The EPQ with partial backordering and phase-dependent backordering rate

Mathematical Programming Approach to the Optimality of the Solution for Deterministic Inventory Models with Partial Backordering

“…Assume that ≥ * . From Proposition 3 we have that ( * , * ) is a KKT point for the problem (6). We are going to prove that the Hessian matrix of the Lagrangian calculated at the point ( * , * ) is positive definite.…”

“…In the remaining part of this subsection, we will prove the optimality of the solution ( * , * ) by examining the optimality conditions for the problem (6). We will use the optimality conditions that deal with the linear independenceconstrained qualification (LICQ), that is, LICQ holds at a point if the set of active constrained gradients in this point is linearly independent, and the KKT point, that is, the point for which there are Lagrangian multipliers such that the KKT system is satisfied, which are not always easily verified; see, for example [18].…”

“…Assume that ≥ * , where * is defined by (4). Then ( * , * ) defined by (2) and 3is the only KKT point for the problem (6).…”

“…But when the assumption of instantaneous replenishment is replaced with the assumption that the replenishment order is received at a constant finite rate over time, EOQ is extended to the economic production quantity (EPQ) model, and by allowing stockouts with partial backordering it will extend to EPQ with partial backordering (EPQ-PBO). There are many papers that propose an inventory model with partial backordering; a very few of them are [1][2][3][4][5][6][7]. For a survey of the deterministic models for the EOQ and EPQ with partial backordering, see Penticio's and Drake's paper [8].…”

“…The paper is organized as follows. In Section 2 Pentico et al 's EPQ-PBO is formulated as a mathematical programming problem, then mathematical programming approach for proving the optimality of the solution is presented, and the same approach is used for two other inventory models with partial backordering, an extension of Pentico et al 's EPQ-PBO [6] and Hu et al 's inventory model [2]. There are comments on how mathematical programming approach can be used for the models presented in Taleizadeh et al [16], Taleizadeh et al [10], Taleizadeh and Pentico [9], and Cárdenas-Barrón [12].…”