Optimal profit for fuzzy demand in the fuzzy sense

Abstract

The linear relationship of the crisp demand function is represented by When the crisp revenue function is R(x) = a + bx2 and crisp cost function as π(x) = u+vx , the profit function is N(x) = ax+bx 2 −u−vx . The demand quantity with respect to price p is x(= (p − a)/b) . In a perfect competition market, there can be slight fluctutation and for the demand quantity d at price p , the points of demand quantities in a straight line vary in the interval between x Δ and x + Δ (0 < Δ < x , in which Δ is to be selected), it means that the fuzzification of d is [Dtilde] . For demand quantity d , the profit function is N(d) = ad + bd 2 − u − vd . If bd 2 + (a − v)d − u = z , the membership function μ N(ᵭ)(z) and its centroid M(Δ, x) of fuzzy profit N(ᵭ) can be obtained by the extension principle. This centroid value is the estimate value of the profit for demand quantity x in the fuzzy sense. By numerical method, the maximum profit for fuzzy number (x** Δ** , x ** + Δ**) is M(Δ**, x **) . If actual fuzzy situation is not (x ** Δ**, x ** +Δ**) , the estimate value of profit is not good, then the optimal solution of crisp function can be compared with the maximum profit M(Δ**, x **) for the optimum quantity of demand x¤¤ in fuzzy sense.