Stochastic Supplier Rejections

Updated 23 October 2025

Stochastic supplier rejections are supply chain scenarios where supplier acceptance is uncertain due to random fluctuations in operational variables.
These models employ probability distributions such as Pareto, Gaussian, and lognormal with optimization frameworks like MILP and MIECP to manage supplier matching.
Practical applications use scenario-based and rolling-horizon approaches to mitigate risks, reduce costs, and improve service levels in dynamic supply chains.

Stochastic supplier rejections describe situations in supply chain, logistics, or resource allocation where the participation or acceptance response of suppliers is subject to uncertainty, resulting in probabilistic supplier engagement, explicit rejection events, or contingent matching. This concept is crucial to modern supply chain optimization, dynamic lot-sizing, procurement, and logistics matching, where stochastic fluctuations determine both cost-effectiveness and service feasibility. Theoretical models rigorously embed randomness into decision structures, employing probability distributions, scenario-based optimization, and probabilistic policy design to anticipate, mitigate, and exploit these rejections for optimal system performance.

1. Mathematical Modeling of Stochastic Supplier Rejections

Rigorous modeling of stochastic supplier rejections involves explicit incorporation of random variables into decision policies, optimization objectives, and system constraints. In supply chain network design, operational variables such as production quantity ( $P_i$ ) or transported quantity ( $Q_{ij}$ ) are perturbed by additive stochastic noise ( $\eta$ ) from a chosen distribution (Pareto, normal, lognormal), i.e., $P_i \leftarrow P_i + \eta$ (Petridis et al., 2015). This captures supply-side uncertainties, information mismatches, and operational disruptions.

In matching and logistics systems, acceptance behavior of each supplier is modeled via Bernoulli random variables $\tilde{\Xi}_{ij}$ with parameters $p_{ij}$ , representing the independent probability that supplier $j$ accepts demand $i$ (Liu et al., 21 Oct 2025). The stochastic optimization objective is typically

$\max \sum_{i \in \mathcal{A}_D} \mathbb{E} \left[ \max_{j \in \mathcal{A}_S} (u_{ij} \cdot \tilde{\Xi}_{ij} x_{ij}) \right]$

subject to constraints on recommendations and final matches.

Scenario-based models for two-stage supplier problems introduce scenario sets $Q$ (samples from the demand or supply distribution) and solve for facility opening or allocation strategies that hedge against high-cost or infeasible scenarios (Brubach et al., 2020). In dynamic lot-sizing with supplier-driven substitution, the feasible set of joint inventory-backlog allocations $Q(D)$ for demand vector $D$ is defined so that for each realization, supply can be substituted across products within a feasible design (Sereshti et al., 2022).

2. Statistical Distributions and Their Impact

The choice of underlying noise distribution is fundamental to cost prediction, selection, and rejection strategies. Pareto distributions with low exponents ( $\alpha \approx 0.01$ ) result in fluctuations that closely mimic deterministic predictions and lead to cost optimization (reduced inlay and outlay costs), facilitating favorable supplier selection (Petridis et al., 2015). In contrast, normal (Gaussian) and lognormal distributions induce symmetric/asymmetric variability that upscales costs, increases deviations from ideal predictions, and drives higher supplier rejection rates.

In matching platforms, homogeneous acceptance (constant $p_{ij}$ ) potentiates tractable linearizations in the optimization objective, whereas heterogeneous or correlated acceptance necessitates more complex approximation schemes due to non-trivial dependencies (Liu et al., 21 Oct 2025). The probability mass function for the $r$ th best accepted supplier is $p_r = p (1-p)^{r-1}$ in the homogeneous case.

3. Optimization Frameworks and Decision Policies

Cost optimization under stochastic supplier rejections typically minimizes a composite cost function subject to constraints and penalty terms for unsatisfied demand or extended lead times. For example, the supply chain objective under noise is

$TC_1 = TC + \sum_{jk} c^{POE}_{jk} + \sum_{jk} c^{PU}_{jk} R_{jk} + \sigma ELD_k$

where $ELD_k$ is the expected lead time, $R_{jk}$ is the required additional production due to stockout, and $c^{PU}_{jk}$ encapsulates unsatisfied penalties (Petridis et al., 2015).

In recommend-to-match with stochastic rejections, two primary optimization models arise:

Exact Mixed-Integer Linear Programming (MILP) formulations for homogeneous, independent supplier acceptance (Liu et al., 21 Oct 2025).
Mixed-Integer Exponential Cone Programming (MIECP), leveraging log-sum-exp inequalities to approximate maximum expected utility under more general, heterogeneous settings.

For two-stage supplier clustering, correlated LP-rounding and greedy clustering iteratively select facilities, ensuring budget and coverage constraints are satisfied with constant-factor approximation guarantees, sometimes discarding high-cost scenarios to guarantee tractability and probabilistic coverage (Brubach et al., 2020).

