty in period-by-period arrival rates. We show that the stochastic formulation, in general, calculates a higher cost optimal schedule than a model which ignores variability, but that the expected cost of this schedule is lower. We conduct extensive experimentation to compare the solutions of the stochastic program with the deterministic programs, based on mean valued arrivals. We find that, in general, the stochastic model provides a significant reduction in the expected cost of operation. The stochastic model also allows the manager to make informed risk management decisions by evaluating the probability that the Service Level Agreement will be achieved.     

A Stochastic Capital Rationing Model

This paper addresses the issue of optimal project selection for capital expenditures assuming uncertain budgetary allocations. A critical review of the historical development of capital budgeting models indicates two major deficiencies: (i) deterministic models ignore the uncertain nature of capital budgeting problems; and (ii) those models which do incorporate the concept of uncertainty have serious computational problems when applied to larger problems. A stochastic capital rationing model (SCRM) is proposed which makes use of recent developments in stochastic programmes with recourse. This model remains computationally tractable despite the explicit incorporation of uncertainty and the application of theoretically sound penalties for constraint violations. Two problems are introduced which illustrate the model's superiority over comparable deterministic formulations. By varying both the probability distributions of the stochastic constraints and the borrowing rates, it was possible to identify the impact these factors have on project selections.     

Planning logistics operations in the oil industry

In this paper we apply stochastic programming modelling and solution techniques to planning problems for a consortium of oil companies. A multiperiod supply, transformation and distribution scheduling problem—the Depot and Refinery Optimization Problem (DROP)—is formulated for strategic or tactical level planning of the consortium's activities. This deterministic model is used as a basis for implementing a stochastic programming formulation with uncertainty in the product demands and spot supply costs (DROPS), whose solution process utilizes the deterministic equivalent linear programming problem. We employ our STOCHGEN general purpose stochastic problem generator to ‘recreate’ the decision (scenario) tree for the unfolding future as this deterministic equivalent. To project random demands for oil products at different spatial locations into the future and to generate random fluctuations in their future prices/costs a stochastic input data simulator is developed and calibrated to historical industry data. The models are written in the modelling language XPRESS-MP and solved by the XPRESS suite of linear programming solvers. From the viewpoint of implementation of large-scale stochastic programming models this study involves decisions in both space and time and careful revision of the original deterministic formulation. The first part of the paper treats the specification, generation and solution of the deterministic DROP model. The stochastic version of the model (DROPS) and its implementation are studied in detail in the second part and a number of related research questions and implications discussed.     

Stochastic programming for optimizing bidding strategies of a Nordic hydropower producer

From the point of view of a price-taking hydropower producer participating in the day-ahead power market, market prices are highly uncertain. The present paper provides a model for determining optimal bidding strategies taking this uncertainty into account. In particular, market price scenarios are generated and a stochastic mixed-integer linear programming model that involves both hydropower production and physical trading aspects is developed. The idea is to explore the effects of including uncertainty explicitly into optimization by comparing the stochastic approach to a deterministic approach. The model is illustrated with data from a Norwegian hydropower producer and the Nordic power market at Nord Pool.     

Strategic facility location: A review

Facility location decisions are a critical element in strategic planning for a wide range of private and public firms. The ramifications of siting facilities are broadly based and long-lasting, impacting numerous operational and logistical decisions. High costs associated with property acquisition and facility construction make facility location or relocation projects long-term investments. To make such undertakings profitable, firms plan for new facilities to remain in place and in operation for an extended time period. Thus, decision makers must select sites that will not simply perform well according to the current system state, but that will continue to be profitable for the facility's lifetime, even as environmental factors change, populations shift, and market trends evolve. Finding robust facility locations is thus a difficult task, demanding that decision makers account for uncertain future events. The complexity of this problem has limited much of the facility location literature to simplified static and deterministic models. Although a few researchers initiated the study of stochastic and dynamic aspects of facility location many years ago, most of the research dedicated to these issues has been published in recent years. In this review, we report on literature which explicitly addresses the strategic nature of facility location problems by considering either stochastic or dynamic problem characteristics. Dynamic formulations focus on the difficult timing issues involved in locating a facility (or facilities) over an extended horizon. Stochastic formulations attempt to capture the uncertainty in problem input parameters such as forecast demand or distance values. The stochastic literature is divided into two classes: that which explicitly considers the probability distribution of uncertain parameters, and that which captures uncertainty through scenario planning. A wide range of model formulations and solution approaches are discussed, with applications ranging across numerous industries.     

