Second-stage Decisions Research Articles

Hybrid manufacturing and remanufacturing lot-sizing problem with stochastic demand, return, and setup costs

This paper presents mixed-integer programming models to hybrid manufacturing and remanufacturing lot-sizing problems in which remanufactured products are treated as new ones, so that both manufactured and remanufactured products compete to meet the demands. Differently from previous studies, we consider an environment with multiple products, both manufacturing and remanufacturing costs, disposal, backlogging, and the inherent uncertainties of demands, return rates of usable products, and setup costs. In order to deal with these uncertainties, we propose a scenario-based two-stage stochastic programming model that assumes production and setup as first-stage decision variables, whereas inventory, disposal, and backlogging are defined as second-stage decision variables. We also analyze a risk-averse model in an attempt to reduce the dispersion of the second-stage costs. The main results of the present study indicate that setup costs for remanufacturing can be decisive in choosing between manufacturing or remanufacturing. Even though remanufacturing costs are lower, the process is still largely dependent on return rates and low storage costs for returned products.

Robust Distribution Network Reconfiguration

We propose a two-stage robust optimization model for the distribution network reconfiguration problem with load uncertainty. The first-stage decision is to configure the radial distribution network and the second-stage decision is to find the optimal a/c power flow of the reconfigured network for given demand realization. We solve the two-stage robust model by using a column-and-constraint generation algorithm, where the master problem and subproblem are formulated as mixed-integer second-order cone programs. Computational results for 16, 33, 70, and 94-bus test cases are reported. We find that the configuration from the robust model does not compromise much the power loss under the nominal load scenario compared to the configuration from the deterministic model, yet it provides the reliability of the distribution system for all scenarios in the uncertainty set.

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.

Modeling and Optimization Techniques for Yield-Aware SRAM Post-Silicon Tuning

SRAM cell design is driven by the need to satisfy several stability and performance criteria for all cells in the array in an energy-efficient manner. Significant randomness of FET threshold voltages makes achieving this difficult and limits both the minimum cell size and minimum array supply voltage. Post-silicon adaptivity in the form of an adaptive-voltage scheme in a partitioned SRAM array can be used to reduce impact of variability despite lack of any spatial correlation in realizations. This paper develops a novel optimization flow for yield-aware cell sizing and voltage selection under variability given the availability of post-silicon voltage tuning. We formulate a two-stage stochastic optimization problem in which the first-stage decision is to select cell size and possible voltage levels, and the second-stage decision is to assign each partition to an optimal voltage after manufacturing. We develop closed-form statistical models of array margin behavior and yield as a function of V dd , cell size, and array size. We solve the problem using dynamic programming that minimizes power while meeting yield constraints on read, write, and static noise margins. The proposed flow allows designs that are on average 8% and up to 17% more power-efficient than the designs in which voltages are selected uniformly. The results also indicate that at high-yield levels power savings can be up to 32% in the active mode and 71% in the standby mode.

Discussion of “Adaptive Robust Optimization for the Security Constrained Unit Commitment Problem”

We commend the authors of [1] for application of an interesting optimization methodology, i.e., adaptive robust optimization to solve the security constrained unit commitment problem (SCUC). They present a two-stage adaptive robust unit commitment model for SCUC considering the uncertainty of nodal net injection in power system. The two-stage adaptive model is intended to obtain the first-stage unit commitment (UC) decision and the second-stage dispatch decision to become robust against all uncertain nodal net injection realizations. In [1], the first-stage UC decision is a vector of binary variables, which includes the on/off and start-up/shut-down status of each generation unit for each time interval of the commitment period. The second-stage dispatch decision is a vector of continuous variables, composed of the generation output, load consumption levels, resource reserve levels, and power flows in the transmission network for each time interval. In addition, the authors have pointed out that generation output, resource reserve levels, and power flow take positive sign, whereas load consumption levels take negative sign. Therefore, the discusser has certain doubt about [1, (8)]. If each element of the second-stage dispatch decision can take any real number, then equation (8) is of no question. However, actually, each element of takes either positive or negative sign as the authors pointed out. As a result, the discusser doubts about equation (8). This question is quite important because the dual of the dispatch problem, (8), is the mathematical basis of the proposed two-stage adaptive robust UC model.

