In this paper, a method is developed to minimize the expected warranty servicing cost per item sold. In this method, to be decided to repair or replace a failed item. These kinds of items work in multi-state deteriorating and they may fail in every state. The last decades some researches for minimizing the expected warranty servicing cost under one-dimensional (Time) and nonrenewable warranty policy have been published. But here, the warranty policy has Two-dimensions (Time & Usage). The decision to repair or replace a failed item depends on Two parameters, the deterioration degree of the item and the length of the residual warranty period. The optimality of these Two parameters for minimizing the expected warranty servicing cost are examined by Two methods, analytically and simulation. Then an offered algorithm is used as an approximating method for performing the warranty policy.

warranty is a powerful implement for marketing strategy that is used by manufacturers and creates satisfaction for consumers by ensuring to compensate for incorrect operation of the product. warranty service results in a cost named warranty cost for a manufacturer. This cost is a function of warranty policy, regions, and product failures pattern. Since this service covers the cost of uncertain failure of the product, it makes some utility for customers. In this paper, we developed a novel customer utility function that is used as a customer objective to be maximized. In addition to the manufacturer objective, minimizing the warranty cost is considered simultaneously. There are four restrictions on warranty parameters such as time, usage, unit product price and the R&D expenditure to be considered. Finally, we will propose a novel bi-objective model that maximizes the utility function for customers and minimizes the warranty cost for the manufacturer. This model will be solved with an evolutionary algorithm called Non-Dominated Sorting Genetic Algorithm (NSGA-II) and non-dominated Pareto solutions will be gained from this method. To give a numerical instance, for a certain usage rate’ s range of custumers, different warranties are provided and compared. It is believed that the computational results can help manufacturers to determine optimal solutions for the objective functions and consequently warranty parameters.

In this paper, a Two-dimensional multi-wavelet is constructed in terms of Chebyshev polynomials. The constructed multi-wavelet is an orthonormal basis for L2 [0,1]2 space. By discretizing Two-dimensional Fredholm integral equation reduce to a algebraic system. The obtained system is solved by the Galerkin method in the subspace of L2 [0, 1]2 by using Twodimensional multi-wavelet bases. Because the bases of subspaces are orthonormal, so the above mentioned system has a small dimension and also high accuracy in approximating solution of integral equations. For one-dimensional case, a similar works are done in [4,5], which they have small dimension and high accuracy. In this article, we extend one-dimensional case to Twodimensional by extending and by choosing good functions on Two axes. Numerical results show that the above mentioned method has a good accuracy.

Exact solution for steady Two-dimensional flow of an incompressible magma is obtained. The magmatic flow is studied by considering the magma as a second grade fluid. The governing partial differential equations are transformed to ordinary differential equations by symmetry transformations. Results are discussed through graphs to understand the rheology of the flowing magma.

A subset W of the vertices of a graph G is a resolving set for G when for each pair of distinct vertices u; v 2 V (G) there exists w 2 W such that d(u; w) ̸ = d(v; w). The cardinality of a minimum resolving set for G is the metric dimension of G. This concept has applications in many diverse areas including neTwork discovery, robot navigation, image processing, combinatorial search and optimization. The problem of  nding metric dimension is NP-complete for general graphs but the metric dimension of trees can be obtained using a polynomial time algorithm. In this paper, we investigate the metric dimension of Cayley graphs on dihedral groups and we characterize a family of them.