Stochastic Programming Formulation Research Articles

Sourcing strategies for a capacitated firm subject to supply and demand uncertainty

We study sourcing strategies for a capacitated manufacturer subject to supply and demand uncertainty. The manufacturer produces two products: low- and high-quality versions of the same product that differ in only one critical component. For the low-quality product, one supplier of the critical component is unreliable, failing to deliver ordered components with some positive probability. To mitigate the risk associated with the unreliable supplier, the manufacturer may: i) order from a perfectly reliable, but expensive secondary supplier (i.e., dual sourcing); ii) produce some or all of the component in-house (i.e., concurrent sourcing); or iii) substitute a higher quality component (i.e., downward substitution). All production activities share a limited capacity resource. We develop a stochastic programming formulation in a single period setting to study the optimal combination of these sourcing tactics, which we refer to as sourcing strategies, under a variety of conditions. An extensive screening experiment reveals that five of the 63 possible sourcing strategies occur as the optimal sourcing strategy in nearly 75% of problem instances. Additional experiments provide insight into the impact of limited capacity, the value of each available sourcing option, and the evolution of the optimal sourcing strategy as key problem parameters change. Most interestingly, our experiments reveal that the ability to produce a component in-house is a highly effective means for mitigating supply and demand uncertainty in a wide range of problem environments.

Simultaneous Stochastic Optimization of Mining Complexes and Mineral Value Chains

Recent developments in modelling and optimization approaches for the production of mineral and energy resources have resulted in new simultaneous stochastic optimization frameworks and related digital technologies. A mining complex is a type of value chain whereby raw materials (minerals) extracted from various mineral deposits are transformed into a set of sellable products, using the available processing streams. The supply of materials extracted from a group of mines represents a major source of uncertainty in mining operations and mineral value chains. The simultaneous stochastic optimization of mining complexes, presented herein, aims to address major limitations of past approaches by modelling and optimizing several interrelated aspects of the mineral value chain in a single model. This single optimization model integrates material extraction from a set of sources along with their uncertainty, the related risk management, blending, stockpiling, non-linear transformations that occur in the available processing streams, the utilization of processing streams, and, finally, the transportation of products to customers. Uncertainty in materials extracted from the related mineral deposits of a mining complex is represented by a group of stochastic simulations. This paper presents a two-stage stochastic mixed integer nonlinear programming formulation for modelling and optimizing a mining complex, along with a metaheuristic-based solver that facilitates the practical optimization of exceptionally large mathematical formulations. The distinct advantages of the approach presented herein are demonstrated through two case studies, where the stochastic framework is compared to past approaches that ignore uncertainty. Results demonstrate major improvements in both meeting forecasted production targets and net present value. Concepts and methods presented in this paper for the simultaneous stochastic optimization for mining complexes may be adopted and applied to the optimization of smart oil fields.

Public R&D project portfolio selection problem with cancellations

In this study, we address a public R&D project portfolio selection problem with project cancellations. For several reasons, a funded R&D project may be halted before finishing the planned research. When a project is canceled, most of its budget is usually unused and also some of the spendings can return to the funding organization. In the call-based R&D programs, usually project selection decisions are made in one go, and, in the current call, it is not possible to award new projects with the unused budget. Decision-makers (DMs) of funding organizations can benefit from considering possible project cancellation situations to improve the budget utilization. We consider two cases. In the first case, we assume that cancellation probability of a project cannot be assessed but the DM can estimate the number of projects that will be canceled. In the second case, we assume that for each project, a cancellation probability can be assessed. For the first problem, we develop a mixed-integer linear programming formulation and a dynamic programming algorithm. For the second problem, we develop a chance-constrained stochastic programming formulation that can be solved as a mixed-integer second-order cone program. Our computational results show that practical-size problems can be solved by the proposed solution approaches.

