Willem K. Klein Haneveld

Stochastic integer programming:General models and algorithms

We survey structural properties of and algorithms for stochastic integer programmingmodels, mainly considering linear two‐stage models with mixed‐integer recourse (and theirmulti‐stage extensions).

Duality in stochastic linear and dynamic programming

1 Introduction and Summary.- 2 Mathematical Programming and Duality Theory.- 3 Stochastic Linear Programming Models.- 4 Some Linear Programs in Probabilities and Their Duals.- 5 On Integrated Chance Constraints.- 6 On The Behaviour of the Optimal Value Operator of Dynamic Programming.- 7 Robustness against Dependence in Pert.- 8 A Dual of a Dynamic Inventory Control Model: The Deterministic and the Stochastic Case.- List of Optimization Problems.

Integrated chance constraints: Reduced forms and an algorithm

We consider integrated chance constraints (ICCs), which provide quantitative alternatives for traditional chance constraints. We derive explicit polyhedral descriptions for the convex feasible sets induced by ICCs, for the case that the underlying distribution is discrete. Based on these reduced forms, we propose an efficient algorithm for this problem class.The relation to conditional value-at-risk models and (simple) recourse models is discussed, leading to a special purpose algorithm for simple recourse models with discretely distributed technology matrix. For both algorithms, numerical results are presented.

Dynamic inventory rationing strategies for inventory systems with two demand classes, Poisson demand and backordering

We study inventory systems with two demand classes (critical and non-critical), Poisson demand and backordering. We analyze dynamic rationing strategies where the number of items reserved for critical demand depends on the remaining time until the next order arrives. Different from results in the literature, we do not discretize demand but derive a set of formulae that determine the optimal rationing level for any possible value of the remaining time. Moreover, we show that the cost parameters can be captured in a single relevant dimension, which allows us to present the optimal rationing levels in charts and lookup tables that are easy to implement. Numerical examples illustrate that the optimal dynamic rationing strategy outperforms all static strategies with fixed rationing levels.

Effects of discounting and demand rate variability on the EOQ

The Economic Order Quantity (EOQ) formula is probably the most well-known formula in inventory theory. It determines the optimal value of the ordering quantity by minimizing the average cost in an ordering cycle. It is based on the assumptions that there is a fixed lead time and a constant demand rate. These assumptions are usually not fulfilled in practice. Especially when dealing with slow-moving items, demand seems to fluctuate. Furthermore, since the time between two successive orderings is often large for slow moving items, discounted costs instead of average costs should be minimized. In this paper we determine the optimal ordering quantity if demand is modelled by a Poisson process and the expected discounted costs are minimized. We also analyze the case where demand is uniform and/or the average cost criterion is used. We give graphical presentations of the optimal ordering quantities. Besides easily determining the optimal values, these allow us to show what the effects of changing the objective function and of changing the demand distribution are. We derive a simple (discounted cost) ordering quantity formula, denoted by EOQ(d), for slow moving items. We show that for items with a small demand rate and high fixed ordering costs the EOQ(d) is much closer to the optimal (discounted cost) ordering quantity than the traditional EOQ-formula

Inventory control of service parts in the final phase

We consider an appliance manufacturers problem of controlling the inventory of a service part in its final phase. That phase begins when the production of the appliance containing that part is discontinued (time 0), and ends when the last service contract on that appliance expires. Thus, the planning horizon is deterministic and known. There is no setup cost for ordering. However, if a part is not ordered at time 0, its price will be higher. The objective is to minimize the total expected undiscounted costs of replenishment, inventory holding, backorder, and disposal (of unused parts at the end of the planning horizon). We propose an ordering policy consisting of an initial order-up-to level at time 0, and a subsequent series of decreasing order-up-to levels for various intervals of the planning horizon. We present a method of calculating the optimal policy, along with a numerical example and sensitivity analysis.

On the convex hull of the simple integer recourse objective function

We consider the objective function of a simple integer recourse problem with fixed technology matrix.Using properties of the expected value function, we prove a relation between the convex hull of this function and the expected value function of a continuous simple recourse program.We present an algorithm to compute the convex hull of the expected value function in case of discrete right-hand side random variables. Allowing for restrictions on the first stage decision variables, this result is then extended to the convex hull of the objective function.

The `final orderrs problem

When the service department of a company selling machines stops producing and supplying spare parts for certain machines, customers are offered an opportunity to place a so-called final order for these spare parts. We focus on one customer with one machine. The customer plans to use this machine up to a fixed horizon. Based on this horizon, and on the failure rates of the components, the prices of spare components and the consequences of the machine failing before the critical lifetime is reached, the size of the final order is determined.

Stochastic Linear Programming Models

Linear programming has proven to be a suitable framework for the quantitative analysis of many decision problems. The reasons for its popularity are obvious: many practical problems can be modeled, at least approximately, as linear programs, and powerful software is available. Nevertheless, even if the problem has the necessary linear structure it is not sure that the linear programming approach works. One of the reasons is that the model builder must be able to provide numerical values for each of the coefficients. But in practical situations one often is not sure about the “true” values of all coefficients. Usually the uncertainty is exorcized by taking reasonable guesses or maybe by making careful estimates. In combination with a sensitivity analysis with respect to the most inaccurate coefficients this approach is satisfactory in many cases. However, if it appears that the optimal solution depends heavily on the value of some inaccurate data, it might be sensible to take the uncertainty of the coefficients into consideration in a more fundamental way. Since an evident framework for the quantitative analysis of uncertainty is provided by probability theory it seems only natural to interpret the uncertain coefficient values as realizations of random variables. This approach characterizes stochastic linear programming.

An ALM model for pension funds using integrated chance constraints

We discuss integrated chance constraints in their role of short-term risk constraints in a strategic ALM model for Dutch pension funds. The problem is set up as a multistage recourse model, with special attention for modeling short-term risk prompted by the development of new guidelines by the regulating authority for Dutch pension funds. The paper concludes with a numerical illustration of the importance of such short-term risk constraints.