Integrated Forecasting and Inventory Control for Seasonal Demand: A Comparison with the Holt-Winters Approach Gokhan Metan∗
Aur´elie Thiele†
December 2007
Abstract We present a data-driven forecasting technique with integrated inventory control for seasonal data and compare it to the traditional Holt-Winters algorithm. Results indicate that the datadriven approach achieves a 2-5% improvement in the average regret. Keywords: data-driven algorithm, Holt-Winters algorithm, inventory management.
1
Introduction
Forecasting and optimization have traditionally been approached as two distinct, sequential components of inventory management: the random demand is first estimated using historical data, then this forecast (either a point forecast of the future demand or a forecast of the distribution) is used as input to the optimization module. In particular, the primary objective of time series analysis is to develop mathematical models that explain past data; these models are used in making forecasting decisions where the goal is to predict the next period’ observation as precisely as possible. To achieve this goal, demand model parameters are estimated or a distribution is fitted to the data using a performance metric such as Mean Square Error, which penalizes overestimating and underestimating the demand equally. In practice, however, the optimization model penalizes under- and over-predictions unequally, e.g., in inventory problems backorder is viewed as particularly undesirable while holding inventory is more tolerated. In such a setting, the decision-maker places an order in each time period based on the demand prediction coming from the forecasting model, but the prediction of the forecasting model does not take into account the nature of the penalties in the optimization process and instead minimizes the (symmetric) error between the forecasts and the actual data points. ∗
Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA 18015,
[email protected]. Work supported in part by NSF Grant DMI-0540143. † Department of Industrial and Systems Engineering, Lehigh University, 200 W Packer Ave, Bethlehem, PA 18015, USA,
[email protected]. Corresponding author.
1
In this paper, we investigate the integration of the forecasting and inventory control decisions; in particular, our focus is on comparing the performance of this approach with the traditional HoltWinters algorithm for random demand with a seasonal trend. The goal is not longer to predict future observations as accurately as possible using a problem-independent metric, but to blend the inventory control principles into the analysis to achieve superior inventory management. This work adds to the growing body of literature on data-driven inventory management (Godfrey and Powell 2001, Bertsimas and Thiele 2004, Levi et. al. 2006) by focusing on cyclical demand, and comparing the performance of a novel algorithm based on the clustering of data points (Metan and Thiele 2008) with that of the traditional Holt-Winters approach. Cluster creation and recombination allows the decision-maker to place his order at each time period based only on the most relevant data. To the best of our knowledge, we are the first authors to propose a clustering approach to the data-driven inventory management problem. As pointed out in Metan and Thiele (2008), customer behavior exhibits cyclical trends in many logistics applications where the influence of exogenous drivers is difficult to quantify accurately, which makes the approach particularly appealing. The proposed methodology captures the tradeoff between the various cost drivers and provides the decision-maker with the optimal order-up-to levels, rather than the projected demand. A cost decrease of 2-5% in experiments suggests that inventory managers could greatly benefit from implementing this approach. The rest of the paper is organized as follows. In Section 2, we describe the methodology for integrated forecasting and inventory control in a data-driven framework. We present simulation results and compare the performance of the proposed approach with the traditional Holt-Winters algorithm in Section 3.
2
Integrated data-driven forecasting and inventory control
In this paper, our objective is to determine the optimal order for the newsvendor problem under cyclical demand using the integrated approach; however, the method can extended to other problem structures. The data-driven algorithm seeks to differentiate between the deterministic seasonal effect and the stochastic variability (see Figure 1) so that cycle stock and safety stock levels can be set accurately. We use the following notation: Cost parameters:
2
Figure 1: Differentiation of cyclic behavior and stochastic variability.
c:
the unit ordering cost,
p:
the unit selling price,
s:
the salvage value,
cu :
undershoot cost (cu = p − c),
co :
overshoot cost (co = c − s),
α:
critical ratio (α =
cu cu +co ).