A Solution to the Chance-Constrained Two-Stage Stochastic Program for Unit Commitment With Wind Energy Integration

This paper presents a unit commitment problem with uncertain loads and wind power. A chance-constrained two-stage stochastic program programming formulation is proposed for the stochastic day-ahead scheduling. The loss of load probability (LOLP) and the probability of wind power utilization less than certain level, referred to as loss of wind probability (LOWP), and transmission line overloading probability (TLOP) are modeled as chance constraints, to provide high reliability of power supply and to ensure the high utilization of wind generation. The chance-constrained stochastic programming formulation is converted into an equivalent deterministic formulation by a sequence of approximations and verification. Correlations between loads at different buses and correlations between wind power at different buses are also considered by the proposed approach. Numerical tests are performed on a six-bus system and a modified IEEE 118-bus system. Results validate the viability of the proposed algorithm formulation for the day-ahead stochastic scheduling. The impact of different levels of LOLP and LOWP on the required up/down spinning reserves are also presented in this paper.

Stochastic programming for qualification management of parallel machines in semiconductor manufacturing

This paper is concerned with the qualification management problem of parallel machines under high uncertainties in the semiconductor manufacturing industry. Product–machine qualification, or recipe–machine qualification, is a complicated, time-consuming process that is frequently encountered in semiconductor manufacturing. High uncertainty, a common aspect of the semiconductor manufacturing process, significantly enhances the complexity of this process. This paper mainly focuses on addressing such a complex scheduling problem by presenting a general two-stage stochastic programming formulation, which embeds uncertainty into the qualification management problem. The proposed model considers the capacity loss resulting from traditional random capacity factors, such as tool failures, and recipe–machine qualification, making it more applicable to real systems. To solve this problem, we propose a Lagrangian-relaxation-based surrogate subgradient approach. Numerical experiments indicate that this approach is capable of optimizing the problem in acceptable computation time. In addition, given that obtaining complete distribution information for random variables is unavailable in practice, a simplified approach is also developed to approximate the initial problem.

Optimizing inventory policies in process networks under uncertainty

Abstract We address the inventory planning problem in process networks under uncertainty through stochastic programming models. Inventory planning requires the formulation of multiperiod models to represent the time-varying conditions of industrial process, but multistage stochastic programming formulations are often too large to solve. We propose a policy-based approximation of the multistage stochastic model that avoids anticipativity by enforcing the same decision rule for all scenarios. The proposed formulation includes the logic that models inventory policies, and it is used to find the parameters that offer the best expected performance. We propose policies for inventory planning in process networks with arrangements of inventories in parallel and in series. We compare the inventory planning strategies obtained from the policy-based formulation and the analogous two-stage approximation of the multistage stochastic program. Sequential implementation of the planning strategies in receding horizon simulations shows the advantages of the policy-based model, despite the increase in computational complexity.

Implementing the conditional value at risk approach for even-flow forest management planning

Deciding on a plan of action for a forest holding involves a significant amount of uncertainty. As forest planning involves the use and extraction of resources, uncertainty lies in both the future development of the forest (biological uncertainty) and the development of the market for forest-based products (economic uncertainty). Additionally, natural hazards can be a source of unexpected losses. In traditional forest management planning, the most common way to deal with uncertainty is to ignore it. Growth models are used that are assumed to be correct, and timber prices are assumed to be held constant. By ignoring the fact that these models provide only one representation of what may happen, the forest owner may get an overly optimistic (or pessimistic) view of the potential value of the forest holding. Through a stochastic programming formulation, these uncertainties can be modelled directly into the optimization formulation, and a management plan can be created that incorporates the risk preferences of the decisionmaker. This is highlighted through an example that maximizes the net present value of the holding while minimizing the conditional value at risk of obtaining a stated even flow of income.

