This study investigates and quantifies the bullwhip effect (BWE) as a coordination tool in the supply chain under the revenue sharing contract (RSC). The network considered consists of one or more main and rival suppliers in one layer, and multiple retailers in another layer. RSC contracts are being signed between suppliers and retailers, and retailers adopt the substitution of similar commodities in a competitive environment. This study examines different demand forecast methods, analyzes the largest lower bound for the variance ratio of demand estimation, and presents a new model to reduce BWE and determine the optimal contract parameters. Results show that the variance ratio index of the total demand of suppliers to the variance of the total demand of retailers outperforms other indicators. In the presence of RSC, this ratio is lower than in the case without a contract, there by preventing BWE amplification. Based on the results of the numerical analysis, the variance ratio of demands under the revenue-sharing contract was lower than in the scenario without a contract. The largest lower bound with a contract was 0.026, while it was 0.1 without a contract, indicating that the Bullwhip Effect (BWE) is amplified in the absence of a contract. The analysis of variance changes in retailers' demand estimates by suppliers, based on contract profit percentages, revealed that the BWE decreases when suppliers increase the contract profit percentage for retailers within the intervals of 0.4-0.5 and 0.6-0.8.