Expected Value Problem Research Articles

A stochastic optimization approach for the supply vessel planning problem under uncertain demand

This paper presents a two-stage stochastic programming with recourse methodology to solve the Supply Vessel Planning Problem with Stochastic Demands (SVPPSD), a problem arising in offshore logistics and which generalizes the Periodic Vehicle Routing Problem with Stochastic Demands and Time Windows. In the SVPPSD, a fleet of vessels is used to deliver a regular supply of commodities to a set of offshore installations to ensure continuous production, with each installation requiring one or more visits per week and having stochastic demands. Both the onshore depot where the product to be distributed is kept and the offshore installations have time windows, and voyages are allowed to span more than one day. A solution to the SVPPSD consists in the identification of an optimal fleet of vessels and the corresponding weekly schedule. As a solution methodology, we embed a discrete-event simulation engine within a genetic search procedure to approximate the cost of recourse and arrive at the minimized expected cost solution. We make comparisons with two alternative approaches: an expected value problem with upscaled demand, and a chance-constrained algorithm. While alternative methodologies yield robust schedules, robustness is achieved mainly through an increase in fleet size. In contrast, a two-stage stochastic programming with recourse algorithm, by accounting for the cost of recourse in the search phase, and exploring a wider solution space, allows arriving at robust schedules with a smaller fleet size, thereby yielding significant cost savings. For the tested problem instances, the proposed algorithm leads to savings of approximately 10 to 15 million USD per year.

Accelerating Benders decomposition for stochastic incomplete multimodal hub location problem in many-to-many transportation and distribution systems

To customize the hub location model for application in many-to-many transportation and distribution systems, this study introduces a stochastic incomplete multimodal hub location problem with multiple assignments and delivery-time restrictions. This problem explicitly considers the mode-specific hub and link, incomplete inter-hub connectivity, multiple-assignment pattern of demand nodes to hubs and two types of uncertainties in an expected cost-minimization context with delivery-time restrictions. Using a filtering technique, this study first presents a sophisticated path-based formulation for the problem with uncertain demand embedded in a two-stage stochastic programming framework. More importantly, the stochastic demand model is proven to be equivalent to the corresponding deterministic expected value problem (EVP), which can be solved to optimality using Gurobi. By considering the uncertainty in transportation cost, this study further proposes an associated two-stage stochastic program, in which the EVP equivalence does not hold. To solve the stochastic transportation cost version efficiently, the study implements an improved Benders decomposition algorithm by adopting a sample average approximation approach and a dualization strategy. To accelerate the convergence of the proposed Benders decomposition algorithm, this study also presents a multi-cut reformulation and a cut-loop stabilization strategy for Benders acceleration. Numerical experiments based on the well-studied Turkish network and AP dataset corroborate the advantages of the proposed models and the effectiveness of the developed approaches. Some key managerial insights are summarized to effectively guide incomplete, multimodal hub network designs against uncertainty in many-to-many transportation and distribution practices.

Nonparametric Compositional Stochastic Optimization for Risk-Sensitive Kernel Learning

In this work, we address optimization problems where the objective function is a nonlinear function of an expected value, i.e., compositional stochastic programs. We consider the case where the decision variable is not vector-valued but instead belongs to a Reproducing Kernel Hilbert Space (RKHS), motivated by risk-aware formulations of supervised learning. We develop the first memory-efficient stochastic algorithm for this setting, which we call Compositional Online Learning with Kernels (COLK). COLK, at its core a two time-scale stochastic approximation method, addresses the facts that (i) compositions of expected value problems cannot be addressed by stochastic gradient method due to the presence of an inner expectation; and (ii) the RKHS-induced parameterization has complexity which is proportional to the iteration index which is mitigated through greedily constructed subspace projections. We provide, for the first time, a non-asymptotic tradeoff between the complexity of a function parameterization and its required convergence accuracy for both strongly convex and non-convex objectives under constant step-sizes. Experiments with risk-sensitive supervised learning demonstrate that COLK consistently converges and performs reliably even when data is full of outliers, and thus marks a step towards overfitting. Specifically, we observe a favorable tradeoff between model complexity, consistent convergence, and statistical accuracy for data associated with heavy-tailed distributions.

