(658f) Extensions to the Guaranteed-Service Model for Safety Stock Positioning in Industrial Environments

Achieving on-time fulfillment of customer demand depends in great part on the inventory levels and policies that are set along a supply chain. Efficient inventory planning is particularly challenging in multi-echelon networks, in which customer demand and lead times are uncertain and the decision at one stage impacts inventory decisions at other stages. The purpose of multi-echelon inventory optimization (MEIO) is to allocate safety stocks to meet customer service levels, while minimizing the total capital tied up in inventory throughout the supply chain. The intent of safety stock allocation is to determine an overall strategy for deploying inventory levels across the supply chain in order to buffer it against sources of uncertainty (Graves and Willems, 2003). In contrast to single-echelon inventory optimization (SEIO), which seeks to independently minimize cost at each echelon, MEIO has a holistic approach to support decision-making. From an optimization perspective, the task is also challenging because MEIO models involve non-linearities and nonconvex functions. De Kok et al. (2018) state that multi-echelon inventory systems are still a very active area of research because of their complexity and practical relevance. More recently, Gonçalves et al. (2020) highlight that the number of contributions to MEIO has seen a significant increase from the year 2005 onwards, and they list many potential directions and trends for future research.

The objective of this work is to develop a MEIO model based on the Guaranteed-Service Model (GMS) (Eruguz et al., 2016; Graves and Willems, 2000; Simpson, 1958) that accounts for different issues and characteristics arising in pharmaceutical industrial practice. Many authors have developed some extensions to the GSM, but to the best of our knowledge, nobody has developed a model that can achieve optimum safety stocks on complex supply chains while integrating all the features typical of industrial environments presented in this work. We integrate them into a single model that enables an improved real-world supply chain representation in order to provide support to strategic decision-makers.

One of the novelties in the proposed model is that it combines several features. First, demand can occur at any node in the network. This can result in hybrid nodes that have both dependent and independent demands. To the best of our knowledge, these characteristics, which represent the common operation mode of many real multi-echelon systems, has not been addressed before, as most of the literature on supply chain inventory management considers only external demand at the final nodes of the network. Our proposed formulation also captures risk-pooling effects by consolidating the safety stock inventory of downstream nodes to the upstream nodes in the multi-echelon supply chain. Second, manufacturing plants can be placed at any location in the network, enabling the manufacture of any desired good at those locations. This feature allows generalizing and managing larger supply chains that have grown in their vertical integration. Third, fill rates can be used as an alternative customer service key performance indicator when setting safety stock levels. Fill rates are not considered in the standard GSM, which relies on cycle service levels instead. However, fill rate is the most widely applied service level measure in industry (Teunter et al., 2017). We thus allow the modeler the flexibility of specifying the customer service metric to be used. In order to do this, we propose a quadratic regression to estimate the equivalent Cycle Service Level (CSL) when fill rates are used in the model as the desired customer service measure. In addition, minimum order quantities (MOQ) for replenishment orders are explicitly modeled. Depending on the size of the order, using MOQ can cause overshoot in the inventory levels, impacting service levels and costs. This is frequently observed in almost all supply chains.

Another novelty of the present work is that the resulting nonconvex Nonlinear Programming (NLP) model is reformulated as a Quadratically Constrained Problem (QCP) by exploiting the structure of the constraints of the base model. Nonconvex NLP problems can in principle be solved with global optimization solvers like BARON. However, for medium or large-scale problem sizes, the computational time required to find a global solution may be very extensive. On the other hand, solvers like CPLEX and Gurobi can solve QCP models quite effectively in reasonable computational times. Several examples for illustrative and real industrial systems from a pharmaceutical supply chain are presented to illustrate the application of the proposed model and its resulting improved computational performance.

All results obtained from the developed model are validated using simulation. The aim is to evaluate if safety stocks can meet expected customer service levels (CSL and/or fill rate). This is done using an open-sourced discrete-time inventory simulation software package (Perez, 2021). This simulator allows modeling multiproduct supply networks of any topology (e.g., serial, divergent, convergent, tree, or general). Each of the features included in the extended GSM model can be simulated using this software: hybrid nodes, minimum order quantities (MOQ), bill of materials, stochastic demand, and stochastic lead times. The software allows evaluating any static (continuous or periodic review) and dynamic inventory control policy. The simulations show that the safety stock settings tend to be more accurate for large expected Cycle Service Levels (CSL) in a system with lost sales. Lower values of expected CSL tend to be less accurate, yielding larger effective CSL’s. When the coefficient of variation (CV) and the expected CSL are low, the CSL is generally underestimated and this effect is amplified when there are large MOQs active. In practice, target CSL’s are generally larger than 90%, so approximations tend to be accurate in that range. Nevertheless, the equivalent CSL resulting from selecting a fill rate as target measure brings the opportunity to largely decrease z safety factors, and the gap in CSL estimations arises. Customer service is generally ensured while reductions in safety stocks can be performed.

Future research may explore adjustments on safety stocks to deal with CV and MOQ to obtain more precise estimations under lost sales assumption, include capacity constraints and extend the model to account for other demand distributions. In conclusion, the validation of results through simulation allows demonstrating the accuracy of the model to obtain safety stocks that can meet specified customer service levels.

REFERENCES

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