Comparing the Performances of Two Techniques for the Optimization Under Parametric Uncertainty of the Simultaneous Design and Planning of a Multiproduct Batch Plant

This paper addresses the comparison between two techniques for the optimization under parametric uncertainty of multiproduct batch plants integrating design and production planning decisions. This problem has been conceived as a two-stage stochastic Mixed Integer Linear Programming (MILP) in which the first-stage decisions consist of design variables that allow determining the batch plant structure, and the second-stage decisions consist of production planning continuous variables in a multi-period context. The objective function maximizes the expected net present value. In the first solving approach, the problem has been tackled through mathematical programming considering a discrete set of scenarios. In the second solving approach, the multi-scenario MILP problem has been reformulated by adopting a simulation-based optimization scheme to accommodate the variables belonging to different management levels. Advantages and disadvantages of both approaches are demonstrated through a case study. Results allow concluding that a simulation-based optimization strategy may be a suitable technique to afford two-stage stochastic programming problems.

Integration of Scheduling and Dynamic Optimization of Batch Processes under Uncertainty: Two-Stage Stochastic Programming Approach and Enhanced Generalized Benders Decomposition Algorithm

Integration of scheduling and dynamic optimization significantly improves the overall performance of a production process compared to the traditional sequential method. However, most integrated methods focus on solving deterministic problems without explicitly taking process uncertainty into account. We propose a novel integrated method for sequential batch processes under uncertainty. The integrated problem is formulated into a two-stage stochastic program. The first-stage decisions are modeled with binary variables for assignment and sequencing while the second-stage decisions are the remaining ones. To solve the resulting complicated integrated problem, we develop two efficient algorithms based on the framework of generalized Benders decomposition. The first algorithm decomposes the integrated problem according to the scenarios so that the subproblems can be optimized independently over each scenario. Besides the scenario decomposition, the second algorithm further decomposes dynamic models from the sc...

Economic and environmental optimization of a large scale sustainable dual feedstock lignocellulosic-based bioethanol supply chain in a stochastic environment

This work proposes a two-stage stochastic optimization model to maximize the expected profit and simultaneously minimize carbon emissions of a dual-feedstock lignocellulosic-based bioethanol supply chain (LBSC) under uncertainties in supply, demand and prices. The model decides the optimal first-stage decisions and the expected values of the second-stage decisions. A case study based on a 4-state Midwestern region in the US demonstrates the effectiveness of the proposed stochastic model over a deterministic model under uncertainties. Two regional modes are considered for the geographic scale of the LBSC. Under co-operation mode the 4 states are considered as a combined region while under stand-alone mode each of the 4 states is considered as an individual region. Each state under co-operation mode gives better financial and environmental outcomes when compared to stand-alone mode. Uncertainty has a significant impact on the biomass processing capacity of biorefineries. While the location and the choice of conversion technology for biorefineries i.e. biochemical vs. thermochemical, are insensitive to the stochastic environment. As variability of the stochastic parameters increases, the financial and environmental performance is degraded. Sensitivity analysis shows that levels of tax credit and carbon price have a major impact on the choice of conversion technology for a selected biorefinery. Biochemical pathway is preferred over the thermochemical as carbon price increases. Thermochemical pathway is preferred over the biochemical as the level of tax credit increases. In addition, bioethanol production in the US is shown to be unviable without adequate governmental subsidy in the form of tax credits.

Energy efficiency and risk management in public buildings: strategic model for robust planning

Due to deregulations of the energy sector and the setting of targets such as the 20/20/20 in the EU, operators of public buildings are now more exposed to instantaneous (short-term) market conditions. On the other hand, they have gained the opportunity to play a more active role in securing long-term supply, managing demand, and hedging against risk while improving existing buildings’ infrastructures. Therefore, there are incentives for the operators to develop and use a Decision Support System to manage their energy sub-systems in a more robust energy-efficient and cost-effective manner. In this paper, a two-stage stochastic model is proposed, where some decisions (so-called first-stage decisions) regarding investments in new energy technologies have to be taken before uncertainties are resolved, and some others (so-called second-stage decisions) on how to use the installed technologies will be taken once values for uncertain parameters become known, thereby providing a trade-off between long- and short-term decisions.

A two-stage robust programming approach to demand-driven disassembly planning for a closed-loop supply chain system

The closed-loop supply chain system, which integrates forward and reverse logistics, is a desirable policy for retaining recoverable resources and extending the life cycles of products. In this study, we propose a methodology to contend with a demand-driven disassembly planning problem under a closed-loop supply chain system. A two-stage robust programming model is developed correspondingly, such that multiple products with a hierarchical product's structure are disassembled to satisfy uncertain demands in multiple periods. The objective of the model is to determine a robust decision for recycle volume and timing of each type of end-of-life (EOL) product, as well as recovery strategies. The results provide two-stage decisions by considering future scenarios of periodic demands at the beginning of a planning horizon. The first-stage decision is to determine a compromise solution that is close to the optimal solution for every scenario while retaining a certain level of infeasibility of constraints, such as unsatisfied demand. Afterward, when the outcome of a scenario has been realised, the second-stage decision, such as, inventory volume, is conducted to become a buffer for mitigating uncertain impacts. Furthermore, the computational results confirm the trade-off relationship between solution robustness and model robustness, which are core results of the robust model apart from expected profit. The different types of decision makers’ preferences toward risk can be accounted for to determine a compromise robust solution.