Modeling Demand Uncertainty in Two-Tier City Logistics Tactical Planning

We consider the complex and not-yet-studied issue of building the tactical plan of a two-tiered city logistics system while explicitly accounting for the uncertainty in the forecast demand. We describe and formally define the problem and then propose a general modeling framework, which takes the form of a two-stage stochastic programming formulation, the first stage selecting the first-tier service network design and the general workloads of the intertier transfer facilities, and the second stage determines the actual vehicle routing on the second tier as well as some limited adjustments of the first-stage service design decisions. Four different strategies of adapting the plan to the observed demand are introduced together with the associated recourse formulations. These strategies are then experimentally compared through an evaluation procedure that, based on Monte Carlo principles, mimics the decision process of a priori planning followed by repetitively applying the adjusted plan to the periods of the planning horizon. The performances of the city logistics system under the adjustment strategies are contrasted through performance measures relative to the costs of operating the system, including those of additional vehicle capacity and movements required when the plan does not provide sufficient transportation means, the utilization of the various types of vehicles, the intensity of the vehicle presence within the city, and the utilization of the intertier transfer facilities. The comparisons are discussed both based on the numerical figures obtained through simulation and from the point of view of managerial insights into the implication for managing city logistics physical and human resources. The analysis emphasizes the interest of flexibility in managing resources and operations for the overall performance of the system, discusses the associated trade-offs, and underlines the benefits of consolidation in terms of system efficiency and impact on the city. The comparisons also show that even when demand variability and management constraints are explicitly taken into account, our approach is still able to build good tactical plans.

Optimal placement of gas detectors: A P‐median formulation considering dynamic nonuniform unavailabilities

A stochastic programming formulation (SPqt), based on the P‐median problem, is proposed for determining the optimal placement of detectors in mitigation systems while considering nonuniform dynamic detector unavailabilities. Unlike previously proposed formulations, SPqt explicitly considers backup detection levels. This allows the modeller to determine the maximum degree of the nonlinear products to be used based on the trade‐off between computational complexity and solution accuracy. We analyze this trade‐off on formulation SPqt results by using 4 real data sets for the gas detector placement problem while using unavailability values obtained from real industry gas detector data. For this data, our results show that two detection levels are sufficient to find objective values within 1% of the optimal solution. Using two detection levels reduces the nonlinear formulation to a quadratic formulation. Three solution strategies are proposed for this quadratic formulation and then compared from the computational efficiency perspective. © 2016 American Institute of Chemical Engineers AIChE J, 62: 2728–2739, 2016

Forward thresholds for operation of pumped-storage stations in the real-time energy market

Pumped-storage hydroelectric plants are very valuable assets on the electric grid and in electric markets as they are able to pump and store water for generation, thus allowing for grid-level storage. Within the realm of short-term energy markets, we present a model for determining forward-looking thresholds for making generation and pumping decisions at such plants. A multistage stochastic programming framework is developed to optimize the thresholds with uncertain system prices over the next three days. Tractability issues are discussed and a novel method based on an implementation of the scatter search algorithm is proposed. Given the size of the multistage stochastic programming formulation, we argue that this novel method is a more accurate representation of the decision process. We demonstrate model stability and quality, and show that the forward thresholds obtained using a stochastic programming framework outperform the forward thresholds from a deterministic model, and thus can lead to efficiency gains for both the generation unit owner and the overall system in the real-time market.

Joint Initial Stocking and Transshipment—Asymptotics and Bounds

A common problem faced by many firms in their supply chains can be abstracted as follows. Periodically, or at the beginning of some selling season, the firm needs to distribute finished goods to a set of stocking locations, which, in turn, supply customer demands. Over the selling season, if and when there is a supply‐demand mismatch somewhere, a re‐distribution or transshipment will be needed. Hence, there are two decisions involved: the one‐time stocking decision at the beginning of the season and the supply/transshipment decision throughout the season. Applying a stochastic dynamic programming formulation to a two‐location model with compound Poisson demand processes, we identify the optimal supply/transshipment policy and show that the optimal initial stocking quantities can be obtained via maximizing a concave function whereas the contribution of transshipment is of order square‐root‐of T. Hence, in the context of high‐volume, fast‐moving products, the initial stocking quantity decision is a much more important contributor to the overall profit. The bounds also lead to a heuristic policy, which exhibits excellent performance in our numerical study; and we further prove both the bounds and the heuristic policy are asymptotically optimal when T approaches infinity. Extension to multiple locations is also discussed.

