An improved quadratic program for unweighted Euclidean 1-center location problem

AbstractIn this paper, an improved quadratic programing formulation for the solution of unweighted Euclidean 1-center location problem is presented. The original quadratic program is proposed by Nair and Chandrasekaran in 1971. Besides, they proposed a geometric approach for problem solving. Then, they concluded that the geometric approach is more efficient than the quadratic program. This conclusion is true only when all decision variables are treated as nonnegative variables. To improve the quadratic program, one of those variables should be an unrestricted variable as it is presented here. Numerically we proved that the improved quadratic program leads to the optimal solution of the problem in parts of second regardless of the size of the problem. Moreover, constrained version of the problem is solved optimally via the improved quadratic program in parts of second

A Two-Step Approach to Scheduling a Class of Two-Stage Flow Shops in Automotive Glass Manufacturing

Driven from real-life applications, this work aims to cope with the scheduling problem of automotive glass manufacturing systems, that is characterized as a two-stage flow-shop with small batches, inevitable setup time for different product changeover at the first stage, and un-interruption requirement at the second stage. To the best knowledge of the authors, there is no report on this topic from other research groups. Our previous study presents a method to assign all batches to each machine at the first stage only without sequencing the assigned batches, resulting in an incomplete schedule. To cope with this problem, if a mathematical programming method is directly applied to minimize the makespan of the production process, binary variables should be introduced to describe the processing sequence of all the products, not only the batches, resulting in huge number of binary variables for the model. Thus, it is necessary and challenging to search for a method to solve the problem efficiently. Due to the mandatory requirement that the second stage should keep working continuously without interruption, solution feasibility is essential. Therefore, the key to solve the addressed problem is how to guarantee the solution feasibility. To do so, we present a method to determine the minimal size of each batch such that the second stage can continuously work without interruption if the sizes of all batches are same. Then, the conditions under which a feasible schedule exists are derived. Based on the conditions, we are able to develop a two-step solution method. At the first step, an integer linear program (ILP) is formulated for handling the batch allocation problem at the first stage. By the ILP, we need then to distinguish the batches only, greatly reducing the number of variables and constraints. Then, the batches assigned to each machine at the first stage are optimally sequenced at the second step by an algorithm with polynomial complexity. In this way, by the proposed method, the computational complexity is greatly reduced in comparison with the problem formulation without the established feasibility conditions. To validate the proposed approach, we carry out extensive experiments on a real case from an automotive glass manufacturer. We run ILP on CPLEX for testing. For large-size problems, we set 3600 s as the longest time for getting a solution and a gap of 1% for the lower bound of solutions. The results show that CPLEX can solve 96.83% cases. Moreover, we can obtain good solutions with the maximum gap of 4.9416% for the unsolved cases