Wilco van den Heuvel

Dynamic lot sizing with product returns and remanufacturing

We address the dynamic lot sizing problem for systems with product returns. The demand and return amounts are deterministic over the finite planning horizon. Demands can be satisfied by manufactured new items, but also by remanufactured returned items. The objective is to determine those lot sizes for manufacturing and remanufacturing that minimize the total cost composed of holding cost for returns and (re)manufactured products and set-up costs. Two different set-up cost schemes are considered: there is either a joint set-up cost for manufacturing and remanufacturing (single production line) or separate set-up costs (dedicated production lines). For the joint set-up cost case, we present an exact, polynomial-time dynamic programming algorithm. For both cases, we suggest modifications of the well-known Silver Meal (SM), Least Unit Cost (LUC) and Part Period Balancing (PPB) heuristics. An extensive numerical study reveals a number of insights. The key ones are that, under both set-up cost schemes: (1) the SM and LUC heuristics perform much better than PPB, (2) increased variation in the demand amounts can lead to reduced cost, showing that predictability is more important than variation, and (3) periods with more returns than demand should, if possible, be avoided by ‘matching’ demand and return.

Economic Lot-Sizing Games

In this paper we introduce a new class of OR games: economic lot-sizing (ELS) games. There are a number of retailers that have a known demand for a fixed number of periods. To satisfy demand the retailers order products at the same manufacturer. By placing joint orders instead of individual orders, costs can be reduced and a cooperative game arises. In this paper we show that ELS games are balanced. Furthermore, we show that two special classes of ELS games are concave.

The economic lot-sizing problem with an emission capacity constraint

We consider a generalisation of the lot-sizing problem that includes an emission capacity constraint. Besides the usual financial costs, there are emissions associated with production, keeping inventory and setting up the production process. Because the capacity constraint on the emissions can be seen as a constraint on an alternative objective function, there is also a clear link with bi-objective optimisation. We show that lot-sizing with an emission capacity constraint is NP-hard and propose several solution methods. Our algorithms are not only able to handle a fixed-plus-linear cost structure, but also more general concave cost and emission functions. First, we present a Lagrangian heuristic to provide a feasible solution and lower bound for the problem. For costs and emissions such that the zero inventory property is satisfied, we give a pseudo-polynomial algorithm, which can also be used to identify the complete set of Pareto optimal solutions of the bi-objective lot-sizing problem. Furthermore, we present a fully polynomial time approximation scheme (FPTAS) for such costs and emissions and extend it to deal with general costs and emissions. Special attention is paid to an efficient implementation with an improved rounding technique to reduce the a posteriori gap, and a combination of the FPTASes and a heuristic lower bound. Extensive computational tests show that the Lagrangian heuristic gives solutions that are very close to the optimum. Moreover, the FPTASes have a much better performance in terms of their actual gap than the a priori imposed performance, and, especially if the heuristic’s lower bound is used, they are very fast.

A polynomial time algorithm for a deterministic joint pricing and inventory model

In this paper we consider the uncapacitated economic lot-size model, where demand is a deterministic function of price. In the model a single price need to be set for all periods. The objective is to find an optimal price and ordering decisions simultaneously. In 1973 Kunreuther and Schrage proposed an heuristic algorithm to solve this problem. The contribution of our paper is twofold. First, we derive an exact algorithm to determine the optimal price and lot-sizing decisions. Moreover, we show that our algorithm boils down to solving a number of lot-sizing problems that is quadratic in the number of periods, i.e., the problem can be solved in polynomial time.

Economic lot-sizing with remanufacturing: Complexity and efficient formulations

Within the framework of reverse logistics, the classic economic lot-sizing problem has been extended with a remanufacturing option. In this extended problem, known quantities of used products are returned from customers in each period. These returned products can be remanufactured so that they are as good as new. Customer demand can then be fulfilled from both newly produced and remanufactured items. In each period, one can choose to set up a process to remanufacture returned products or produce new items. These processes can have separate or joint setup costs. In this article, it is shown that both variants are NP-hard. Furthermore, several alternative mixed-integer programming (MIP) formulations of both problems are proposed and compared. Because “natural” lot-sizing formulations provide weak lower bounds, tighter formulations are proposed, namely, shortest path formulations, a partial shortest path formulation, and an adaptation of the (l, S, WW) inequalities used in the classic problem with Wagner–Whitin costs. Their efficiency is tested on a large number of test data sets and it is found that, for both problem variants, a (partial) shortest path–type formulation performs better than the natural formulation, in terms of both the linear programming relaxation and MIP computation times. Moreover, this improvement can be substantial.

A comparison of methods for lot-sizing in a rolling horizon environment

We argue that the superior performance of a recent method for lot-sizing in a rolling horizon scheme is to a large extent due to the assumption that quite accurate future demand estimates are available. We show that other methods, including a straightforward one, can use this information just as effectively.

Four equivalent lot-sizing models

textabstractWe study the following lot-sizing models that recently appeared in the literature: a lot-sizing model with a remanufacturing option, a lot-sizing model with production time windows, and a lot-sizing model with cumulative capacities. We show the equivalence of these models with a classical model: the lot-sizing model with inventory bounds.

Worst-Case Analysis for a General Class of Online Lot-Sizing Heuristics

In this paper, we analyze the worst-case performance of heuristics for the classical economic lot-sizing problem with time-invariant cost parameters. We consider a general class of online heuristics that is often applied in a rolling-horizon environment. We develop a procedure to systematically construct worst-case instances for a fixed time horizon and use it to derive worst-case problem instances for an infinite time horizon. Our analysis shows that any online heuristic has a worst-case ratio of at least 2.

Note on “An efficient approach for solving the lot-sizing problem with time-varying storage capacities”

In a recent paper Gutierrez et al. (2008) show that the lot-sizing problem with inventory bounds can be solved in O(T log T) time. In this note we show that their algorithm does not lead to an optimal solution in general.

Integrated market selection and production planning: complexity and solution approaches

Emphasis on effective demand management is becoming increasingly recognized as an important factor in operations performance. Operations models that account for supply costs and constraints as well as a supplier’s ability to influence demand characteristics can lead to an improved match between supply and demand. This paper presents a class of optimization models that allow a supplier to select, from a set of potential markets, those markets that provide maximum profit when production/procurement economies of scale exist in the supply process. The resulting optimization problem we study possesses an interesting structure and we show that although the general problem is