Supply chain optimization of petroleum organization under uncertainty in market demands and prices

A multi-period stochastic planning model has been developed and implemented for a supply chain network of a petroleum organization operating in an oil producing country under uncertain market conditions. The proposed supply chain network consists of all activities related to crude oil production, processing and distribution. Uncertainties were introduced in market demands and prices. A deterministic optimization model was first developed and tested. The impact of uncertainty on the supply chain was studied by performing a sensitivity analysis in which ±20% deviations were introduced in market demands and prices of different commodities. A stochastic formulation was then proposed, which is based on the two-stage problem with finite number of realizations. The proposed stochastic programming approach proved to be quite effective in developing resilient production plans in light of high degree of uncertainty in market conditions. The anticipated production plans have a considerably lower expected value of perfect information (EVPI). The main conclusion of this study is that for an oil producing country with oil processing capabilities, the impact of economic uncertainties may be tolerated by an appropriate balance between crude exports and processing capacities.     

Valuation of interest rate ceiling and floor in uncertain financial market

The interest rate ceiling and floor are the popular interest rate derivatives in a financial market. In this paper, the valuation of interest rate ceiling and floor is investigated by using uncertainty theory. Different from the classical stochastic interest rate models, the uncertain interest rate model is used in this paper as the basis of evaluating the interest rate ceiling and floor. Based on the assumption that the short interest rate follows uncertain differential equations, the price formulas of interest rate ceiling and floor are derived.     

Gas storage valuation applying numerically constructed recombining trees

The liberalization of European natural gas markets forces market participants to base their decisions on market prices. For owners and operators of natural gas storage facilities it is therefore necessary to take market prices into account for their decisions. In this framework this paper provides a new approach for the valuation of natural gas storage facilities. Using stochastic dynamic programming on multinomial recombining trees, the optimal storage strategy and value are determined. For this we (i) estimate the deterministic and random impacts on natural gas prices, (ii) simulate gas prices considering the results of the first step, (iii) construct numerically the recombining tree using the simulation results, (iv) determine the optimal storage strategy and value. Besides the determination of the optimal storage value and operation schedule the value quantiles are calculated. Via the quantiles relevant risk measures like value at risk and conditional value at risk are determined.     

A new distributionally robust p-hub median problem with uncertain carbon emissions and its tractable approximation method

The p-hub median problem is to determine the optimal location for p hubs and assign the remaining nodes to hubs so as to minimize the total transportation costs. Under the carbon cap-and-trade policy, we study this problem by addressing the uncertain carbon emissions from the transportation, where the probability distributions of the uncertain carbon emissions are only partially available. A novel distributionally robust optimization model with the ambiguous chance constraint is developed for the uncapacitated single allocation p-hub median problem. The proposed distributionally robust optimization problem is a semi-infinite chance-constrained optimization model, which is computationally intractable for general ambiguity sets. To solve this hard optimization model, we discuss the safe approximation to the ambiguous chance constraint in the following two types of ambiguity sets. The first ambiguity set includes the probability distributions with the bounded perturbations with zero means. In this case, we can turn the ambiguous chance constraint into its computable form based on tractable approximation method. The second ambiguity set is the family of Gaussian perturbations with partial knowledge of expectations and variances. Under this situation, we obtain the deterministic equivalent form of the ambiguous chance constraint. Finally, we validate the proposed optimization model via a case study from Southeast Asia and CAB data set. The numerical experiments indicate that the optimal solutions depend heavily on the distribution information of carbon emissions. In addition, the comparison with the classical robust optimization method shows that the proposed distributionally robust optimization method can avoid over-conservative solutions by incorporating partial probability distribution information. Compared with the stochastic optimization method, the proposed method pays a small price to depict the uncertainty of probability distribution. Compared with the deterministic model, the proposed method generates the new robust optimal solution under uncertain carbon emissions.     

Strategic real options with stochastic volatility in a duopoly model

The investment-timing problem has been considered by many authors under the assumption that the instantaneous volatility of the demand shock is constant. Recently, Ting et al. (2013) [12] carried out an asymptotic approach in a monopoly model by letting the volatility parameter follow a stochastic process. In this paper, we consider a strategic game in which two firms compete for a new market under an uncertain demand, and extend the analysis of Ting et al. to duopoly models under different strategic game structures. In particular, we investigate how the additional uncertainty in the volatility affects the investment thresholds and payoffs of players. Several numerical examples and comparison of the results are provided to confirm our analysis.     

