The value of aggregate service levels in stochastic lot sizing problems☆
Introduction
In dynamic lot sizing problems, proper inventory control and production decisions are crucial to achieve a balance between customer demand satisfaction and cost management. While insufficient inventory will lead to shortages, unnecessary stocks will increase the holding cost. An inventory holding cost is charged for the quantity being stored at the end of each period. Furthermore, in each period in which production occurs, a setup has to be performed which incurs a fixed setup cost. The basic lot sizing problem hence considers the trade-off between setup costs and inventory holding costs. The goal of the standard lot sizing problem is to determine the optimal timing and production quantities in order to satisfy a known demand over a finite and discrete time horizon [16]. The lot sizing problem has been extended to include several practical cases such as multiple products, capacitated machines, or backlog costs [10].
While the standard assumption in lot sizing problems is that all the parameters are deterministic, it is inevitable that some parameters are actually uncertain in practice. From a practical point of view, even a small level of uncertainty may heavily affect the nominal solution determined by a deterministic model and make it infeasible or more costly than anticipated [2]. To deal with the uncertainty in demand, safety stock levels are usually predetermined for each item under strict assumptions such as stationary demand, normality, as well as the independence of demand. The decisions resulting from models that do not incorporate uncertainty are known to be sub-optimal compared to the solution of the models in which the uncertainty has explicitly been taken into account [15]. Consequently, there is a need to have methods to mitigate the risk of uncertainty and simultaneously determine the time-dependent lot size and buffer stock decisions in the dynamic lot sizing problem.
The stochastic lot sizing problem is an extension of the deterministic case in which the problem is to determine the production schedules and quantities to satisfy stochastic demand over a finite planning horizon. In the context where the planner must ensure that a service level is satisfied, the objective is to minimize the total expected cost whereas the decisions are subject to certain demand fulfillment criteria [22]. These criteria are usually modeled as chance constraints in which the probability of reaching a service level must be greater than or equal to a predefined value [4]. In this paper, several service level measures have been investigated. The α service level considers the probability of no stock out during the production or procurement cycle, the β service level or the fill rate is the proportion of the demand directly filled from stock, the γ service level limits the proportion of expected backlog to expected demand, and the δ service level limits the proportion of total expected backlog to the maximum expected backlog. These service levels are typically defined for each product separately.
In this research, we investigate an aggregate service level which is defined aggregately for multiple products in addition to individual service levels when uncertainty in the demand is present [1]. Such a situation is relevant in practical applications where there is a lot of product variety. For a specific type of clothing that comes in different colors or sizes, an aggregate service level can be imposed at the product level (i.e., for a specific piece of clothing), while specific service levels are imposed at the individual levels (i.e., for the different sizes). Consider a situation where a firm is concerned with its aggregate service level across multiple products. While it is clear that an aggregate service level of, for example, 95% can be achieved by imposing an individual service level of 95% for each item, this solution does not take advantage of the possible flexibility to have different individual service levels. The firm can impose a specific aggregate service level (e.g. 95%) while also imposing individual service levels which are less strict (e.g. 90%). This provides the flexibility to have a solution in which the resulting individual service levels for some products are less strict than the imposed aggregate level, while others are stricter. This flexibility can result in an overall cost reduction.
Different mathematical models are proposed to approximate this problem when considering different types of service level. These formulations are a piece-wise linear approximation and a quantile-based formulation. The contributions of this paper are as follows. The first contribution of this paper is to propose the idea of an aggregate service level for the stochastic lot sizing problem which generalizes the formulations presented in the literature. Next, we propose mathematical formulations to model an aggregate service level for different types of service levels considered in the literature (i.e., the α, β, γ, and δ service levels). In this problem, the imposed aggregate service levels allow the flexibility to choose a separate service level for each item and can be used in conjunction with minimum individual service level constraints used in the literature. While the aggregate service levels are parameters in the model, the individual service levels are a result of the optimization process. The proposed piece-wise linear approximation formulations for the β, γ, and δ service levels in this research are extensions of existing formulations, while the quantile-based formulation is newly proposed for these problems. Finally, we provide computational experiments and investigate the value of the aggregate service level in different situations. Another strength of the paper is the use of a unified simulation procedure for the evaluation of the approximation formulations in order to have a fair comparison of the different models and service levels. In the situations in which some level of nervousness is acceptable, the formulations under the static uncertainty can be used in a rolling/receding horizon environment to overcome some of their inherent limitations. The analysis of the application of the proposed formulations in a receding horizon fashion is also another contribution of this paper.
