Economic Production Quantity (EPQ) Model

• The EPQ inventory model is a quantitative framework that optimizes production lot sizes by balancing production rates with inventory holding, setup, and shortage costs.
• Extensions incorporate imperfect production, rework processes, and deterioration, using mathematical formulations and convex analysis to derive optimal cycle times.
• Numerical examples and sensitivity analyses demonstrate the model’s practical use in manufacturing, supply chain management, and green logistics to enhance operational efficiency.

The Economic Production Quantity (EPQ) inventory model is a quantitative framework for optimizing lot sizes in production systems with periodic replenishment, originally formulated to balance production rates with inventory holding, setup, and shortage costs. Contemporary research has extended EPQ to address imperfect product quality, rework processes, deterioration, stochastic demand, multi-depot reverse logistics, capacity constraints, environmental objectives, and dynamic control regimes. EPQ models now serve as an essential methodological base for operational optimization in manufacturing, supply chain management, green logistics, and service operations.

1. Extensions for Imperfect Production, Deterioration, and Rework

Recent EPQ models incorporate imperfect production yields and rework operations to more accurately reflect industrial realities (Tai, 2012). In a representative formulation, a single plant produces goods at rate $p$, of which only a fraction $\alpha$ is initially non-defective. Imperfect items are earmarked for rework, processed at rate $pr$ with recovery fraction $ar$. During storage, serviceable items deteriorate at rate $\theta$, with potential for defective screening and inadvertent sale of deteriorated units—invoking penalty costs and reputational impacts.

Inventory states are described by piecewise differential equations across production, inspection, rework, and shortage periods. For example,

$I_1(t_1) = (\alpha p - X)t_1$

and

$$I_2(t_2) = (\alpha p - X)\left[1 - \exp(-\theta t_2)\right]$$

encapsulate inventory build-up and decay, where $X$ is the immediate production allocated to backlog. Allowance for partial backlogging is parameterized by $\beta$, such that only a fraction of unmet demand is delayed rather than lost.

For multi-plant settings, imperfect locally-produced items are pooled at a central rework facility, enabling instant conversion and streamlined cost accounting, but also increasing the risk of unsold "salvaged" items impacting brand perception.

2. Mathematical Formulations and Convexity Analysis

EPQ extensions formalize total cost per unit time as a function $TC$, which aggregates operating and penalty costs:

• Holding costs for serviceable and imperfect inventory,
• Deterioration and inspection costs,
• Setup costs per batch,
• Penalties for shortages, backlogged demand, and unrecoverable items.

A general cost function, e.g., Equation (16) in (Tai, 2012), integrates costs over production-rework cycles,

$$TC = \sum_i \text{cost}_i(T_1, T_2, T_3, T_4, T_5, \dots)$$

with exponential decay factors and boundary conditions. Analytical tractability is improved via Taylor approximations (e.g., $\exp(x) \approx 1 + x + \frac{x^2}{2}$ for small $x$).

Optimal cycle times $(T_4^*, T^*)$ are obtained by solving for stationary points of the cost function. Convexity is demonstrated under regularity conditions (e.g., strict positivity of key coefficients, Theorem 1 in (Tai, 2012)), guaranteeing existence of a unique global minimum.

3. Approximate Analytic Results and Solution Procedure

By employing approximate expansions for exponential terms, closed-form solutions for optimal cycle times and lot sizes are derived—crucial for practical application. For example, the optimal cycle length $T^*$ and the corresponding economic production quantity $Q^* = p T^*$ are computable for prescribed parameter regimes. Parameter sensitivities (e.g., to $\alpha$, $\theta$, $ar$, $pr$, cost rates) are directly accessible from analytic expressions and facilitate real-time decision adjustment.

For full backlogging ($\beta = 1$), the simplification leads to an explicit optimality formula of the type

$(T_4^*, T^*) = \frac{-B \pm \sqrt{4AC - B^2}}{2C}$

where $A, B, C$ are parameter functions encompassing both cost and process rates.

4. Numerical Examples and Error Validation

Numerical instances (e.g., $p = 6000$, $\alpha = 0.7$, $\theta = 0.1$, $pr = 4000$, $K = \$300$,$h_s = \$5$,$C_a = \$100$$) quantify the models’ recommendations. In single plant scenarios, the computed$$T^* \approx 0.055$time units yields$Q^* \approx 330$units, with total cycle cost$\approx \$5837.6$$per unit time, and analytic error under 1$$n = 5$,$K_c = \$250$,$h_c = \$3$$), separate feasibility cases are assessed; optimal lot sizes can increase substantially ($$Q^* \approx 479$$units) according to remanufacturing setup.</p> <p>The computational pipeline involves plugging parameter values into simplified expressions, solving for feasible case conditions, and verifying optimality by direct computation or search algorithms.</p> <h2 class='paper-heading' id='practical-implications-corporate-image-multi-plant-operations-and-sensitivity'>5. Practical Implications: Corporate Image, Multi-Plant Operations, and Sensitivity</h2> <p>These revised EPQ models enable managers in industries dealing with perishables, electronics, and fashion to optimally synchronize production and rework cycles, reflecting the realities of product deterioration and defective output. By explicitly modeling imperfect inspections and penalizing the sale of deteriorated products, companies are incented to implement improved screening and feedback procedures, mitigating image risk.</p> <p>When multiple local production plants are coordinated with a centralized rework operation, cost trade-offs in logistics and remanufacturing setup emerge—key in distributed supply chains. Sensitivity analysis to key parameters (e.g., improvement in$$ar$and$pr$ leads to more favorable cost structures) directly guides investment in upstream process quality and downstream rework technology.

6. Generalization and Research Directions

Current EPQ models are being extended to hybrid frameworks incorporating fuzzy parameters (Malik et al., 2016), stochastic demand and continuous review (Maitra et al., 2023), dynamic control (Golui et al., 2022), and reverse logistics optimization with sustainability targets (Rizvi et al., 23 Sep 2025). Aspects such as rework under learning effect and machine breakdown (Fekri, 2019), finite capacity regime switching (Azcue et al., 2020), and environmental cost embedding (e.g., GHG emissions) (Rizvi et al., 23 Sep 2025) indicate a trajectory toward increasingly integrated, multi-objective, and computationally tractable EPQ frameworks.

Ongoing research is focused on scalable solution algorithms for high-dimensional mixed-integer nonlinear programs, improved risk quantification under uncertainty, and the use of heuristic/metaheuristic methods in realistic, non-convex optimization landscapes.

7. Epistemic Considerations and Limitations

While analytic EPQ solutions provide tractable decision tools, their accuracy depends on the fit between parameter regimes and model approximations (notably the validity of Taylor expansions for decay and deterioration). Under rapid deterioration, large cycle times, or highly volatile backlogging fractions, more robust numerical algorithms—potentially leveraging simulation or convex relaxation—may be needed. In multi-site supply chains and complex repair/recovery loops, empirical validation against operational data remains essential.

In summary, modern EPQ inventory models reflect an evolution from static lot-sizing toward dynamic, multi-objective frameworks integrating defective production, rework, deterioration, partial backlogging, supply chain coordination, and environmental impacts. Their mathematical rigor and sensitivity-adaptive formulations underpin both strategic and operational decision making in contemporary production and inventory management.