Production planning under uncertainty in textile manufacturing

Textile manufacturing consists of yarn production, fabric formation, and finishing and dyeing stages. The subject of this paper is the yarn production planning problem, although the approach is directly applicable to the fabric production planning problem due to similarities in the respective models. Our experience at an international textile manufacturer indicates that demand uncertainty is a major challenge in developing yarn production plans. We develop a stochastic programming model that explicitly includes uncertainty in the form of discrete demand scenarios. This results in a large-scale mixed integer model that is difficult to solve with off-the-shelf commercial solvers. We develop a two-step preprocessing algorithm that improves the linear programming relaxation of the model and reduces its size, consequently improving the computational requirements. We illustrate the benefits of a stochastic programming approach over a deterministic model and share our initial application experience.     

Stochastic models for air cargo terminal manpower supply planning in long‐term operations

In this research, based on two deterministic‐demand planning models, we established two long‐term stochastic‐demand planning models by incorporating the stochastic disturbances of manpower demands that occur in actual operations. The models are formulated as mixed integer linear programs that are solved using a mathematical programming solver. To compare the performance of the two stochastic‐demand and two deterministic‐demand planning models under the stochastic demands that occur in actual operations, we further develop a simulation‐based evaluation method. Finally, we perform numerical tests using real operating data from a Taiwan air cargo terminal. The preliminary results show that the stochastic models could be useful for planning air cargo terminal manpower supply. Copyright © 2008 John Wiley & Sons, Ltd.     

Stochastic multi-site capacity planning of TFT-LCD manufacturing using expected shadow-price based decomposition

This paper presents a stochastic optimization model and efficient decomposition algorithm for multi-site capacity planning under the uncertainty of the TFT-LCD industry. The objective of the stochastic capacity planning is to determine a robust capacity allocation and expansion policy hedged against demand uncertainties because the demand forecasts faced by TFT-LCD manufacturers are usually inaccurate and vary rapidly over time. A two-stage scenario-based stochastic mixed integer programming model that extends the deterministic multi-site capacity planning model proposed by Chen et al. (2010) [1] is developed to discuss the multi-site capacity planning problem in the face of uncertain demands. In addition a three-step methodology is proposed to generate discrete demand scenarios within the stochastic optimization model by approximating the stochastic continuous demand process fitted from the historical data. An expected shadow-price based decomposition, a novel algorithm for the stage decomposition approach, is developed to obtain a near-optimal solution efficiently through iterative procedures and parallel computing. Preliminary computational study shows that the proposed decomposition algorithm successfully addresses the large-scale stochastic capacity planning model in terms of solution quality and computation time. The proposed algorithm also outperforms the plain use of the CPLEX MIP solver as the problem size becomes larger and the number of demand scenarios increases.     

A Scalable Parallel Interior Point Algorithm for Stochastic Linear Programming and Robust Optimization

We present a computationally efficient implementation of an interior point algorithm for solving large-scale problems arising in stochastic linear programming and robust optimization. A matrix factorization procedure is employed that exploits the structure of the constraint matrix, and it is implemented on parallel computers. The implementation is perfectly scalable. Extensive computational results are reported for a library of standard test problems from stochastic linear programming, and also for robust optimization formulations.The results show that the codes are efficient and stable for problems with thousands of scenarios. Test problems with 130 thousand scenarios, and a deterministic equivalent linear programming formulation with 2.6 million constraints and 18.2 million variables, are solved successfully.     

Models and model value in stochastic programming

Finding optimal decisions often involves the consideration of certain random or unknown parameters. A standard approach is to replace the random parameters by the expectations and to solve a deterministic mathematical program. A second approach is to consider possible future scenarios and the decision that would be best under each of these scenarios. The question then becomes how to choose among these alternatives. Both approaches may produce solutions that are far from optimal in the stochastic programming model that explicitly includes the random parameters. In this paper, we illustrate this advantage of a stochastic program model through two examples that are representative of the range of problems considered in stochastic programming. The paper focuses on the relative value of the stochastic program solution over a deterministic problem solution.The author's work was supported in part by the National Science Foundation under Grant DDM-9215921.     

