In this paper we analyze a known relaxation for the Sparsest Cut problem based on positive semidefinite constraints, and we present a branch and bound algorithm and heuristics based on this relaxation. The relaxed formulation and the algorithms were tested on small and moderate sized instances. It leads to values very close to the optimum solution values. The exact algorithm could obtain solutions for small and moderate sized instances, and the best heuristics obtained optimum or near optimum solutions for all testedinstances. The semidefinite relaxation gives a lower bound $\frac{C}{W}$ and each heuristic produces a cut S with a ratio $\frac{c_S}{w_S}$, where either cS is at most a factor of C or wS is at least a factor of W. We solved the semidefinite relaxation using a semi-infinite cut generation with a commercial linear programming package adapted to the sparsest cut problem. We showed that the proposed strategy leads to a better performance compared to the use of a known semidefinite programming solver.

Uncertainty in optimization is not a new ingredient. Diverse modelsconsidering uncertainty have been developed over the last 40 years.In our paper we essentially discuss a particular uncertainty modelassociated with combinatorial optimization problems, developed inthe 90's and broadly studied in the past years. This approach namedminmax regret (in particular our emphasis is on the robustdeviation criteria) is different from the classical approach for handlinguncertainty, stochastic approach, where uncertainty is modeledby assumed probability distributions over the space of all possiblescenarios and the objective is to find a solution with good probabilisticperformance. In the minmax regret (MMR) approach, the set of all possible scenariosis described deterministically, and the search is for a solution thatperforms reasonably well for all scenarios, i.e., that has the bestworst-case performance. In this paper we discuss the computational complexity of some classiccombinatorial optimization problems using MMR approach, analyze thedesign of several algorithms for these problems, suggest the studyof some specific research problems in this attractive area, and alsodiscuss some applications using this model.

This paper concerns a discrete time Geo[X]/G/1 retrial queue with general retrial time in which all the arriving customers require first essential service with probability $\alpha_{0}$ while only some of them demand one of other optional services: type − r (r = 1, 2, 3,...M) service with probability $\alpha_{r}$. The system state distribution, the orbit size and the system size distributions are obtained in terms of generating functions. The stochastic decomposition law holds for the proposed model. Performance measures of the system in steady state are obtained.  Finally, some numerical illustrations are presented to justify the influence of parameters on several performance characteristics.

Various topics related to reverse logistics have been discussed over the years. Most of them have assumed that facilities are kept open once they are established, and no returned products or recovery parts are stocked in intermediate recycling stations. However, firms may have the right to repeatedly open or close their facilities according to their economic benefits if they can acquire their facilities by lease. It also turns out that intermediate recycling stations like collection centers and disassembly centers usually stock returned products or parts in their facilities. By simultaneously relaxing these two assumptions, this study explores a logistics system with multiple items, each of which consists of some components among a variety of spare parts. The purpose is to maximize the total logistics costs by establishing a production schedule and reverse logistics framework over finite time periods for a logistics system. The mathematical model established in this study is a constrained linear integer programming problem. A genetic based algorithm is developed with the help of linear programming to find solutions to this problem. Limited computational experiments show that the proposed approach can produce better feasible solutions than the well-known CPLEX 10.0 software.