Stochastic programming approach for unidirectional quay crane scheduling problem with uncertainty

Quay crane scheduling is a key aspect of container terminal operation, which can be regarded as a decision-making process with uncertainty. Each task involves stochastic loading and unloading operation times owing to the existence of uncertainty. In this study, we investigate the unidirectional quay crane scheduling problem for a stochastic processing time, which requires that all the quay cranes move in the same direction either from bow to stern, or vice versa, throughout the planning horizon. The problem is formulated as a two-stage stochastic mixed-integer programming model, where the binary first-stage decision variables correspond to the assignment of tasks to quay cranes, and the mixed-integer second-stage decision variables are related to the generation of detailed schedules. To make the model solvable, we develop an alternative equivalent reformulation with a special structure that involves binary variables in the first stage and continuous variables in the second stage. To solve this reformulated model, an integer L-shaped method is presented for small-size instances, and a simulated annealing algorithm is presented for large-size instances to obtain near-optimal solutions. Numerical experiments show that the integer L-shaped method and simulated annealing algorithm could efficiently solve the unidirectional quay crane scheduling problem with uncertainty. The results also indicate that the stochastic model has distinct advantages in terms of shortening the completion time of vessels and improving the service level of container terminals compared with the expected value problem solutions.

Bounds in multi-horizon stochastic programs

In this paper, we present bounds for multi-horizon stochastic optimization problems, a class of problems introduced in Kaut et al. (Comput Manag Sci 11:179–193, 2014) relevant in many industry-life applications typically involving strategic and operational decisions on two different time scales. After providing three general mathematical formulations of a multi-horizon stochastic program, we extend the definition of the traditional Expected Value problem and Wait-and-See problem from stochastic programming in a multi-horizon framework. New measures are introduced allowing to quantify the importance of the uncertainty at both strategic and operational levels. Relations among the solution approaches are then determined and chain of inequalities provided. Numerical experiments based on an energy planning application are finally presented.

A novel data-driven scenario generation framework for transmission expansion planning with high renewable energy penetration

Transmission expansion planning (TEP) is facing unprecedented challenges with the rise of integrated renewable energy resources (RES), flexible load elements, and the potential electrification of transport and heat sectors. Under this reality, the inadequate information of the stochastic parameters’ behavior may lead to inefficient expansion decisions, especially in the context of very high renewable penetration. This paper proposes a novel data-driven scenario generation framework for the TEP problem to generate unseen but important load and wind power scenarios while capturing inter-spatial dependencies between loads and wind generation units’ output in various locations, using a vine-copula based high-dimensional stochastic variable modeling approach. The superior performance of the proposed model is demonstrated through a case study on a modified IEEE 118-bus system. The expected result of using the expected value problem solution (EEV) and the net benefits of transmission expansion (NBTE) are used as the evaluation metrics to quantitatively illustrate the advantages of the proposed approach. In addition, the case of very high wind penetration is carried out to further highlight the importance of the multivariate stochastic dependence of load and wind power generation. The results demonstrate that the proposed scenario generation method can result in near-optimal investment decisions for the TEP problem that make more net benefits than using limited number of historical data.

The time window assignment vehicle routing problem with product dependent deliveries

This paper presents a new formulation for a time window assignment vehicle routing problem where time windows are defined for multiple product segments. This two-stage stochastic optimization problem is solved by means of a fix-and-optimize based matheuristic. The first stage assigns product dependent time windows while the second stage defines delivery schedules. Our approach outperforms a general-purpose solver and achieves an average cost decrease of 5.3% over expected value problem approaches. Furthermore, a sensitivity analysis on three operational models shows that it is possible to obtain significant savings compared to the solutions provided by a large European food retailer.

