We study a decentralized assembly supply chain in which an assembler (she) assembles a set of n components, each produced by a different supplier (he), into a final product to satisfy an uncertain market demand. Each supplier holds private cost information to himself, for which the assembler only has a subjective estimate. Furthermore, the assembler believes that the suppliers' costs follow a joint discrete probability distribution. The assembler aims to design an optimal menu of contracts to maximize her own expected profit. The assembler's problem is a complex multi-dimensional constrained optimization problem. We prove that there exists a unique optimal menu of contracts for the assembler, and we further develop an efficient algorithm with a complexity of O(n) to compute the optimal contract. In addition, we conduct a comprehensive sensitivity analysis to analyze how environmental parameters affect individual firm's performance and the value of information to the assembler, to each supplier, and to the supply chain. Our results suggest that each supplier's private cost information becomes more valuable to the assembler and each supplier when the average market demand increases or when the final product unit revenue increases. Surprisingly, when a supplier's cost volatility increases and its mean remains the same, the value of information to the assembler or to each supplier does not necessarily increase. Furthermore, we show that when the suppliers' cost distributions become more positively correlated, the suppliers are always worse off, but the assembler is better off. However, the value of information for the assembler might increase or decrease.