Minimal Forecast Horizons Research Articles

Forecast and rolling horizons under demand substitution and production changeovers: analysis and insights

For most multi-period decision-making problems, it is generally well accepted that the influence of information about later periods on the optimal decision in the current period reduces as we move farther into the future. If and when this influence reduces to zero, the corresponding problem horizon is referred to as a forecast horizon. For real businesses, the problem of obtaining a minimal forecast horizon becomes relevant because the task of estimating reliable data for future periods gets progressively more challenging and expensive. This article investigates forecast horizons for a two-product dynamic lot-sizing model under (i) the possibility of substitution in one direction; that is, one product can be used to satisfy the demand of the other product but not vice versa; and (ii) a changeover cost when production switches from one product to the other. It is assumed that only one of the two products can be produced in a period. The notion of substitution, due to the inherent flexibility it offers, has recently been recognized as an effective tool to improve the efficiency of multi-product inventory systems. The concept of regeneration points is used to justify the use of a practically relevant restricted version of the problem to obtain forecast horizons. A dynamic programming-based polynomial-time algorithm for the restricted version is developed and, subsequently, an efficient procedure for obtaining minimal forecast horizons by establishing the monotonicity of the regeneration points is obtained. Using a comprehensive test bed of instances, useful insights are obtained on the impact of substitution and production changeovers on the length of the minimal forecast horizons. Finally, for infinite-horizon problems, a practical rolling-horizon procedure is developed that uses forecasting costs to balance the benefit of additional information. It is shown that, instead of fixing the duration of the rolling horizon at a predetermined value, changing it dynamically based on the lengths of the minimal forecast horizons can significantly reduce the combined production and forecasting cost. [Supplemental materials are available for this article. Go to the publisher’s online edition of IIE Transactions to view the supplemental file.]

Forecast and Rolling Horizons Under Demand Substitution and Production Changeovers: Analysis and Insights

For most multi-period decision-making problems, it is generally well-accepted that the influence of information about later periods on the optimal decision in the current period reduces as we move farther into the future. If and when this influence reduces to zero, the corresponding problem horizon is referred to as a forecast horizon. For real businesses, the problem of obtaining a minimal forecast horizon becomes relevant because the task of estimating reliable data for future periods gets progressively challenging and expensive. We investigate forecast horizons for a two-product dynamic lot-sizing problem under (a) the possibility of substitution in one direction; that is, one product can be used to satisfy the demand of the other product but not vice-versa and (b) a changeover cost when production switches from one product to the other. The notion of substitution, due to the inherent flexibility it offers, has recently been recognized as an effective tool to improve the efficiency of multi-product inventory systems. Using the concept of regeneration points, we first justify the use of a practically-relevant restricted version of the problem for obtaining forecast horizons. Next, we develop a dynamic programming-based polynomial-time algorithm for the restricted version and, subsequently, an efficient procedure for obtaining minimal forecast horizons by establishing the monotonicity of the regeneration points. Using a comprehensive test bed of instances, we obtain useful insights on the impact of substitution and production changeovers on the length of the minimal forecast horizons. Finally, for infinite-horizon problems, we develop a practical rolling-horizon procedure that uses forecasting costs to balance the benefit of additional information. We show that, instead of fixing the duration of rolling horizon at a pre-determined value, changing it dynamically based on the lengths of the minimal forecast horizons can significantly reduce the combined production and forecasting cost.

Discrete forecast horizons for two-product variants of the dynamic lot-size problem

Motivated by the recent success of integer programming based procedures for computing discrete forecast horizons, we consider two-product variants of the classical dynamic lot-size model. In the first variant, we impose a warehouse capacity constraint on the total ending inventory of the two products in any period. In the second variant, the two products have both individual and joint setup costs for production. To our knowledge, there are no known procedures for computing forecast horizons for these variants. Under the assumption that future demands are discrete, we characterize forecast horizons for these two variants as feasibility/optimality questions in 0–1 mixed integer programs. A detailed computational study establishes the effectiveness of our approach and enables us to gain valuable insights into the behavior of minimal forecast horizons.