Stochastic versus Deterministic Approach to Coordinated Supply Chain Scheduling

The purpose of this paper is to consider coordinated selection of supply portfolio and scheduling of production and distribution in supply chains under regional and local disruption risks. Unlike many papers that assume the all-or-nothing supply disruption pattern, in this paper, only the regional disruptions belong to the all-or-nothing disruption category, while for the local disruptions all disruption levels can be considered. Two biobjective decision-making models, stochastic, based on the wait-and-see approach, and deterministic, based on the expected value approach, are proposed and compared to optimize the trade-off between expected cost and expected service. The main findings indicate that the stochastic programming wait-and-see approach with its ability to handle uncertainty by probabilistic scenarios of disruption events and the much simpler expected value problem, in which the random parameters are replaced by their expected values, lead to similar expected performance of a supply chain under multilevel disruptions. However, the stochastic approach, which accounts for all potential disruption scenarios, leads to a more diversified supply portfolio that will hedge against a variety of scenarios.

A stochastic approach for solving the operating room scheduling problem

We address a stochastic operating room scheduling problem which consists of assigning an intervention date and operating room to surgeries on the waiting list, minimizing the under- and overtime costs of the operating rooms, and the cost of exceeding the capacity constraints of the system. Uncertainties in surgeries duration, in the arrivals of emergency surgeries and in surgeons’ capacity are considered. To solve the problem we propose a Monte Carlo optimization method based on the sample average approximation method, which combines an iterative greedy local search method and Monte Carlo simulation. The performance of the iterative greedy local search method is evaluated against an exact method and two existing heuristic methods for solving the deterministic version of the problem, using testbeds generated based on the literature. Finally, numerical experiments are presented to evaluate the performance of the Monte Carlo optimization method in a stochastic setting. The results show that the objective function value converges with exponential rates when the number of samples increases, obtaining an optimality index value around 1 %. By comparing the results against the solution obtained by the corresponding deterministic expected value problem, we conclude that an important cost reduction can be obtained by solving the stochastic problem rather than the deterministic one.

The operating room case-mix problem under uncertainty and nurses capacity constraints.

Surgery is one of the key functions in hospitals; it generates significant revenue and admissions to hospitals. In this paper we address the decision of choosing a case-mix for a surgery department. The objective of this study is to generate an optimal case-mix plan of surgery patients with uncertain surgery operations, which includes uncertainty in surgery durations, length of stay, surgery demand and the availability of nurses. In order to obtain an optimal case-mix plan, a stochastic optimization model is proposed and the sample average approximation method is applied. The proposed model is used to determine the number of surgery cases to be weekly served, the amount of operating rooms' time dedicated to each specialty and the number of ward beds dedicated to each specialty. The optimal case-mix selection criterion is based upon a weighted score taking into account both the waiting list and the historical demand of each patient category. The score aims to maximizing the service level of the operating rooms by increasing the total number of surgery cases that could be served. A computational experiment is presented to demonstrate the performance of the proposed method. The results show that the stochastic model solution outperforms the expected value problem solution. Additional analysis is conducted to study the effect of varying the number of ORs and nurses capacity on the overall ORs' performance.

Production Optimization under Uncertainty with Constraint Handling Kristian

To maximize the daily production from an oil and gas field, mathematical optimization may be used to find the optimal operating point. When optimizing, a model of the system is used to predict the outcome for different operating points. The model is, however, subject to uncertainty, e.g., the gas oil ratio estimates may be imprecise. The uncertainty is often ignored, and what is known as the expected value problem is solved. Because of inherent uncertainties, there is a great chance that constraints will be violated when implemented. In this paper, we formulate the production optimization problem as a stochastic programming problem, and use Conditional Value at Risk to handle the constraints. This allows us to control the conservativeness of the solution in an efficient manner.

Generalization on optimal multiple stopping with application to swing options with random exercise rights number

This paper develops the theory of optimal multiple stopping times expected value problems by stating, proving, and applying a dynamic programming principle for the case in which both the reward process and the number of stopping times are stochastic. This case comes up in practice when valuing swing options, which are somewhat common in commodity trading. We believe our results significantly advance the study of option pricing.

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.