Mathematical programming for network revenue management revisited

Mathematical programming models for airline seat inventory control provide booking limits and bid-prices for all itineraries and fare classes. E.L. Williamson [Airline network seat inventory control: methodologies and revenue impacts, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1992] finds that simple deterministic approximation methods based on average demand often outperform more advanced probabilistic heuristics. We argue that this phenomenon is due to a booking process that includes nesting of the fare classes, which is ignored in the modeling phase. The differences in the performance between these approximations are studied using a stochastic programming model that includes the deterministic model as a special case. Our study carefully examines the trade-off between computation time and the aggregation level of demand uncertainty with examples of a multi-leg flight and a single-hub network.     

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.     

A two-echelon stochastic facility location model for humanitarian relief logistics

We develop a two-stage stochastic programming model for a humanitarian relief logistics problem where decisions are made for pre- and post-disaster rescue centers, the amount of relief items to be stocked at the pre-disaster rescue centers, the amount of relief item flows at each echelon, and the amount of relief item shortage. The objective is to minimize the total cost of facility location, inventory holding, transportation and shortage. The deterministic equivalent of the model is formulated as a mixed-integer linear programming model and solved by a heuristic method based on Lagrangean relaxation. Results on randomly generated test instances show that the proposed solution method exhibits good performance up to 25 scenarios. We also validate our model by calculating the value of the stochastic solution and the expected value of perfect information.     

A two-stage stochastic programming with recourse model for determining robust planting plans in horticulture

A two-stage stochastic programming with recourse model for the problem of determining optimal planting plans for a vegetable crop is presented in this paper. Uncertainty caused by factors such as weather on yields is a major influence on many systems arising in horticulture. Traditional linear programming models are generally unsatisfactory in dealing with the uncertainty and produce solutions that are considered to involve an unacceptable level of risk. The first stage of the model relates to finding a planting plan which is common to all scenarios and the second stage is concerned with deriving a harvesting schedule for each scenario. Solutions are obtained for a range of risk aversion factors that not only result in greater expected profit compared to the corresponding deterministic model, but also are more robust.     

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.     

Building and Solving Large-Scale Stochastic Programs on an Affordable Distributed Computing System

We present an integrated procedure to build and solve big stochastic programming models. The individual components of the system – the modeling language, the solver and the hardware – are easily accessible, or a least affordable to a large audience. The procedure is applied to a simple financial model, which can be expanded to arbitrarily large sizes by enlarging the number of scenarios. We generated a model with one million scenarios, whose deterministic equivalent linear program has 1,111,112 constraints and 2,555,556 variables. We have been able to solve it on the cluster of ten PCs in less than 3 hours.     

Probability maximization models for portfolio selection under ambiguity

This paper considers several probability maximization models for multi-scenario portfolio selection problems in the case that future returns in possible scenarios are multi-dimensional random variables. In order to consider occurrence probabilities and decision makers’ predictions with respect to all scenarios, a portfolio selection problem setting a weight with flexibility to each scenario is proposed. Furthermore, by introducing aspiration levels to occurrence probabilities or future target profit and maximizing the minimum aspiration level, a robust portfolio selection problem is considered. Since these problems are formulated as stochastic programming problems due to the inclusion of random variables, they are transformed into deterministic equivalent problems introducing chance constraints based on the stochastic programming approach. Then, using a relation between the variance and absolute deviation of random variables, our proposed models are transformed into linear programming problems and efficient solution methods are developed to obtain the global optimal solution. Furthermore, a numerical example of a portfolio selection problem is provided to compare our proposed models with the basic model.     

A stochastic programming approach to cash management in banking

The treasurer of a bank is responsible for the cash management of several banking activities. In this work, we focus on two of them: cash management in automatic teller machines (ATMs), and in the compensation of credit card transactions. In both cases a decision must be taken according to a future customers demand, which is uncertain. From historical data we can obtain a discrete probability distribution of this demand, which allows the application of stochastic programming techniques. We present stochastic programming models for each problem. Two short-term and one mid-term models are presented for ATMs. The short-term model with fixed costs results in an integer problem which is solved by a fast (i.e. linear running time) algorithm. The short-term model with fixed and staircase costs is solved through its MILP equivalent deterministic formulation. The mid-term model with fixed and staircase costs gives rise to a multi-stage stochastic problem, which is also solved by its MILP deterministic equivalent. The model for compensation of credit card transactions results in a closed form solution. The optimal solutions of those models are the best decisions to be taken by the bank, and provide the basis for a decision support system.     

