Research Article: Modeling lot-size with time-dependent demand based on stochastic programming and case study of drug supply in Chile

Date Published: March 1, 2019

Publisher: Public Library of Science

Author(s): Fernando Rojas, Víctor Leiva, Peter Wanke, Camilo Lillo, Jimena Pascual, Yossiri Adulyasak.


The objective of this paper is to propose a lot-sizing methodology for an inventory system that faces time-dependent random demands and that seeks to minimize total cost as a function of order, purchase, holding and shortage costs. A two-stage stochastic programming framework is derived to optimize lot-sizing decisions over a time horizon. To this end, we simulate a demand time-series by using a generalized autoregressive moving average structure. The modeling includes covariates of the demand, which are used as predictors of this. We describe an algorithm that summarizes the methodology and we discuss its computational framework. A case study with unpublished real-world data is presented to illustrate the potential of this methodology. We report that the accuracy of the demand variance estimator improves when a temporal structure is considered, instead of assuming time-independent demand. The methodology is useful in decisions related to inventory logistics management when the demand shows patterns of temporal dependence.

Partial Text

The use of inventory models is important when managing a logistically efficient organization [1–3]. A typical objective for evaluating an inventory system is to minimize the total cost (TC), which is a function of purchase cost, ordering cost per lot (or setup), inventory holding cost, and shortage cost [4]. The inventory system should establish an economic order quantity (EOQ) or lot size to satisfy demand [5, 6]. The EOQ model often considers a single period of decision and assumes a constant rate of demand per unit time (DPUT). Note that this model can also consider multiple periods with more than one level for the decision stages, but in that case the DPUT rate is frequently non-constant. However, the multi-period EOQ model may conduct to an inventory cost greater than that obtained with single-period EOQ model [4].

In this section, we introduce the statistical methodology utilized here to represent time-dependent random demands in the probabilistic ELS problem. Specifically, we discuss GLM and GARMA structures and then a novel result on generating random numbers that follow a GARMA model is presented. These results have been implemented computationally in a programming language of the R software; see details of this software in Subsection 4.2.

In this section, we present the SP methodology to find the ELS in T periods of decision stages assuming a time-dependent random DPUT. This methodology has been implemented in the R software.

In this section, we condense, in an algorithm, the proposed methodology constructed from Sections 2 and 3. Then, the computational framework used to implement this methodology is described.

In this section, we first conduct a Monte Carlo simulation study, which allows us to compare the performance of ELS and its inventory TC for different methods when describing DPUT based on GARMA and GLM structures. Second, real-world DPUT data from a drug supply case study in a Chilean public hospital are analyzed with the proposed methodology.

This research proposed a methodology to solve a probabilistic ELS problem under time-dependent demand. A two-stage SP approach and a GARMA model for generation of scenarios were considered. Our results reported that scenarios with temporal dependence provide more accurate estimates for lot-sizing and smaller amounts of stored and shortage items in the different periods of the decision stages, when compared to time-independent DPUT, represented by GLM. The methodology proposed in this work showed an interesting approach to achieve savings in inventory total costs, assuming the variability of DPUT scenarios linked to time-series. This approach improved the existing results, preventing both unnecessary stock-outs and inventory holding. The approach offered the advantage of considering a more realistic and accurate situation of the DPUT. The proposed ELS policy is useful in organizations that have a single supplier to meet their requirements and that are generally characterized by high bureaucracy in their administrative systems. This is the case of public hospitals, where such a behavior in pharmaceutical product DPUT is frequent, and therefore, the supply system can be facilitated [51]. GARMA models give the possibility of achieving an efficient characterization of the mean, as well as the adjustment of other parameters, which may be used in probabilistic ELS problems. Although we employ a normal distribution in this paper, the proposed methodology is valid for any distribution that may be parameterized with respect to the conditional mean of a time-series model, considering linear and non-linear link functions for describing the covariates. On the one hand, this parametrization provides a basis for generating useful scenarios in SP. On the other hand, forecasting to future values based on GARMA models also gives the possibility of evaluating the quality of the prediction.




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