Forecast Horizons in the Discounted Dynamic Lot Size Model

We derive a sharp upper bound on the minimal forecast horizon in the discounted dynamic lot size model with constant initial demand. This bound is given by m(m 1), where m is the EOQ's worth, i.e., the number of periods for which the total demand equals Economic Order Quantity. Our results do not require the solution of the infinite horizon problem to be unique. Nor do they require the infinite horizon problem to be well defined. We also prove some sensitivity results with respect to the discount factor and the setup cost.

Forecast Horizons for a Class of Dynamic Lot-Size Problems Under Discrete Future Demand

We present structural and computational investigations of a new class of weak forecast horizons—minimal forecast horizons under the assumption that future demands are integer multiples of a given positive real number—for a specific class of dynamic lot-size (DLS) problems. Apart from being appropriate in most practical instances, the discreteness assumption offers a significant reduction in the length of a minimal forecast horizon over the one using the classical notion of continuous future demands. We provide several conditions under which a discrete-demand forecast horizon is also a continuous-demand forecast horizon. We also show that the increase in the cost resulting from using a discrete minimal forecast horizon instead of the classical minimal forecast horizon is modest. The discreteness assumption allows us to characterize forecast horizons as feasibility/optimality questions in 0-1 mixed-integer programs. On an extensive test bed, we demonstrate the computational tractability of the integer programming approach. Owing to its prevalence in practice, our computational experiments emphasize the special case of integer future demands.

Computing Minimal Forecast Horizons:An Integer Programming Approach

In this paper, we use integer programming (IP) to compute minimal forecast horizons for the classical dynamic lot-sizing problem (DLS). As a solution approach for computing forecast horizons, integer programming has been largely ignored by the research community. It is our belief that the modelling and structural advantages of the IP approach coupled with the recent significant developments in computational integer programming make for a strong case for its use in practice. We formulate some well-known sufficient conditions, and necessary and sufficient conditions (characterizations) for forecast horizons as feasibility/optimality questions in 0–1 mixed integer programs. An extensive computational study establishes the effectiveness of the proposed approach.

Solving Nonstationary Infinite Horizon Dynamic Optimization Problems

Monotonicity of optimal solutions to finite horizon dynamic optimization problems is used to prove the existence of a forecast horizon, i.e., a long enough planning horizon that ensures that a first-period optimal action for the infinite horizon and the finite horizon problem agree, regardless of problem parameter changes in the tail. The existence of extremal monotone optimal solutions motivates a stopping rule that is ensured to detect the minimal forecast horizon to be used in a rolling horizon procedure to exactly solve the problem.

Learning in Setups: Analysis, Minimal Forecast Horizons, and Algorithms

We analyze the dynamic lot-sizing model in which the cost of a setup depends on the number of setups that have occurred prior to it. This arises, for example, when there exist learning effects in setups. Our model is more general than most learning models in the literature since it allows the total setup cost to be a general nondecreasing (but not necessarily concave) function of the number of setups. We explore tight relationships between our model and special cases of the classical dynamic lot-sizing model. On the basis of these we find minimal forecast and planning horizons for our model, which determine the first decision when the model is solved on a rolling horizon basis. When a forecast horizon cannot be found, we provide guidelines regarding the optimal first decision. We also provide an algorithm to solve the finite horizon problem, which uses as sub-problems variations of the classical dynamic lot-sizing problem. The advantage of this approach is the ability to use the extensive literature available on the latter, to generalize the results of this paper. As many of our results are qualitative in nature, they provide insights which can be useful for other models with a similar setup cost behavior.

Detection of minimal forecast horizons in dynamic programs with multiple indicators of the future

Many sequential planning problems can be represented as a shortest path problem in an acyclic network. This includes all deterministic dynamic programs as well as certain stochastic sequential decision problems. In this article, we identify a large class of shortest path problems for which a general efficient algorithm for the simultaneous solution and detection of minimal forecast horizons is developed. Detection of a such minimal forecast horizons is essential when accurate information regarding various relevant parameters is obtained progressively, i.e., when the initial information is restricted to a limited horizon of “future” stages only. We describe five classes of planning problems which can be efficiently addressed by the general algorithm. These classes deal with multi-item joint replenishment systems, combined inventory and routing problems, machine scheduling issues, single item stochastic inventory settings and routing problems in the plane and in space. © 1996 John Wiley & Sons, Inc.

