S. Armagan Tarim

Synthesizing filtering algorithms for global chance-constraints

Stochastic Constraint Satisfaction Problems (SCSPs) are a powerful modeling framework for problems under uncertainty. To solve them is a P-Space task. The only solution approach to date compiles down SCSPs into classical CSPs. This allows the reuse of classical constraint solvers to solve SCSPs, but at the cost of increased space requirements and weak constraint propagation. This paper tries to overcome some of these drawbacks by automatically synthesizing filtering algorithms for global chance-constraints. These filtering algorithms are parameterized by propagators for the deterministic version of the chance-constraints. This approach allows the reuse of existing propagators in current constraint solvers and it enhances constraint propagation. Experiments show the benefits of this novel approach.

The stochastic dynamic production/inventory lot-sizing problem with service-level constraints

Abstract This paper addresses the multi-period single-item inventory lot-sizing problem with stochastic demands under the “static–dynamic uncertainty” strategy of Bookbinder and Tan (Manage. Sci. 34 (1988) 1096). In the static-dynamic uncertainty strategy, the replenishment periods are fixed at the beginning of the planning horizon, but the actual orders are determined only at those replenishment periods and will depend upon the demand that is realised. Their solution heuristic was a two-stage process of firstly fixing the replenishment periods and then secondly determining what adjustments should be made to the planned orders as demand was realised. We present a mixed integer programming formulation that determines both in a single step giving the optimal solution for the “static–dynamic uncertainty” strategy. The total expected inventory holding, ordering and direct item costs during the planning horizon are minimised under the constraint that the probability that the closing inventory in each time period will not be negative is set to at least a certain value. This formulation includes the effect of a unit variable purchase/production cost, which was excluded by the two-stage Bookbinder–Tan heuristic. An evaluation of the accuracy of the heuristic against the optimal solution for the case of a zero unit purchase/production cost is made for a wide variety of demand patterns, coefficients of demand variability and relative holding cost to ordering cost ratios. The practical constraint of non-negative orders and the existence of the unit variable cost mean that the replenishment cycles cannot be treated independently and so the problem cannot be solved as a stochastic form of the Wagner–Whitin problem, applying the shortest route algorithm.

Stochastic Constraint Programming: A Scenario-Based Approach

To model combinatorial decision problems involving uncertainty and probability, we introduce scenario based stochastic constraint programming. Stochastic constraint programs contain both decision variables, which we can set, and stochastic variables, which follow a discrete probability distribution. We provide a semantics for stochastic constraint programs based on scenario trees. Using this semantics, we can compile stochastic constraint programs down into conventional (non-stochastic) constraint programs. This allows us to exploit the full power of existing constraint solvers. We have implemented this framework for decision making under uncertainty in stochastic OPL, a language which is based on the OPL constraint modelling language [Van Hentenryck et al., 1999]. To illustrate the potential of this framework, we model a wide range of problems in areas as diverse as portfolio diversification, agricultural planning and production/inventory management.

The temporal knapsack problem and its solution

This paper introduces a problem called the temporal knapsack problem, presents several algorithms for solving it, and compares their performance. The temporal knapsack problem is a generalisation of the knapsack problem and specialisation of the multidimensional (or multiconstraint) knapsack problem. It arises naturally in applications such as allocating communication bandwidth or CPUs in a multiprocessor to bids for the resources. The algorithms considered use and combine techniques from constraint programming, artificial intelligence and operations research.

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

In this paper, we develop a unified mixed integer linear modelling approach to compute near-optimal policy parameters for the non-stationary stochastic lot sizing problem under static–dynamic uncertainty strategy. The proposed approach applies to settings in which unmet demand is backordered or lost; and it can accommodate variants of the problem for which the quality of service is captured by means of backorder penalty costs, non-stockout probabilities, or fill rate constraints. This approach has a number of advantages with respect to existing methods in the literature: it enables seamless modelling of different variants of the stochastic lot sizing problem, some of which have been previously tackled via ad hoc solution methods and some others that have not yet been addressed in the literature; and it produces an accurate estimation of the expected total cost, expressed in terms of upper and lower bounds based on piecewise linearisation of the first order loss function. We illustrate the effectiveness and flexibility of the proposed approach by means of a computational study.