Demand parameters and decision variable: Q: the order quantity, D:
the demand level,
φ(.):
pdf of standard normal distribution,
Φ(.):
cdf of standard normal distribution,
S:
the set of data points obtained from previous observations,
N:
the number of historical observations in S,
di :
the ith demand observation in set S, with i = 1, 2, . . . , N ,
d :
the ith smallest demand in set S, with i = 1, 2, . . . , N ,
Additional parameters required for algorithm definition: T : periodicity of time series, Cj :
list of data points assigned to cluster j, j = 1, . . . , T , ranked in increasing order,
m:
number of data points assigned to each cluster. 3
The methodology that we propose here is a dynamic and data-driven approach that builds directly upon the historical observations and addresses seasonality by creating and recombining clusters of past demand points. Our description follows Metan and Thiele (2008) closely, although at a less technical level, since our purpose here is not to demonstrate the algorithm’s absolute performance but its relative performance with respect to the Holt-Winters method. Clustering can be defined as grouping a set of instances into a given number of subsets, called clusters, so that instances in the same cluster share similar characteristics. Most research efforts in clustering focus on developing efficient algorithms, in particular in the context of marketing applications and customer data (Jain et. al. 1999), which are beyond the scope of this paper. We only mention here the method of k-means, which was originally developed by MacQueen (1967) and is one of the simplest clustering approaches available in the literature: it partitions the data into k clusters by minimizing the sum of squared differences of objects to the centroid or mean of the cluster. The objective of minimizing total intra-cluster variability can be formulated as: min
Pk
i=1
P
j∈Si
|xj − µi |2 , where there are k
clusters Si , i = 1, 2, ..., k and µi is the mean point of all the instances xj in Si . This method presents similarities with the clustering approach we propose; in particular, we sort the demand and assign the data points to clusters in a manner that also attempts to minimize intra-cluster variability. A difference is that, while the k-means algorithm proceeds iteratively by updating the centroids of each cluster but keeps the same number of clusters throughout, we update the number of clusters by merging subsets as needed. Techniques for cluster aggregation/disaggregation are not discussed in this note; the reader is referred to Metan and Thiele (2008) for more details. The master algorithm consists of five main steps, as described in Algorithm 2.1. For simplicity, we assume that the periodicity of the time series is known to the master algorithm; for settings where the true periodicity is unknown, an initial period estimation subroutine can be used to initiate the master algorithm. Algorithm 2.1 (Master algorithm for cyclical demand (Metan and Thiele 2008)) b Step 1. Sort the historical demand data in set S in ascending order to form the list S.
Step 2. Create T data clusters using Sb (Algorithm 2.2). repeat (until the end of the planning horizon) Step 3. Detect the phase ϕ and the corresponding cluster Cj of the demand at the next time period (j = 1, 2, . . . , T ). Step 4. Select the next order level using inventory control policy π defined over the set Cj . Step 5. Assign the new demand observation to appropriate cluster Cj (j = 1, 2, . . . , T ). Update the phase for the next decision point (ϕ ←− (ϕ + 1) mod T ) end repeat. Category and objective of each step of the master algorithm is summarized in Table 1. In Step 4
b which is then used by 1 of Algorithm 2.1 the historical data is sorted and stored in the list S,
Algorithm 2.2 to create the initial clusters (Step 2). Algorithm 2.2 creates as many clusters as the periodicity of the seasonal demand function, i.e., one cluster for each phase of the seasonality, and assigns m =
j k N T
data points to each cluster, with the N − mT oldest data points being discarded.
Alternatively, the last cluster could receive more points than the others; for the sake of clarity, we present here the version where all the clusters have the same number of points. Table 1: Categorization of master algorithm’s steps. Activity Category Forecasting Inventory control
Objective Initialize forecasting method Forecast cyclic effect Update forecasts Make optimal ordering decision given the forecast
Steps Steps 1 & 2 Step 3 Step 5 Step 4
The output of Step 2 is a set of clusters {Cj |j = 1, 2, . . . , T } and each cluster can be thought of as a set that defines a group of possible forecasts corresponding to a given phase of the time series. The algorithm selects such a set Cj among the available sets {Cj |j = 1, 2, . . . , T } in Step 3, which is equivalent to defining a set of predictions rather than having a single point forecast. Then, Step 4 concludes the decision-making process by selecting the order-up-to level using inventory control policy π : Cj →