Is the Willingness to Take Financial Risk a Sex-Linked Trait? Evidence from National Surveys of Household Finance

We investigate whether the willingness to take investment risk is a sex-linked trait and link the results to the country's gender equality regime. Our empirical analysis involves household data on financial asset holdings as well as on self-reported risk tolerance for Austria, Italy, the Netherlands and Spain. Of those countries, Italy is by far the country with the greatest degree of gender inequality according to the 2009 Global Gender Gap Report. Two stages of building a portfolio of financial assets are analyzed. For the first-stage decision of whether to invest in risky assets in the first place, gender is found to have no effect in Austria, the Netherlands and Spain but does have an impact in Italy. However, even for Italy, it seems to be irrelevant in the second-stage decision about the share of wealth invested in the risky assets. We infer from these findings that, for countries with a high degree of gender equality, it is inappropriate to base financial advice primarily on gender.

Real option games with R&D and learning spillovers

We demonstrate to decision makers how to optimally make costly strategic pre-investment R&D decisions in the presence of spillover effects in an option pricing framework with analytic tractability. Decisions are modeled as impulse-type controls with random outcome. Two firms face two decisions that are solved interdependently in a two-stage game. The first-stage decision is: What is the optimal level of coordination (optimal policy/technology choice)? The second-stage decision is: What is the optimal effort for a given level of the spillover effects and the cost of information acquisition? The framework is extended to a two-period closed-loop stochastic game with (path-dependency inducing) switching costs that make strategy revisions harder. When conditions of learning-by-doing exist, we find that strategy shifts are easier to observe in market environments of high growth and high volatility.

Disjunctive Decomposition for Two-Stage Stochastic Mixed-Binary Programs with Generalized Upper Bound Constraints

The disjunctive decomposition (D2) algorithm has emerged as a powerful tool to solve stochastic integer programs. In this paper, we consider two-stage stochastic integer programs with binary first-stage and mixed-binary second-stage decisions and present several computational enhancements to D2. First, we explore the use of a cut generation problem restricted to a subspace of the variables, which yields significant computational savings. Then, we examine problems with generalized upper bound constraints in the second stage and exploit this structure to generate cuts. We establish convergence of D2 variants. We present computational results on a new stochastic scheduling problem with an uncertain number of jobs motivated by companies in industries such as consulting and defense contracting, where these companies bid on future contracts but may or may not win the bid. The enhancements reduced computation time on average by 45% on a set of test problems.

Evaluating the Effectiveness of Pre-Positioning Policies in Response to Natural Disasters

Recent natural disasters highlight the complexities associated with planning, coordination and distribution of supplies in a manner which provides timely and effective response. In this paper, the authors present a model to quantify the benefits associated with pre-positioning local supplies. They assume the supplies are in a high-risk location and may be destroyed if an appropriate strategy to protect the supplies is not implemented. A stochastic linear programming model is developed where the first-stage decision pre-positions existing supplies to minimize the supply loss. Second-stage decisions attempt to maximize the responsiveness of the system by allocating supplies to satisfy demand. The benefits associated with pre-positioning versus non-pre-positioning are discussed.

Corporate dashboards for integrated business and engineering decisions in oil refineries: An agent-based approach

It is generally very challenging for an oil refinery to make integrated decisions encompassing multiple functions based on a traditional Decision Support System (DSS), given the complexity and interactions of various decisions. To overcome this limitation, we propose an integrated DSS framework by combining both business and engineering systems with a dashboard. The dashboard serves as a human–computer interface and allows a decision maker to adjust decision variables and exchange information with the DSS. The proposed framework provides a two-stage decision making mechanism based on optimization and agent-based models. Under the proposed DSS, the decision maker decides on the values of a subset of decision variables. These values, or the first-stage decision, are forwarded through the dashboard to the DSS. For the given set of first-stage decision variables, a multi-objective robust optimization problem, based on an integrated business and engineering simulation model, is solved to obtain the values for a set of second-stage decision variables. The two-stage decision making process iterates until a convergence is achieved. A simple oil refinery case study with an example dashboard demonstrates the applicability of the integrated DSS.

