Abstract
The Multiple-Square-Root Minimization Problem (MSR) has an objective function that consists of a sum of a linear term and at least two square root terms. The Lagrangian sub-problem for the LMRP is a typical MSR problem and there are other MSR problems in real life. A simple example is that we add other concave costs besides the safety stock cost to the LMRP, such as the labor cost and even minimize the negation of the revenue. We tested a sort of heuristic involved, similar to the method to solve the problem of the LMRP vibrational days, and we explore the heuristic is probably the most optimal condition. The accuracy of this approach is declining at a slow rate, because the number of square roots is increasing, and when the number is not too large, it stays at a higher level.
Highlights
The Multiple-Square-Root Minimization Problem (MSR) has an objective function that consists of a sum of a linear term and at least two square root terms
A simple example is that we add other concave costs besides the safety stock cost to the LMRP, such as the labor cost and even minimize the negation of the revenue
As for the multiple square root (MSR) problem, we discuss a sufficient condition that guarantees the sorting method works, and an alternate analytical approach that makes the sorting method often needs the optimal solution while the average error is low
Summary
The Multiple-Square-Root Minimization Problem (MSR) has an objective function that consists of a sum of a linear term and at least two square root terms. We apply this method to the LMRP problem They listed problems like which all had at least two concave (square root) terms in the objective function. We try to find under what condition the sorting algorithm for the LMRP Lagrangian sub-problem can be applied to the general two square root pricing problem. The LMRP is structured much like the UFLP model, with two extra non-linear terms in the objective function. Despite its concave objective function, the LMRP problem can be solved by Lagrangian relaxation quite efficiently, just like the UFLP, assuming that the ratio of the customer demand rate and the standard deviation of daily demand are constant.
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