Models for single-sector stochastic air traffic flow management under reduced airspace capacity

Air traffic efficiency is heavily influenced by unanticipated factors that result in capacity reduction. Of these factors, weather is the most significant cause of delays in airport and airspace operations. Considering weather-related uncertainty, air traffic flow management involves controlling air traffic through allocation of available airspace capacity to flights. The corresponding decision process results in a stochastic dynamic problem where aircraft on the ground and in the air are controlled based on the evolution of weather uncertainty. We focus on the single-sector version of the problem that is applicable to a majority of cases where a volume of airspace has reduced capacity due to convective weather. We model the decision process through stochastic integer programming formulations and computationally analyse it for tractability. We then demonstrate through actual flight schedule data that a simplistic but practically implementable approximation procedure is a generally effective solution approach for these models.

Integrated crude selection and refinery optimization under uncertainty

Crude oil selection and procurement is the most important step in the refining process and impacts the profit margin of the refinery significantly. Due to uncertain quality of the crudes, conventional deterministic modeling and optimization methods are not suitable for refinery profitability enhancement. Therefore, a novel optimization scheme for crude oil procurement integrated with refinery operations in the face of uncertainties is presented. The decision process comprises two stages and is solved using a scenario‐based stochastic programming formulation. In Stage I, the optimal crude selections and purchase amounts are determined by maximizing the expected profit across all scenarios. In Stage II, the uncertainties are realized and optimal operations for the refinery are determined according to this realization. The resulting large‐scale mixed‐integer nonlinear programming formulation incorporates integer variables for crude selection and continuous variables for refinery operations, as well as bilinear terms for pooling processes. Nonconvex generalized Benders decomposition is used to solve this problem to obtain an global optimum efficiently. © 2015 American Institute of Chemical Engineers AIChE J, 62: 1038–1053, 2016

A Progressive Hedging Approach for Surgery Planning Under Uncertainty

We propose a multistage stochastic mixed-integer programming formulation for the assignment of surgeries to operating rooms over a finite planning horizon. We consider the demand for and the duration of surgery to be random variables. The objective is to minimize three competing criteria: expected cost of surgery cancellations, patient waiting time, and operating room overtime. We discuss properties of the model and an implementation of the progressive hedging algorithm to find near-optimal surgery schedules. We conduct numerical experiments using data from a large hospital to identify managerial insights related to surgery planning and the avoidance of surgery cancellations. We compare the progressive hedging algorithm to an easy-to-implement heuristic for practical problem instances to estimate the value of the stochastic solution. Finally, we discuss an implementation of the progressive hedging algorithm within a rolling horizon framework for extended planning periods.

Stochastic-Predictive Energy Management System for Isolated Microgrids

This paper presents the mathematical formulation and control architecture of a stochastic-predictive energy management system for isolated microgrids. The proposed strategy addresses uncertainty using a two-stage decision process combined with a receding horizon approach. The first stage decision variables (unit commitment) are determined using a stochastic mixed-integer linear programming formulation, whereas the second stage variables (optimal power flow) are refined using a nonlinear programming formulation. This novel approach was tested on a modified CIGRE test system under different configurations comparing the results with respect to a deterministic approach. The results show the appropriateness of the method to account for uncertainty in the power forecast.

Production control problem integrating overhaul and subcontracting strategies for a quality deteriorating manufacturing system

We study an unreliable deteriorating manufacturing system that produces conforming and non-conforming parts to satisfy a constant demand product rate. The manufacturing system is comprised of a failure-prone machine. Due to the combined effect of random availability variations and deterioration, the system is not able to fulfill long-term product demand. In particular, when finished goods inventory is positive, clients demand are fulfilled on-time and without delay. When backlog exists, subcontracting options are available at a higher cost to supplement the limited production capacity of the manufacturing system. The effect of deterioration is observed mainly in the quality of the parts produced, since the rate of defectives increases as the machines deteriorate. Overhaul activities can be conducted to mitigate the effect of deterioration. We propose a joint feedback control policy based on a stochastic dynamic programming formulation which aims simultaneously to determine the production and overhaul rates, and the rate at which subcontractors are requested. The proposed joint control policy minimizes the total cost, including the inventory, backlog, repair, overhaul, defectives, production and subcontracting costs, over an infinite planning horizon. To determine the optimal control parameters, we adopt a numerical scheme to solve the optimality equations and a numerical example is presented as an illustration of the approach. The structure of the joint control policy is validated through an extensive sensitivity analysis.