A two-stage stochastic programming model for scheduling replacements in sow farms

This paper presents a formulation and resolution of a two-stage stochastic linear programming model with recourse for sow farms producing piglets. The proposed model considers a medium-term planning horizon and specifically allows optimal replacement and schedule of purchases to be obtained for the first stage. This model takes into account sow herd dynamics, housing facilities, reproduction management, herd size with initial and final inventory of sows and uncertain parameters such as litter size, mortality and fertility rates. These last parameters are explicitly incorporated via a finite set of scenarios. The proposed model is solved by using the algebraic modelling software OPL Studio from ILOG, in combination with the solver CPLEX to solve the linear models resulting from different instances considered. The article also presents results obtained with previous deterministic models assessing the suitability of the stochastic approach. Finally, the conclusions drawn from the study including an outlook are presented.      

A simple recourse model for power dispatch under uncertain demand

Optimal power dispatch under uncertainty of power demand is tackled via a stochastic programming model with simple recourse. The decision variables correspond to generation policies of a system comprising thermal units, pumped storage plants and energy contracts. The paper is a case study to test the kernel estimation method in the context of stochastic programming. Kernel estimates are used to approximate the unknown probability distribution of power demand. General stability results from stochastic programming yield the asymptotic stability of optimal solutions. Kernel estimates lead to favourable numerical properties of the recourse model (no numerical integration, the optimization problem is smooth convex and of moderate dimension). Test runs based on real-life data are reported. We compute the value of the stochastic solution for different problem instances and compare the stochastic programming solution with deterministic solutions involving adjusted demand portions.This research is supported by the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung of the Deutsche Forschungsgemeinschaft.     

Probabilistic Networks and R & D Portfolio Selection

Mathematical programming methods have been suggested and used as an aid to R & D project portfolio selection. One of the main criticisms of the use of such models is that the stochastic nature of the problem has been largely ignored. This paper presents a method which takes into account the stochastic nature of resource requirements and project benefits, using a combination of probabilistic networks, simulation and mathematical programming. A case study based on data from an industrial R & D laboratory is presented and compared with the use of expected value methods. The results of the study indicate that in this particular case the deterministic linear programming solution is robust.     

Solving the logit-based stochastic user equilibrium problem with elastic demand based on the extended traffic network model

This paper proposes a novel extended traffic network model to solve the logit-based stochastic user equilibrium (SUE) problem with elastic demand. In this model, an extended traffic network is established by properly adding dummy nodes and links to the original traffic network. Based on the extended traffic network, the logit-based SUE problem with elastic demand is transformed to the SUE problem with fixed demand. Such problem is then further converted to a linearly constrained convex programming and addressed by a predictor–corrector interior point algorithm with polynomial complexity. A numerical example is provided to compare the proposed model with the method of successive averages (MSA). The numerical results indicate that the proposed model is more efficient and has a better convergence than the MSA.     

A multiple objective goal programming approach to the truckload routing problem

Truckload (TL) routing has always been a challenge. The TL routing problem (TRP) itself is hard, but the complexity of solving the problem increases due to the stochastic nature of TL demand. It is traditionally approached using single objective solution methodologies that range from linear programming to dynamic programming techniques. This paper presents a deterministic multiple objective formulation of the TRP. A ‘route algebra’ is developed to facilitate the solution procedure, paving the way for the use of goal programming and tabu search techniques.     

Cost/risk balanced management of scarce resources using stochastic programming

We consider the situation when a scarce renewable resource should be periodically distributed between different users by a Resource Management Authority (RMA). The replenishment of this resource as well as users demand is subject to considerable uncertainty. We develop cost optimization and risk management models that can assist the RMA in its decision about striking the balance between the level of target delivery to the users and the level of risk that this delivery will not be met. These models are based on utilization and further development of the general methodology of stochastic programming for scenario optimization, taking into account appropriate risk management approaches. By a scenario optimization model we obtain a target barycentric value with respect to selected decision variables. A successive reoptimization of deterministic model for the worst case scenarios allows the reduction of the risk of negative consequences derived from unmet resources demand. Our reference case study is the distribution of scarce water resources. We show results of some numerical experiments in real physical systems.