This paper considers the general lot sizing and scheduling problem (GLSP) in single level capacitated environments with sequence dependent item changeovers. The proposed model simultaneously determines the production sequence of multiple items with capacity-constrained dynamic demand and lot size to minimize overall costs. First, a mixed-integer programming (MIP) model for the GLSP is developed in order to solve smaller size problems. Afterwards, a matheuristic algorithm that integrates Simulated Annealing (SA) algorithm and the proposed MIP model is devised for solving larger size problems. The proposed matheuristic approach decomposes the GLSP into sub-problems. The proposed SA algorithm plays the controller role. It guides the search process by determining values for some of the decision variables and calls the MIP model to identify the optimal values for the remaining decision variables at each iteration. Extensive numerical experiments on randomly generated test instances are performed in order to evaluate the performance of the proposed matheuristic method. It is observed that the proposed matheuristic based solution method outperforms the MIP and SA, if they are used alone for solving the present GLSP.

The set covering problem is to find the minimum cardinality set of locations to site the facilities which cover all of the demand points in the network. In this classical problem, it is assumed that the potential facility locations and the demand points are limited to the set of vertices. Although this problem has some applications, there are some covering problems in which the facilities can be located along the edges and the demand exists on the edges, too. For instance, in the public service environment the demand (population) is distributed along the streets. In addition, in many applications (like bus stops), the facilities are not limited to be located at the vertices (intersections), rather they are allowed to be located along the edges (streets). For the first time, this paper develops a novel integer programming formulation for the set covering problem wherein the demand and facility locations lie continuously along the edges. In order to find good solutions in a reasonable time, a matheuristic algorithm is developed which iteratively adds dummy vertices along the edges and solves a simpler problem which does not allow non-nodal facility locations. Finally, a Benders decomposition reformulation of the problem is developed and the lower bounds generated by the Benders algorithm are used to evaluate the quality of the heuristic solutions. Numerical results show that the Benders lower bounds are tight and the matheuristic algorithm generates good quality solutions in short time.