Manufacturer vs. Retailer: Who Should Manage Returns in Revenue-Sharing Contracts under Multiplicative Demand?

Technical Summary: Manufacturer vs. Retailer: Who Should Manage Returns in Revenue-Sharing Contracts under Multiplicative Demand?

Problem Formulation
This study investigates pricing and ordering decisions within a two-echelon supply chain (SC) consisting of an independent manufacturer and a retailer. The system faces price-sensitive, multiplicative stochastic demand and is subject to product returns and exchanges, specifically focusing on non-defective returns (false failure returns). The core problem addresses the strategic choice of who should manage these returns: the manufacturer (Strategy M) or the retailer (Strategy R).

The interaction is modeled as a Stackelberg game where the manufacturer acts as the leader and the retailer as the follower. Before the game begins, the manufacturer selects a contract scenario from four possibilities, combining two sharing policies (Revenue Sharing [RS] and Cost-Revenue Sharing [CRS]) with the two return handling strategies.

• Revenue Sharing (RS): Revenue is shared according to a ratio $r$, but return handling costs are borne solely by the party managing the returns.
• Cost-Revenue Sharing (CRS): Both revenue and return handling costs are shared according to the same ratio $r$.

The demand function is defined as $D_k(p, \varepsilon) = a(p + h_c^k)^{-b}\varepsilon$, where $p$ is the retail price, $\varepsilon$ is a random variable, and $h_c^k$ represents the consumer's average return handling cost, which varies depending on whether the manufacturer ($k=M$) or retailer ($k=R$) manages the process.

Methodology
The authors employ a game-theoretic approach using backward induction to solve for the Stackelberg equilibrium.

1. Centralized Benchmark: First, the optimal decisions for a centralized channel (where a single authority maximizes total SC profit) are derived to serve as a benchmark for coordination.
2. Decentralized Analysis: The study analyzes the four decentralized scenarios (RS-M, RS-R, CRS-M, CRS-R). The retailer first determines the optimal retail price ($p$) and inventory factor ($z$) to maximize their expected profit, given the wholesale price ($w$) set by the manufacturer. The manufacturer then anticipates this response and sets $w$ to maximize their own expected profit.
3. Analytical Derivation: The paper derives closed-form expressions for optimal prices and inventory factors. A critical mathematical feature identified is the behavior of the pricing functions under multiplicative demand. The authors define a critical inventory factor $z_0$ derived from the equation $(b-1)(z_0 - \Lambda(z_0)) - bz_0\bar{F}(z_0) = 0$. They demonstrate that the pricing functions possess a vertical asymptote at $z_0$, creating a structural singularity that restricts the feasible decision space to the interval $[z_0, B]$.
4. Numerical and Sensitivity Analysis: Using parameters calibrated from the electronics industry (where return rates are high), the study conducts numerical simulations to compare the performance of the four scenarios. Sensitivity analyses are performed regarding return handling costs ($h_k$) and the revenue sharing ratio ($r$).

Key Contributions
The paper makes four primary contributions to the literature on supply chain contracts and returns management:

1. Structural Singularity in Multiplicative Demand: The study demonstrates that multiplicative demand introduces a structural singularity (a vertical asymptote in pricing functions) that is absent in additive demand models. This fundamentally alters the analytical structure of the Stackelberg game and limits the feasible range of decisions, distinguishing it from previous works on additive demand.
2. Asymmetric Impact of Return Costs: The analysis reveals that the party managing returns does not always bear the full economic impact of the costs. Under RS-M, higher return costs lead the manufacturer to increase the wholesale price. Conversely, under RS-R, the retailer raises the retail price, prompting the manufacturer to lower the wholesale price to mitigate the negative impact on order quantity.
3. Coordination Mechanisms: The paper establishes that standard RS contracts do not fully coordinate the supply chain when return handling costs are not shared. However, CRS contracts can achieve full coordination if the wholesale price is set at $w = rc$ (below production cost) and costs are shared in the same proportion as revenue. The study notes that while coordination is theoretically possible, the manufacturer's profit-maximizing choice in the Stackelberg equilibrium may not naturally select this coordinating wholesale price.
4. Profit vs. Quantity Trade-off: Numerical results indicate that decentralization leads to a relatively small loss in total channel profit (approx. 4% in the baseline) but a significantly larger reduction in order quantity (approx. 37%). This highlights that efficiency metrics based solely on profit may obscure substantial distortions in inventory and sales volume.

Results and Findings

• Contract Performance: The CRS contract generally outperforms the RS contract in terms of channel efficiency and coordination potential. However, full coordination requires specific wholesale price settings that may not align with the manufacturer's self-interested equilibrium decisions.
• Strategy Preference (M vs. R): The preference for who manages returns is not absolute; it depends heavily on the sharing ratio $r$. The study identifies a critical threshold $r^*$.
  • Below $r^*$, the manufacturer prefers Strategy R (retailer-managed), while the retailer prefers Strategy M, creating a conflict of interest.
  • Above $r^*$, Strategy R Pareto-dominates Strategy M, benefiting both parties.
• Consumer Convenience: In scenarios with asymmetric costs, a party with higher direct handling costs (e.g., the retailer) may still be the preferred return handler if their process offers greater convenience to consumers (lowering consumer return costs), thereby stimulating demand enough to increase total channel profit, even if the retailer's individual profit decreases slightly.

Significance and Claims
The authors claim that these findings offer practical guidelines for designing contracts, particularly in e-commerce and electronics supply chains where returns are frequent. The study emphasizes that:

• The form of demand uncertainty (multiplicative vs. additive) critically affects the economic properties of contracts and the feasible decision space.
• Managers must consider the sharing ratio and return cost allocation jointly, rather than in isolation.
• A strategy that maximizes total channel profit may not be preferred by the individual firm responsible for returns, potentially requiring contractual compensation to align incentives.
• The decision of who manages returns should balance direct handling costs against the indirect benefits of consumer convenience and demand stimulation.

The paper concludes that while retailer-managed returns can be beneficial for the entire channel under specific conditions (high sharing ratios or high consumer convenience), they can also create conflicts that require careful contract design to resolve.