The best prices of three mutually complementary merchandises in the fuzzy sense (Q1595200)

Three mutually replaceable merchandises \(X_1,X_2,X_3\) are given, with the demand functions in a monopoly market \(x_i=a_{i0}+ \sum_{1\leq j\leq 3,j \neq i}a_{ij}P_j- a_{ii}P_i\), \(i=1,2,3\). Here \(a_{ij}>0\) are known, and \(P_i\) is the price for \(X_i\). The total revenue is \(R=x_1P_1+ x_2P_2+x_3P_3\). The monopolists can find the best prices \(P_i^*\) that make \(R(P_1,P_2,P_3)\) reach its maximum. In the present paper, a perfect competition market is considered, where the demand functions usually deviate from the model. The best prices for the three merchandises are found in the fuzzy sense. The same problem for two merchandises has been solved by \textit{J.-S. Yao} and \textit{K. Wu} [Fuzzy Sets Syst. 111, 433-454 (2000; Zbl 0978.91056)].