Optimal Design of Plane Frames under Uncertainty: A Comparison of Two Approaches

AbstractUsing the first collapse theorem, the necessary and sufficient survival conditions of an elasto‐plastic structure consist of the yield condition and the equilibrium condition. In practical applications several random model parameters have to be taken into account. This leads to a stochastic optimization problem which cannot be solved using the traditional methods. Instead of that, appropriate (deterministic) substitute problems must be formulated. Here, the design of plane frames is considered, where the applied load is supposed to be stochastic. In the first approach the recourse problem will be formulated in the standard form of stochastic linear programming (SLP). In order to apply efficient numerical solution procedures (LP‐solvers), approximate recourse problems based on discretization (RPD) and the expected value problem (EVP) are introduced. In the second approach – based on the yield condition – a quadratic cost function will be introduced. After the formulation of the stochastic optimization problem, the expected cost based optimization problem (ECBOP) and the minimum expected cost problem (MECP) are formulated as representatives of appropriate substitute problems. Subsequently, comparative numerical results using these methods are presented. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The impact of flexibility on operational supply chain planning

We study in this paper the effects of volume flexibility, delivery flexibility and operational decision flexibility in operational supply chain planning under uncertain demand. We use a rolling schedule to plan supply chain operations for a whole year. The planning horizon is 4 weeks with deterministic demand in the first week and predicted for the following 3 weeks. Using a case from the Norwegian meat industry, we compare the annual operating results of using a two-stage stochastic programming model to the deterministic expected value problem in order to discuss the impact of flexibility in the supply chain.

Topology optimization of plane frames in case of random external loadings

AbstractUsing the first collapse–theorem, the necessary and sufficient survival conditions of an elasto–plastic structure consist of the yield condition and the equilibrium condition. In practical applications several random model parameters have to be taken into account. This leads to a stochastic optimization problem which cannot be solved using the traditional methods. Instead of that, appropriate (deterministic) substitute problems must be formulated. Here, the topology optimization of frames is considered, where the external load is supposed to be stochastic. The recourse problem will be formulated in general and in the standard form of stochastic linear programming (SLP). After the formulation of the stochastic optimization problem, the recourse problem with discretization and the expected value problem are introduced as representatives of substitute problems. Subsequently, numerical results using these methods are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Stochastic Dynamic Optimization with Discounted Stochastic Dominance Constraints

We introduce a stochastic dynamic optimization problem, where risk aversion is expressed by a stochastic ordering constraint. The constraint requires that a random reward sequence depending on our decisions dominates a given benchmark random sequence. The dominance is defined by discounting both processes with a family of discount sequences, and by applying a univariate order. We describe the generator of this order. We develop necessary and sufficient conditions of optimality for convex stochastic control problems with the new ordering constraint, and we derive an equivalent control problem featuring implied utility functions. Furthermore, we prove the existence of an optimal random discount sequence such that the solution of the risk averse problem is also a solution of an expected value problem with this discount. Finally, we derive a version of the maximum principle for the problem with discounted dominance constraints.

Optimal design of frames under stochastic uncertainties

AbstractIn the following the design of frames is treated where the load is supposed to be stochastic. The recourse problem will be formulated in the standard form of stochastic linear programming (SLP). After this formulation the recourse problem with discretization and the expected value problem are introduced as representatives of substitute problems. Subsequently, optimization results using these methods for the robust optimal design of a two‐storey frame are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Coherent risk measures in inventory problems

We analyze an extension of the classical multi-period, single-item, linear cost inventory problem where the objective function is a coherent risk measure. Properties of coherent risk measures allow us to offer a unifying treatment of risk averse and min–max type formulations. For the single period newsvendor problem, we show that the structure of the optimal solution of the risk averse model is similar to that of the classical expected value problem. For a finite horizon dynamic inventory model, we show that, again, the optimal policy has a similar structure as that of the expected value problem. This result carries over even to the case when there is a fixed ordering cost. We also analyze monotonicity properties of the optimal order quantity with respect to the degree of risk aversion for certain risk measures.

Applying the Minimum Risk Criterion in Stochastic Recourse Programs

In the setting of stochastic recourse programs, we consider the problem of minimizing the probability of total costs exceeding a certain threshold value. The problem is referred to as the minimum risk problem and is posed in order to obtain a more adequate description of risk aversion than that of the accustomed expected value problem. We establish continuity properties of the recourse function as a function of the first-stage decision, as well as of the underlying probability distribution of random parameters. This leads to stability results for the optimal solution of the minimum risk problem when the underlying probability distribution is subjected to perturbations. Furthermore, an algorithm for the minimum risk problem is elaborated and we present results of some preliminary computational experiments.