Natural frequency analysis of functionally graded material beams with axially varying stochastic properties

The functionally graded material (FGM) has a potential to replace ordinary ones in engineering reality due to its superior thermal and dynamical characteristics. In this regard, the paper presents an effective approach for uncertain natural frequency analysis of composite beams with axially varying material properties. Rather than simply assuming the material model as a deterministic function, we further extend the FGM property as a random field, which is able to account for spatial variability in laboratory observations and in-field data. Due to the axially varying input uncertainty, natural frequencies of the stochastically FGM (S-FGM) beam become random variables. To this end, the Karhunen–Loève expansion is first introduced to represent the composite material random field as the summation of a finite number of random variables. Then, a generalized eigenvalue function is derived for stochastic natural frequency analysis of the composite beam. Once the mechanistic model is available, the brutal Monte-Carlo simulation (MCS) similar to the design of experiment can be used to estimate statistical characteristics of the uncertain natural frequency response. To alleviate the computational cost of the MCS method, a generalized polynomial chaos expansion model developed based on a rather small number of training samples is used to mimic the true natural frequency function. Case studies have demonstrated the effectiveness of the proposed approach for uncertain natural frequency analysis of functionally graded material beams with axially varying stochastic properties.     

Toward net-zero carbon manufacturing operations: an onsite renewables solution

A growing number of manufacturing firms are striving to achieve eco-friendly operations through onsite wind or solar generation. This paper proposes a zero-carbon power supply model to guide the integration of onsite renewable energy into manufacturing facilities. We intend to address two fundamental questions: (1) Is it cost-effective to deploy onsite wind turbines and solar photovoltaics (PVs) systems to achieve net-zero carbon environmental performance? (2) Is the renewable generation system able to meet the electricity demand despite the power intermittency? To answer these questions, we formulate a stochastic optimization model to minimize the levelized cost of onsite renewable energy. The goal is achieved by optimizing the sizing of wind and solar generating units. The proposed energy solution is tested in ten cities around the world under diverse climatic conditions. While PV is still expensive, we conclude that manufacturers could realize zero-carbon emissions at affordable cost provided the local wind speed is above 5 m/s.     

STOCHASTIC ENERGY DEMAND AND THE STABILIZATION VALUE OF ENERGY STORAGE

The economic value of energy storage to meet peak electricity demand is analyzed with an emphasis on the role of demand uncertainty. The concept of the stabilization value, which measures that part of the benefit of the storage project which is due solely to the stochastic demand components, is defined. The magnitude of the stabilization value, relative to the overall value of energy storage, is evaluated in terms of a simple model that accounts for the relevant characteristics of the electric power utility's production mix. It is found that neglecting the demand uncertainty can seriously bias the benefit assessment of the storage project as well as the determination of the optimal storage capacity.     

Solving non-linear portfolio optimization problems with interval analysis

Estimation errors or uncertainities in expected return and risk measures create difficulties for portfolio optimization. The literature deals with the uncertainty using stochastic, fuzzy or probability programming. This paper proposes a new approach to treating uncertainty. By assuming that the expected return and risk vary within a bounded interval, this paper uses interval analysis to extend the classical mean-variance portfolio optimization problem to the cases with bounded uncertainty. To solve the interval quadratic programming problem, the paper adopts order relations to transform the uncertain programme into a deterministic programme, and includes the investors’ risk preference into the model. Numerical analysis illustrates the advantage of this new approach against conventional methods.     

Stochastic uncapacitated hub location

We study stochastic uncapacitated hub location problems in which uncertainty is associated to demands and transportation costs. We show that the stochastic problems with uncertain demands or dependent transportation costs are equivalent to their associated deterministic expected value problem (EVP), in which random variables are replaced by their expectations. In the case of uncertain independent transportation costs, the corresponding stochastic problem is not equivalent to its EVP and specific solution methods need to be developed. We describe a Monte-Carlo simulation-based algorithm that integrates a sample average approximation scheme with a Benders decomposition algorithm to solve problems having stochastic independent transportation costs. Numerical results on a set of instances with up to 50 nodes are reported.