This paper is structured into nine different sections. In the second section, we review the existing literature. In Section 3, different aggregate service levels are introduced. Sections 4 to 6 are dedicated to the formulations for the various aggregate service levels. In each of these sections, the mathematical models and different approximations are presented. Section 7 discusses the implementation of the model in a receding horizon environment. Section 8 discusses the experimental results. Section 9 concludes the paper and discusses possible future research.
Section snippets
Literature review
Although imposing individual demand fulfillment criteria is the most common approach to deal with multiple products in inventory management, the idea of defining an integrated service level has been investigated in the inventory management literature. Kelle [12] stated that having different items with different demand, cost, and delivery characteristics, requires different service levels, and defining fixed service levels for different groups of items to insure an aggregate service level is a
Individual and aggregate service levels
In all the models proposed in this paper, a static strategy in which all the decisions are made at the beginning of the planning horizon is considered and the production quantity decisions cannot be changed when demands are realized. In addition to the deterministic multi-item lot sizing problem assumptions, we assume that the demand for different products in different periods is not known, but the distributions are known and they are independent for each product. In the case of a stock out,
Models with aggregate β service level
In this section, we investigate the aggregate β service level, imposing that the total expected amount of backorder divided by the total expected demand should be less than a predefined percentage. The expected inventory and backorder in each planning period is a non-linear function of the cumulative production in each planning period. To solve this problem we use a piece-wise linear approximation.
Models with aggregate γ and δ service level
In this section, we investigate an aggregate version of time and quantity oriented service levels (i.e., γ and δ). First, we explain the model for the γ service level which imposes that the total expected backlog divided by the total expected demand should be less than a predefined percentage. We then modify the model to consider an aggregate δ service level.
Models with αc aggregate service level
In this section, we present the model for the αc aggregate service level with a capacity constraint. First we define the mathematical model for this case. Next we present a quantile-based mathematical model to approximate the actual model.
Receding horizon model
The models which are presented in this paper, assume the static strategy in which the setup and production quantity decisions remain unchanged during the planning horizon. Although this characteristic is important in some applications to avoid nervousness, it will reduce the responsiveness of the plan and potentially incur additional costs to the system. In order to deal with the case when the setup and quantity decisions can be continuously adjusted once the demand information is revealed, we
Computational experiments
To investigate the effect of an aggregate service level and gain computational insights into the benefits of the solutions based on different types of service levels, we conduct different computational experiments. First, the data generation procedure is explained. Next, the parameters of the different models such as the number of service level options in the quantile-based approximation and the number of segments in the piece-wise linear models are analysed. In the third section, we evaluate
Conclusion and future directions
In this research, different aggregate service levels have been investigated in the context of multi-item capacitated lot-sizing problems. Such aggregate service levels allow the planner to flexibly assign different service levels to individual products so that they collectively satisfy the aggregate service level measures. These aggregate service levels can be used in conjunction with the commonly adopted service levels imposed on individual products. These service levels are the extensions of
CRediT authorship contribution statement
Narges Sereshti: Conceptualization, Writing - original draft, Methodology, Data curation, Visualization, Software. Yossiri Adulyasak: Conceptualization, Methodology, Writing - review & editing. Raf Jans: Conceptualization, Methodology, Writing - review & editing.
Acknowledgments
The authors gratefully acknowledge the support of Calcul Canada and the Natural Sciences and Engineering Research Council of Canada (Grants PGSD3-504732-2017, 2014-03849, and 2016-05822), the HEC Montréal Professorship in Operations Planning, and GERAD (Doctoral Fellowships Conference Fees). We would also like to thank professor Stefan Helber and professor Katja Schimmelpfeng for their valuable comments on an earlier version of this paper. We also thank the editor and referees for their helpful
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Area: Production Management, Scheduling and Logistics. This manuscript was processed by Associate Editor Kuhn.