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.

The first order loss function and its complementary function are extensively used in practical settings. When the random variable of interest is normally distributed, the first order loss function can be easily expressed in terms of the standard normal cumulative distribution and probability density function. However, the standard normal cumulative distribution does not admit a closed form solution and cannot be easily linearised. Several works in the literature discuss approximations for either the standard normal cumulative distribution or the first order loss function and their inverse. However, a comprehensive study on piecewise linear upper and lower bounds for the first order loss function is still missing. In this work, we initially summarise a number of distribution independent results for the first order loss function and its complementary function. We then extend this discussion by focusing first on random variable featuring a symmetric distribution, and then on normally distributed random variables. For the latter, we develop effective piecewise linear upper and lower bounds that can be immediately embedded in MILP models. These linearisations rely on constant parameters that are independent of the mean and standard deviation of the normal distribution of interest. We finally discuss how to compute optimal linearisation parameters that minimise the maximum approximation error. keywords: first order loss function; complementary first order loss function; piecewise linear approximation; minimax; Jensen's; Edmundson-Madansky.

Purpose -The purpose of this paper is to assess whether an existing sourcing strategy can effectively supply products of appropriate quality with acceptable levels of product waste if applied to an international perishable product supply chain. The authors also analyse whether the effectiveness of this sourcing strategy can be improved by including costs for expected shelf life losses while generating order policies. Design/methodology/approach -The performance of sourcing strategies is examined in a prototype international strawberry supply chain. Appropriate order policies were determined using parameters both with and without costs for expected shelf life losses. Shelf life losses during transport and storage were predicted using microbiological growth models. The performance of the resulting policies was assessed using a hybrid discrete event chain simulation model that includes continuous quality decay. Findings -The study's findings reveal that the order policies obtained with standard cost parameters result in poor product quality and large amounts of product waste. Also, including costs for expected shelf life losses in sourcing strategies significantly reduces product waste and improves product quality, although transportation costs rise.Practical implications -The study shows that in perishable product supply chain design a trade-off should be made between transportation costs, shortage costs, inventory costs, product waste, and expected shelf life losses. Originality/value -By presenting a generically applicable methodology for perishable product supply chain design, the authors contribute to research and practice efforts to reduce food waste. Furthermore, product quality information is included in supply chain network design, a research area that is still in its infancy.

We consider a class of production/inventory control problems that has a single product and a single stocking location, for which a stochastic demand with a known non-stationary probability distribution is given. Under the widely-known replenishment cycle policy the problem of computing policy parameters under service level constraints has been modeled using various techniques. Tarim and Kingsman introduced a modeling strategy that constitutes the state-of-the-art approach for solving this problem. In this paper we identify two sources of approximation in Tarim and Kingsman's model and we propose an exact stochastic constraint programming approach. We build our approach on a novel concept, global chance-constraints, which we introduce in this paper. Solutions provided by our exact approach are employed to analyze the accuracy of the model developed by Tarim and Kingsman.

In this paper we address the general multi-period production/inventory problem with non-stationary stochastic demand and supplier lead time under service-level constraints. A replenishment cycle policy (R n ,S n ) is modeled, where R n is the n-th replenishment cycle length and S n is the respective order-up-to-level. We propose a Stochastic Constraint Programming approach for computing the optimal policy parameters. In order to do so, a dedicated global chance-constraint and the respective filtering algorithm that enforce the required service level are presented. Our numerical examples show that a stochastic supplier lead time significantly affects the structure of the optimal policy with respect to the case in which the lead time is assumed to be deterministic or absent.