Corrigendum to “Safety stock management in single vendor-single buyer problem under VMI with consignment stock agreement” [Int. J. Prod. Econ. 154 (2014) 16–31] (International Journal of Production Economics (2014) 154 (16–31), (S0925527314001224), (10.1016/j.ijpe.2014.04.007))

The authors regret < that in Braglia et al. (2014) we claimed that [Formula presented] (see Eq. (A.3)) is strictly convex in the reference domain [Formula presented] (see Eq. (A.1)) is strictly convex in [Formula presented]. However, the proof concerning the convexity of [Formula presented] contains an error. In particular, in the expression [Formula presented] used to check the definiteness of the Hessian matrix of [Formula presented], we did not distinguish the generic point inside [Formula presented], for which the definiteness of H has to be evaluated, denoted by [Formula presented], from the vector [Formula presented] used to carry out the test, which was also denoted by [Formula presented]. The adoption of identical notation led to the error. The test to evaluate the definiteness of H (or, equivalently, the convexity/concavity of [Formula presented] should be adjusted as follows. Let [Formula presented] be a non-zero row vector in [Formula presented]. Then, according to a well-established result (Horn and Johnson, 1990), H is positive semi-definite over [Formula presented] (or, equivalently, [Formula presented]) if and only if [Formula presented] for all [Formula presented] and for each [Formula presented]. Now, it can be verified that the above condition is not satisfied for some [Formula presented] and for some combination of parameter values. In other words, contrarily to what we originally affirmed, the proposition “[Formula presented] is convex over the entire domain [Formula presented]” is false, i.e., there exists a non-empty set [Formula presented] such that H is negative definite for all [Formula presented]. To prove this, it is sufficient to show that [Formula presented] in at least one particular case. To this aim, suppose that [Formula presented] units/year, [Formula presented]$3/unit/year, [Formula presented]20%. If we take [Formula presented] 1000 units and [Formula presented]10, it can be observed that the Hessian matrix is not positive semi-definite in correspondence to this point. In fact, condition (1) is not verified for, e.g., [Formula presented], obtaining [Formula presented], which proves our statement. We now show that, in contradiction to our original statement, even the proposition “[Formula presented] is convex over the entire domain [Formula presented]” is false. With the same arguments as above, if we let M be the Hessian matrix of [Formula presented], then M is positive semi-definite over [Formula presented] if and only if [Formula presented] for all [Formula presented]. It is now necessary to show that there exists a non-empty set [Formula presented] such that M is negative definite for all [Formula presented], for some combinations of parameter values. To prove that [Formula presented] is non-empty, it is sufficient to find at least one particular case in which condition (2) is not satisfied. To this aim, consider the same parameter values as above, in addition to the following values, which are taken from Braglia et al. (2014): [Formula presented]900. If we take [Formula presented]1000 units, [Formula presented]10, [Formula presented], and [Formula presented], we get [Formula presented], which proves our statement. Although [Formula presented] has been demonstrated to be non-convex for some [Formula presented], in contradiction to what originally affirmed, we now show that the procedure we presented to minimize [Formula presented] over [Formula presented] is still valid. We firstly recall that [Formula presented] has a single stationary point in n over [Formula presented], for a given q, that was denoted by [Formula presented], and that represents the global minimum in n inside [Formula presented] identifies a curve connecting all global minima in n inside [Formula presented], for fixed q. Hence, we can write [Formula presented] From known results (see Appendix B in Braglia et al. (2014)), we deduce that [Formula presented], for [Formula presented], i.e., [Formula presented] is convex in [Formula presented]. Since [Formula presented] as [Formula presented], then it is clear that the minimum of [Formula presented] in [Formula presented] lies either on a stationary point inside [Formula presented] or on the right boundary of [Formula presented], i.e., on [Formula presented]. These arguments prove that the procedure we developed in Braglia et al. (2014) to minimize [Formula presented] over [Formula presented] is correct, despite the corrections previously made concerning the convexity of [Formula presented] over [Formula presented]>. The authors would like to apologise for any inconvenience caused.