随机存贮模型中两个函数的重要性质

Important Properties of Two Functions in a Stochastic Inventory Model

文献[1]给出了随机存贮系统中每单位时间的平均缺货量函数F_1(x,y)和平均未偿还的延迟交货额函数F_2(x,y)的表达式,即 F_1(x,y)=ρ[f_1(x)-f_1(x+y)]/y;F_2(x,y)=[f_2(x)-f_2(x+y)]/y,(1)其中,f_1(u)=integral from n=u to ∞([1-Φ_D(ξ)]dξ);ρ为系统的平均需求速率,且为大于0的常数;f_2(u)=integral from n=u to ∞((ξ-u)[1-Φ_D(ξ)]dξ)。

Based on the formulas given by Hapley G, this paper proves both the monotonicity and joint convexity with respect to x and y of two commonly used inventory level measuring functions F1(x,y) and F2(x,y), the average out-of-stock per unit time and the average outstanding back order respectively in the stochastic inventory model and their extremum properties are derived. With ΦD(ξ) =[(ξ-μ)/τ] and by replacing x with an equivalent control variable or the safety stock z = x-μ, this paper deduces F2(y,z,τ), another form of the function F2(x,y) and proves its monotonicity and joint convexity. It is also pointed out that when (t)>0 for any t, function F2(y,z,τ) is a strictly convex function with respect to each pair of control variables. However, as a whole, the function is only convex and not strictly convex.