Dynamic lot-sizing policies use rolling-horizon stochastic programs, leveraging joint chance constraints to enforce minimum service levels under supplier-driven substitutions, with deterministic policies as benchmarks that often fail to meet prescribed reliability targets (Sereshti et al., 2022).

4. Selection and Rejection Criteria

Supplier rejection events are endogenous decisions driven by the objective and realized random variables. In equilibrium analysis of two-echelon supply chains, the MRL fixed-point condition $r^* = m(r^*)$ defines an acceptance threshold. If realized demand $\alpha < r^*$ , the transaction does not occur, representing a supplier rejection by the market (Koki et al., 2018).

In logistics DSS, fuzzy multi-attribute decision-making ranks suppliers via closeness coefficients (TOPSIS), allowing resilient order allocation that adapts to stochastic rejections or performance variability (Hassan et al., 2019). The multi-choice goal programming model enables real-time reallocation under supplier failures.

For sequential submodular probing, each supplier (item) is probed, then selected or rejected irrevocably based on the observed state and its associated cost, subject to a global budget constraint. Acceptance probability and value are computed from the fractional LP solution, yielding immediate and theoretically grounded rejection (Tang, 2019).

5. Approximation Methods and Performance Guarantees

Approximation strategies address intractable objective functions arising from stochastic maximum operators and complex scenario structure. In recommend-to-match, the MIECP approach provides a tractable surrogate for the stochastic maximum, with provable parametric performance bounds depending logarithmically on system parameters (e.g., number of recommendations $\theta$ and acceptance probability bounds $[p_{\min}, p_{\max}]$ ) (Liu et al., 21 Oct 2025). Corollary 1 establishes exactness for $\theta = 1$ , and Theorems 3.1–3.2 generalize to bounded gaps in more general settings.

For two-stage supplier selection, scenario discarding algorithms ensure that coverage and budget constraints are maintained for a $(1-\alpha)$ fraction of scenarios, with total cost inflation controlled by the parameter $\varepsilon$ (Brubach et al., 2020). Iterative rounding and greedy clustering offer constant-factor approximation guarantees (3–11x), even in the presence of matroid or multi-knapsack constraints.

Direct assignment policies (DAP) are shown to be arbitrarily poor (relative gap approaching 1), emphasizing the necessity of stochastic and log-sum-exp approaches (Liu et al., 21 Oct 2025).

6. Practical Implementation and Managerial Implications

Empirical studies and real-world applications validate stochastic supplier rejection strategies in supply chain and logistics optimization. MIECP-based recommend-to-match achieves near-optimal matching performance and reduces computation time by over 90% compared to MILP and sample-average approximations (Liu et al., 21 Oct 2025). Simulation-optimization frameworks in large-scale supply chains deliver inventory reductions of 10–35%, yielding hundreds of millions in economic benefit while reliably maintaining service levels (Jin et al., 16 Feb 2025).

Managerial strategies include selective engagement with suppliers whose cost profiles follow favorable distributions (e.g., low-exponent Pareto), explicit incorporation of stochasticity into operational variables, and risk assessment via heatmaps and sensitivity analysis (Petridis et al., 2015, Hassan et al., 2019). Rolling-horizon chance-constrained policies outperform deterministic heuristics in dynamic inventory environments, especially when substitution across suppliers or products is feasible (Sereshti et al., 2022). Real-time probing and immediate rejection/acceptance mechanisms facilitate agile response to supplier performance variability, supporting robust procurement and allocation decisions (Tang, 2019).

7. Quantitative Analysis and Thresholds

Cost and service-level predictions in stochastic supplier rejection models frequently exploit error functions, thresholding, and percentile operators. For stockout probabilities, the error function quantifies the likelihood of meeting demand: $P_k(U) = \frac{1}{2} [ 1 + \operatorname{erf} \left( \frac{Q - \Delta_k}{\sigma \sqrt{2}} \right) ]$ with expected lead time

$ELD_k = T_u P_k(U) \zeta_k + T_l P_k(L) \lambda_k$

(Petridis et al., 2015).

In simulation-optimization for inventory, service levels are computed by evaluating the fraction of simulated inventory trajectories remaining nonnegative; k-iteration procedures adjust safety stock until percentile targets are met (Jin et al., 16 Feb 2025).

Sensitivity analyses explore the trade-off between cost and resiliency through composite indices such as

$SCRI_i = \alpha p_i^R + (1-\alpha) p_i^C$

allowing dynamic prioritization in order allocation under stochastic supplier rejections (Hassan et al., 2019).

Overall, stochastic supplier rejections constitute a multidimensional optimization challenge central to modern supply chain and logistics theory. Precise mathematical modeling, appropriately chosen distributional forms, scenario-aware algorithms, and advanced approximation methods combine to deliver resilient, efficient, and cost-effective system designs, as robustly demonstrated across a spectrum of theoretical and empirical research (Petridis et al., 2015, Koki et al., 2018, Hassan et al., 2019, Tang, 2019, Brubach et al., 2020, Sereshti et al., 2022, Jin et al., 16 Feb 2025, Liu et al., 21 Oct 2025).