Confidence-based optimisation for the newsvendor problem under binomial, Poisson and exponential demand

We consider the problem of controlling the inventory of a single item with stochastic demand over a single period. Most of the research on single-period inventory models has focused on the case in which demand distribution parameters are known. Nevertheless, it is clear that the applicability of these models directly depends on the accuracy of demand parameters estimation. In this work, we introduce a novel strategy to address the issue of demand estimation in single-period inventory optimization problems. Our strategy is based on the theory of statistical estimation. We assume that the decision maker is given a set of past demand samples and we employ confidence interval analysis in order to identify a range of candidate order quantities that, with prescribed confidence probability, includes the real optimal order quantity for the underling stochastic demand process with unknown parameter. In addition, for each candidate order quantity that is identified, our approach can produce an upper and a lower bound for the associated cost. We apply our novel approach to three demand distribution in the exponential family: Binomial, Poisson, and Exponential. For two of these distributions we also discuss the extension to the case of unobserved lost sales. Numerical examples are presented in which we show how our approach complements existing strategies based on maximum likelihood estimators or on Bayesian analysis. keywords: inventory control; newsvendor problem; sampling; confidence interval analysis; demand estimation.We introduce a novel strategy to address the issue of demand estimation in single-item single-period stochastic inventory optimisation problems. Our strategy analytically combines confidence interval analysis and inventory optimisation. We assume that the decision maker is given a set of past demand samples and we employ confidence interval analysis in order to identify a range of candidate order quantities that, with prescribed confidence probability, includes the real optimal order quantity for the underlying stochastic demand process with unknown stationary parameter(s). In addition, for each candidate order quantity that is identified, our approach produces an upper and a lower bound for the associated cost. We apply this approach to three demand distributions in the exponential family: binomial, Poisson, and exponential. For two of these distributions we also discuss the extension to the case of unobserved lost sales. Numerical examples are presented in which we show how our approach complements existing frequentist—e.g. based on maximum likelihood estimators—or Bayesian strategies.

Production, Manufacturing and LogisticsConfidence-based optimisation for the newsvendor problem under binomial, Poisson and exponential demand

We consider the problem of controlling the inventory of a single item with stochastic demand over a single period. Most of the research on single-period inventory models has focused on the case in which demand distribution parameters are known. Nevertheless, it is clear that the applicability of these models directly depends on the accuracy of demand parameters estimation. In this work, we introduce a novel strategy to address the issue of demand estimation in single-period inventory optimization problems. Our strategy is based on the theory of statistical estimation. We assume that the decision maker is given a set of past demand samples and we employ confidence interval analysis in order to identify a range of candidate order quantities that, with prescribed confidence probability, includes the real optimal order quantity for the underling stochastic demand process with unknown parameter. In addition, for each candidate order quantity that is identified, our approach can produce an upper and a lower bound for the associated cost. We apply our novel approach to three demand distribution in the exponential family: Binomial, Poisson, and Exponential. For two of these distributions we also discuss the extension to the case of unobserved lost sales. Numerical examples are presented in which we show how our approach complements existing strategies based on maximum likelihood estimators or on Bayesian analysis. keywords: inventory control; newsvendor problem; sampling; confidence interval analysis; demand estimation.We introduce a novel strategy to address the issue of demand estimation in single-item single-period stochastic inventory optimisation problems. Our strategy analytically combines confidence interval analysis and inventory optimisation. We assume that the decision maker is given a set of past demand samples and we employ confidence interval analysis in order to identify a range of candidate order quantities that, with prescribed confidence probability, includes the real optimal order quantity for the underlying stochastic demand process with unknown stationary parameter(s). In addition, for each candidate order quantity that is identified, our approach produces an upper and a lower bound for the associated cost. We apply this approach to three demand distributions in the exponential family: binomial, Poisson, and exponential. For two of these distributions we also discuss the extension to the case of unobserved lost sales. Numerical examples are presented in which we show how our approach complements existing frequentist—e.g. based on maximum likelihood estimators—or Bayesian strategies.

Cost-Based Filtering Techniques for Stochastic Inventory Control Under Service Level Constraints

This paper1 considers a single product and a single stocking location production/inventory control problem given a non-stationary stochastic demand. Under a widely-used control policy for this type of inventory system, the objective is to find the optimal number of replenishments, their timings and their respective order-up-to-levels that meet customer demands to a required service level. We extend a known CP approach for this problem using three cost-based filtering methods. Our approach can solve to optimality instances of realistic size much more efficiently than previous approaches, often with no search effort at all.

Mean-based error measures for intermittent demand forecasting

To compare different forecasting methods on demand series, we require an error measure. Many error measures have been proposed, but when demand is intermittent some become inapplicable because of infinities, some give counter-intuitive results, and there is no agreement on which is best. We argue that almost all known measures rank forecasters incorrectly on intermittent demand series. We propose several new error measures with almost no infinities, and with correct forecaster ranking on several intermittent demand patterns. We call these ‘mean-based’ error measures because they evaluate forecasts against the (possibly time-dependent) mean of the underlying stochastic process instead of point demands.

Learning market prices in real-time supply chain management

This paper proposes a model for dynamic pricing that combines knowledge of production capacity and existing commitments, reasoning about uncertainty and learning of market conditions in an attempt to optimise expected profits. In particular, the market conditions are represented as a set of probabilities over the success rate of product prices, and those prices are learned online as the market develops. The dynamic pricing model is integrated into a real-time supply chain management agent using the Trading Agent Competition Supply Chain Management game as a test framework. We evaluate the agent experimentally in competition with other supply chain agents, and demonstrate the benefits of incorporating more market data into the dynamic pricing mechanism.