756 research outputs found

    Piecewise linear approximations for the static-dynamic uncertainty strategy in stochastic lot-sizing

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    In this paper, we develop mixed integer linear programming models to compute near-optimal policy parameters for the non-stationary stochastic lot sizing problem under Bookbinder and Tan's static-dynamic uncertainty strategy. Our models build on piecewise linear upper and lower bounds of the first order loss function. We discuss different formulations of the stochastic lot sizing problem, in which the quality of service is captured by means of backorder penalty costs, non-stockout probability, or fill rate constraints. These models can be easily adapted to operate in settings in which unmet demand is backordered or lost. The proposed approach has a number of advantages with respect to existing methods in the literature: it enables seamless modelling of different variants of the above problem, which have been previously tackled via ad-hoc solution methods; and it produces an accurate estimation of the expected total cost, expressed in terms of upper and lower bounds. Our computational study demonstrates the effectiveness and flexibility of our models.Comment: 38 pages, working draf

    Computing (R, S) policies with correlated demand

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    This paper considers the single-item single-stocking non-stationary stochastic lot-sizing problem under correlated demand. By operating under a nonstationary (R, S) policy, in which R denote the reorder period and S the associated order-up-to-level, we introduce a mixed integer linear programming (MILP) model which can be easily implemented by using off-theshelf optimisation software. Our modelling strategy can tackle a wide range of time-seriesbased demand processes, such as autoregressive (AR), moving average(MA), autoregressive moving average(ARMA), and autoregressive with autoregressive conditional heteroskedasticity process(AR-ARCH). In an extensive computational study, we compare the performance of our model against the optimal policy obtained via stochastic dynamic programming. Our results demonstrate that the optimality gap of our approach averages 2.28% and that computational performance is good

    Computing non-stationary (s,S) policies using mixed integer linear programming

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    This paper addresses the single-item single-stocking location stochastic lot sizing problem under the (s,S)(s, S) policy. We first present a mixed integer non-linear programming (MINLP) formulation for determining near-optimal (s,S)(s, S) policy parameters. To tackle larger instances, we then combine the previously introduced MINLP model and a binary search approach. These models can be reformulated as mixed integer linear programming (MILP) models which can be easily implemented and solved by using off-the-shelf optimisation software. Computational experiments demonstrate that optimality gaps of these models are around 0.3%0.3\% of the optimal policy cost and computational times are reasonable

    An extended mixed-integer programming formulation and dynamic cut generation approach for the stochastic lot sizing problem

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    We present an extended mixed-integer programming formulation of the stochastic lot-sizing problem for the static-dynamic uncertainty strategy. The proposed formulation is significantly more time efficient as compared to existing formulations in the literature and it can handle variants of the stochastic lot-sizing problem characterized by penalty costs and service level constraints, as well as backorders and lost sales. Also, besides being capable of working with a predefined piecewise linear approximation of the cost function-as is the case in earlier formulations-it has the functionality of finding an optimal cost solution with an arbitrary level of precision by means of a novel dynamic cut generation approach

    Stochastic lot sizing problem with controllable processing times

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    Cataloged from PDF version of article.In this study, we consider the stochastic capacitated lot sizing problem with controllable processing times where processing times can be reduced in return for extra compression cost. We assume that the compression cost function is a convex function as it may reflect increasing marginal costs of larger reductions and may be more appropriate when the resource life, energy consumption or carbon emission are taken into consideration. We consider this problem under static uncertainty strategy and α service level constraints. We first introduce a nonlinear mixed integer programming formulation of the problem, and use the recent advances in second order cone programming to strengthen it and then solve by a commercial solver. Our computational experiments show that taking the processing times as constant may lead to more costly production plans, and the value of controllable processing times becomes more evident for a stochastic environment with a limited capacity. Moreover, we observe that controllable processing times increase the solution flexibility and provide a better solution in most of the problem instances, although the largest improvements are obtained when setup costs are high and the system has medium sized capacities

    Mathematical programming for single- and multi-location non-stationary inventory control

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    Stochastic inventory control investigates strategies for managing and regulating inventories under various constraints and conditions to deal with uncertainty in demand. This is a significant field with rich academic literature which has broad practical applications in controlling and enhancing the performance of inventory systems. This thesis focuses on non-stationary stochastic inventory control and the computation of near-optimal inventory policies for single- and two-echelon inventory systems. We investigate the structure of optimal policies and develop effective mathematical programming heuristics for computing near-optimal policy parameters. This thesis makes three contributions to stochastic inventory control. The first contribution concerns lot-sizing problems controlled under a staticdynamic uncertainty strategy. From a theoretical standpoint, I demonstrate the optimality of the non-stationary (s,Q) form for the single-item single-stocking location non-stationary stochastic lot-sizing problem in a static-dynamic setting; from a practical standpoint, I present a stochastic dynamic programming approach to determine optimal (s,Q)-type policy parameters, and I introduce mixed integer non-linear programming heuristics that leverage piecewise linear approximation of the cost function. The numerical study demonstrates that the proposed solution method efficiently computes near-optimal parameters for a broad class of problem instances. The second contribution is to develop computationally efficient approaches for computing near-optimal policy parameters for the single-item single-stocking location non-stationary stochastic lot-sizing problem under the static-dynamic uncertainty strategy. I develop an efficient dynamic programming approach that, starting from a relaxed shortest-path formulation, leverages a state space augmentation procedure to resolve infeasibility with respect to the original problem. Unlike other existing approaches, which address a service-level-oriented formulation, this method is developed under a penalty cost scheme. The approach can find a near-optimal solution to any instance of relevant size in negligible time by implementing simple numerical integrations. This third contribution addresses the optimisation of the lateral transshipment amongst various locations in the same echelon from an inventory system. Under a proactive transshipment setting, I introduce a hybrid inventory policy for twolocation settings to re-distribute the stock throughout the system. The policy parameters can be determined using a rolling-horizon technique based on a twostage dynamic programming formulation and a mixed integer linear programme. The numerical analysis shows that the two-stage formulation can well approximate the optimal policy obtained via stochastic dynamic programming and that the rolling-horizon heuristic leads to tight optimality gaps
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