Modelling of containerized air cargo forwarding problems under uncertainty

This paper examines air container renting and cargo loading problems experienced by freight forwarding companies. Containers have to be booked in advance, in order to obtain discounted rental rates from airlines; renting or returning containers on the day of shipping will incur a heavy penalty. We first propose a mixed-integer model for the certain problem, in which shipment information is known with certainty, when booking. We then present a two-stage recourse model to handle the uncertainty problem, in which accurate shipment information cannot be obtained when booking, and all cargoes have to be shipped without delay. The first-stage decision is made at the booking stage, to book specific numbers of different types of containers. The second-stage decision is made on the day of shipping, depending on the extent to which the uncertainty has been realized. The decisions include number of additional containers of different types that are required to be rented, or the number of containers to be returned, under the scenario that might occur on the day of shipping. We then extend the recourse model into a robust model for dealing with the situation in which cargoes are allowed to be shipped later. The robust model provides a quantitative method to measure the trade-off between risk and cost. A series of experiments demonstrate the effectiveness of the robust model in dealing with risk and uncertainty.

Systematic Initialization Techniques for Hybrid Evolutionary Algorithms for Solving Two-Stage Stochastic Mixed-Integer Programs

This paper introduces new initialization approaches for evolutionary algorithms that solve two-stage stochastic mixed-integer problems. The two-stage stochastic mixed-integer programs are handled by a stage decomposition based hybrid algorithm where an evolutionary algorithm handles the first-stage decisions and mathematical programming handles the second-stage decisions. The population of the evolutionary algorithm is initialized by using solutions which are generated in a preprocessing step of the hybrid algorithm. This paper presents three different initialization approaches in which the two-stage stochastic mixed-integer program is exploited in order to obtain potentially good starting solutions for the evolutionary algorithm. In case of infeasible initializations, the population is driven toward feasibility by a penalty function. Comparisons of an evolutionary algorithm with a classical random initialization and the new initialization approaches for two real-world problems show that the new initialization approaches lead to high quality feasible solutions in significantly shorter computing times.

A stochastic model for production loading in a global apparel manufacturing company under uncertainty

This article studies production loading problems with uncertainties of demand and import quotas experienced by a global apparel manufacturing company, whose markets are located in Northern America and Europe, manufacturing factories are in Asia (Mainland China, Thailand, the Philippines and Sri Lanka) and headquarters in Hong Kong. Loading production among different factories in different countries involves many uncertain factors, such as market information and quota premium. This article presents a two-stage stochastic programming model for production loading problems with uncertainties where the first-stage decisions are made before accurate information is available, and the second-stage decisions are made when the stochasticity is realised. By using the two-stage production planning, the company is able to achieve a quick response to the changing market information while minimising the total production cost. A series of experiments, based on the data from the apparel company, are designed to test the effectiveness of the proposed model. Compared with the results of the deterministic model, the stochastic recourse model can provide a more flexible, responsive and cheaper production loading system.

Two-stage Stochastic Programming Formulation for Ship Design Optimisation under Uncertainty

The paper presents a two-stage approach for ship design optimisation under uncertainties related to stochastic parameters that cannot be controlled by the designer. The designer decision is partitioned into two sets: the first contains variables that have to be decided before the onset of any uncertain condition, while the second involves variables that may be decided once the random events have occurred. The first-stage variables are selected so that the sum of the first-stage cost and the expectation of the second-stage cost is minimised. The problem is solved as a two-nested-loop minimisation problem, which combines robustness of the first-stage decision with flexibility of the second-stage decision. The approach is applied to the conceptual optimisation of a bulk carrier.

Stochastic Optimization for Natural Disaster Asset Prepositioning

A key strategic issue in pre‐disaster planning for humanitarian logistics is the pre‐establishment of adequate capacity and resources that enable efficient relief operations. This paper develops a two‐stage stochastic optimization model to guide the allocation of budget to acquire and position relief assets, decisions that typically need to be made well in advance before a disaster strikes. The optimization focuses on minimizing the expected number of casualties, so our model includes first‐stage decisions to represent the expansion of resources such as warehouses, medical facilities with personnel, ramp spaces, and shelters. Second‐stage decisions concern the logistics of the problem, where allocated resources and contracted transportation assets are deployed to rescue critical population (in need of emergency evacuation), deliver required commodities to stay‐back population, and transport the transfer population displaced by the disaster. Because of the uncertainty of the event's location and severity, these and other parameters are represented as scenarios. Computational results on notional test cases provide guidance on budget allocation and prove the potential benefit of using stochastic optimization.