Detection of minimal forecast horizons in dynamic programs with multiple indicators of the future

Detection of minimal forecast horizons in dynamic programs with multiple indicators of the future

Detection of minimal forecast horizons in dynamic programs with multiple indicators of the future

Detection of minimal forecast horizons in dynamic programs with multiple indicators of the future

Fast Solution and Detection of Minimal Forecast Horizons in Dynamic Programs with a Single Indicator of the Future: Applications to Dynamic Lot-Sizing Models

In most dynamic planning problems, one observes that an optimal decision at any given stage depends on limited information, i.e., information pertaining to a limited set of adjacent or nearby stages. This holds in particular for planning problems over time, where an optimal decision in a given period depends on information related to a limited future time horizon, a so-called forecast horizon, only. In this paper we identify a general class of dynamic programs in which an efficient forward algorithm can be designed to solve the problem and to identify minimal forecast horizons. Such a procedure specifies necessary and sufficient conditions for a stage to arise as a forecast horizon. This class of dynamic programs includes the single-item dynamic lot-sizing model with general concave costs, both with and without backlogging, to which special attention is given.

Minimal Forecast Horizons and a New Planning Procedure for the General Dynamic Lot Sizing Model: Nervousness Revisited

We show for the general dynamic lot sizing model how minimal forecast horizons may be detected by a slight adaptation of an earlier O(n log n) or O(n) forward solution method for the model. A detailed numerical study indicates that minimal forecast horizons tend to be small, that is, include a small number of orders. We describe a new planning approach to ensure stability of the lot sizing decisions over an initial interval of time or stability horizon in those (relatively rare) cases where no planning horizon is detected or where the stability horizon extends beyond the planning horizon. To this end, we develop a heuristic, but full horizon-based adaptation of the optimal lot sizing schedule, designed to minimize an upper bound for the worst-case optimality gap under the desired stability conditions. We also show how the basic horizon length n may be chosen to guarantee any prespecified positive optimality gap.

The dynamic lot-sizing model with backlogging: A simpleo(n logn) algorithm and minimal forecast horizon procedure

We develop a simple O(n log n) solution method for the standard lot-sizing model with backlogging and a study horizon of n periods. Production costs are fixed plus linear and holding and backlogging costs are linear with general time-dependent parameters. The algorithm has linear [O(n)] time complexity for several important subclasses of the general model. We show how a slight adaptation of the algorithm can be used for the detection of a minimal forecast horizon and associated planning horizon. The adapted algorithm continues to have complexity O(n log n) or O(n) for the above-mentioned subclasses of the general model. © 1993 John Wiley & Sons, Inc.

Forecast Horizons in the Discounted Dynamic Lot Size Model

We derive a sharp upper bound on the minimal forecast horizon in the discounted dynamic lot size model with constant initial demand. This bound is given by m(m + 1), where m is the EOQ's worth, i.e., the number of periods for which the total demand equals Economic Order Quantity. Our results do not require the solution of the infinite horizon problem to be unique. Nor do they require the infinite horizon problem to be well defined. We also prove some sensitivity results with respect to the discount factor and the setup cost.

Minimal forecast horizons in equipment replacement models with multiple technologies and general switching costs

This article considers the problem of equipment replacement in which the replacement decision at a particular time must take into account (i) the state of the existing machine in use, (ii) the available replacement alternatives at the time, (iii) the future advances in the relevant technologies with regard to the equipment under consideration, and (iv) costs of switching between different technologies. A methodology that attains minimal forecast horizons for the problem is developed. A numerical example illustrates the methodology.

Minimal forecast horizon procedures for dynamic lot size models

AbstractThis article presents new results which should be useful in finding production decisions while solving the dynamic lot sizing problem of Wagner–Whitin on a rolling horizon basis. In a rolling horizon environment, managers obtain decisions for the first period (or the first few periods) by looking at the forecasts for several periods. This article develops procedures to find optimal decisions for any specified number of initial periods (called planning horizon in the article) by using the forecast data for the minimum possible number of future periods. Computational results comparing these procedures with the other procedures reported in the literature are very encouraging.