[Paper Review] An EOQ model for imperfect quality items with multiple screening and shortage backordering

Allen H. Tai|arXiv (Cornell University)|Feb 6, 2013

Supply Chain and Inventory Management6 references3 citations

TL;DR

This paper develops an EOQ model for imperfect quality items with multiple screening processes and backorderable shortages, using the renewal reward theorem to derive closed-form solutions for optimal order size and maximum backorder quantity. The key contribution is a generalized framework that accounts for sequential screening rates and defect rates, improving upon prior models by incorporating multiple inspection stages and shortage backordering.

ABSTRACT

In this paper, we propose an inventory model where items are inspected through multiple screening processes before delivery to customers. Each screening process has independent screening rate and defective percentage. Defective items screened out are stored and then returned to supplier. Shortage backordering are also allowed in the model. Two approaches are used to obtain the closed-form optimal order size and the maximum backordering quantity. Numerical examples are also provided to demonstrate the use of the model.

Motivation & Objective

• To develop a generalized EOQ model that incorporates multiple screening processes for imperfect quality items.
• To integrate shortage backordering into the screening process framework, allowing for optimal backorder management.
• To derive closed-form solutions for optimal order size and maximum backorder quantity using the renewal reward theorem.
• To analyze the impact of varying screening rates, defect percentages, and cost parameters on inventory performance.
• To provide numerical examples demonstrating the model’s applicability and robustness under different configurations.

Proposed method

• The model uses the renewal reward theorem to compute the expected profit per unit time (ETPU) over a replenishment cycle.
• Multiple screening processes are modeled sequentially, with each having independent screening rates $x_i$ and defect proportions $p_i$.
• Defective items are screened out at each stage and stored separately, with total defective proportion $\rho = \sum_{i=1}^n \rho_i$.
• The inventory cycle is divided into phases: screening, demand fulfillment, and backorder liquidation, with time durations derived from $y/x_i$ and demand rate $D$.
• Two approaches are used to solve for optimal $y^*$ and $B^*$, ensuring closed-form solutions for decision variables.
• The model assumes defective items are returned to the supplier and backorders are fully satisfied once inventory is available.

Experimental results

Research questions

• RQ1 How does the inclusion of multiple screening processes affect the optimal order size and backorder level in an EOQ model with imperfect quality?
• RQ2 What is the impact of varying screening rates and defect percentages on the expected profit per unit time?
• RQ3 How do backordering costs and screening costs influence the optimal inventory policy under multiple inspection stages?
• RQ4 Can closed-form solutions be derived for order size and backorder quantity under multiple screening and backordering?
• RQ5 How does the sequence of screening processes affect the overall inventory performance and cost structure?

Key findings

• The optimal order size $y^*$ increases with higher backordering cost $\beta$, while the maximum backorder quantity $B^*$ and expected profit per unit time $ETPU^*$ decrease.
• As screening rate $x_i$ increases, both $y^*$, $B^*$, and $ETPU^*$ increase, indicating improved system efficiency.
• When both backordering cost $\beta$ and screening cost $d$ increase, $y^*$ and $B^*$ rise, but $ETPU^*$ declines due to higher operational costs.
• For multiple screening processes (e.g., $S_1 + S_4 + S_6$), the optimal order size and backorder level are bounded by the slowest screening process, with $y^*$ and $B^*$ approaching those of the slowest stage.
• Numerical results show approximation errors in ETPU are less than 0.01%, confirming the accuracy of the derived closed-form solutions.
• The model demonstrates that adding more screening stages reduces $B^*$ compared to single-stage screening, improving service levels.