Cost Minimisation, Return on Investment, Residual Income: Alternative Criteria for Inventory Models

Profitability Index Maximization in an Inventory Model with a Price- and Stock-Dependent Demand Rate in a Power-Form

“…Otake et al [15], Otake and Min [16], Li et al [17], and Chen and Liao [18] are some papers in this research line. Instead, Morse and Scheiner [19] used the term residual income (RI), defined as the excess of net income over the opportunity cost of invested capital. Dave [20] and Baldenius and Reichelstein [21] also advocated the use of this profitability measurement.…”

“…), i.e., ψ(p * ) = 0. Thus, p * is implicitly defined by the equation Ψ(p * , K) = 0 beingΨ(p, K) = (γ + p) α/(2−β) − α 2 − β p(γ + p) −1+α/(2−β) + c A 3 ,with A 3 dependent on K.Then, by using the implicit differentiation theorem (see, for example, Krantz and Parks[30], Theorem 1.3.1), and taking into account the expression(19) for ψ (p), we have∂p * ∂K = − ∂Ψ(p,K) ∂K (p * ,K) ∂Ψ(p,K) ∂p (p * ,K) β)(2 − β)c α(α + β − 2)A 3 Kp * (γ + p * ) −2+α/(2−β) < 0.Similarly, for the other parameters we obtain:∂p * ∂h = −(2 − β)c α(α + β − 2)A 3 hp * (γ + p * ) −2+α/(2−β) < 0, ∂p * ∂λ = (2 − β)c α(α + β − 2)A 3 λp * (γ + p * ) −2+α/(2−β) > 0, ∂p * ∂c = − ∂Ψ(p,c) ∂c ψ (p * ) = (2 − β) 2 α(α + β − 2)A 3 p * (γ + p * ) −2+α/(2−β) > 0, ∂p * ∂γ = α(γ + p * ) −2+α/(2−β) ψ (p * )(2 − β) 2 [(α − 2(2 − β))p * − (2 − β)γ] = 2 − β (α + β − 2)p * (γ + (2 − α/(2 − β))p * ).Then,∂p * ∂γ < 0 if γ + (2 − α/(2 − β))p * < 0 . Otherwise, ∂p *∂γ ≥ 0.…”