Incorporating nurse absenteeism into staffing with demand uncertainty.

Increased nurse-to-patient ratios are associated negatively with increased costs and positively with improved patient care and reduced nurse burnout rates. Thus, it is critical from a cost, patient safety, and nurse satisfaction perspective that nurses be utilized efficiently and effectively. To address this, we propose a stochastic programming formulation for nurse staffing that accounts for variability in the patient census and nurse absenteeism, day-to-day correlations among the patient census levels, and costs associated with three different classes of nursing personnel: unit, pool, and temporary nurses. The decisions to be made include: how many unit nurses to employ, how large a pool of cross-trained nurses to maintain, how to allocate the pool nurses on a daily basis, and how many temporary nurses to utilize daily. A genetic algorithm is developed to solve the resulting model. Preliminary results using data from a large university hospital suggest that the proposed model can save a four-unit pool hundreds of thousands of dollars annually as opposed to the crude heuristics the hospital currently employs.

Stochastic Scheduling of Renewable and CHP-Based Microgrids

Microgrids (MGs) are considered as a key solution for integrating renewable and distributed energy resources, combined heat and power (CHP) systems, as well as distributed energy-storage systems. This paper presents a stochastic programming framework for conducting optimal 24-h scheduling of CHP-based MGs consisting of wind turbine, fuel cell, boiler, a typical power-only unit, and energy storage devices. The objective of scheduling is to find the optimal set points of energy resources for profit maximization considering demand response programs and uncertainties. The impact of the wind speed, market, and MG load uncertainties on the MG scheduling problem is characterized through a stochastic programming formulation. This paper studies three cases to confirm the performance of the proposed model. The effect of CHP-based MG scheduling in the islanded and grid-connected modes, as well as the effectiveness of applying the proposed DR program is investigated in the case studies.

Medium-term maintenance turnaround planning under uncertainty for integrated chemical sites

Plant maintenance poses extended disruptions to production. Maintenance effects are amplified when the plant is part of an integrated chemical site, as production levels of adjacent plants in the site are also significantly influenced. A challenge in dealing with turnarounds is the difficulty in predicting their duration, due to discovery work and delays. This uncertainty in duration affects two major planning decisions: production levels and maintenance manpower allocation. The latter must be decided several months before the turnarounds occur. We address the scheduling of a set of plant turnarounds over a medium-term of several months using integer programming formulations. Due to the nature of uncertainty, production decisions are treated through stochastic programming ideas, while the manpower aspect is handled through a robust optimization framework. We propose combined robust optimization and stochastic programming formulations to address the problem and demonstrate, through an industrial case study, the potential for significant savings.

A progressive hedging method for the multi-path travelling salesman problem with stochastic travel times

In this paper, we consider a recently introduced problem specifically designed for Smart Cities and City Logistics applications: the multi-path travelling salesman problem with stochastic travel costs (mpTSPs)(mpTSPs). The mpTSPsmpTSPs is a variant of the TSP problem where a set of paths exists between any two nodes and each path is characterized by a random travel time. We propose a two-stage stochastic programming formulation where tour design makes up the first stage, while recourse decisions related to the choice of the path to follow are made in the second stage. To solve this formulation, we propose a heuristic method inspired by the Progressive Hedging (PH) algorithm of Rockafellar & Wets (1991, Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res., 16, 119–147.) We then benchmark the solution method by solving model instances derived from the traffic speed sensor network of the city of Turin. Furthermore, the impact of the stochastic travel time costs on the problem solution is examined, showing the benefits of the proposed methodology in both solution quality and computational effort when compared to solving the deterministic equivalent formulation using a commercial solver.