Traditional cost models ignore the internal structure of decision-making units (DMUs), so, may produce ambiguous outcomes and provide a biased assessment. In this paper, we evaluate the performance of the units by considering their internal structures. We proposed a new cost Malmquist index for measuring the cost productivity change of the units with bi-level structures. The bi-level structure is a special case of hierarchical structures with two levels, where the leader unit is positioned at the upper level and followers are located at the lower level. The overall system of bi-level units tries to use inputs and produce outputs in a cost-efficient way. However, each subunit performs according to its goals and limited resources. This research tries to develop a bi-level cost model that is suitable for measuring the cost efficiency of bi-level units. Based on this model, a new cost Malmquist index (CMI) is suggested to evaluate the productivity changes of bi-level units. This index presents a new aspect of CMI and provides the productivity changes of units by considering the impact of the leader's and the subunits' performance. In addition, similar to the traditional CMI, it decomposes into various components, such as cost efficiency changes and cost technological changes. The developed CMI is applied to a real-world case study to evaluate eight management regions which all together manage 198 branches. The results show that the proposed CMI provides a more meaningful evaluation of DMUs compared to the conventional CMI. Manuscript profile

Generalized solution on Neumann problem of the fourth order ordinary differentialequation in space $W^2_\alpha(0,b)$ has been discussed, we obtain the condition on B.V.P when thesolution is in classical form. Formulation of Quintic Spline Function has been derived and theconsistency relations are given.Numerical method,based on Quintic spline approximation hasbeen developed. Spline solution of the given problem has been considered for a certain valueof $\alpha$. Error analysis of the spline method is given and it has been tested by an example. Manuscript profile

This paper presents a numerical matrix method based on Bernstein polynomials (BPs) for approximate the solution of a system of m-th order nonlinear Volterra integro-differential equations under initial conditions. The approach is based on operational matrices of BPs. Using the collocation points,this approach reduces the systems of Volterra integro-differential equations associated with the given conditions, to a system of nonlinear algebraic equations. By solving such arising non linear system, the Bernstein coefficients can be determined to obtain the finite Bernstein series approach. Numerical examples are tested and the resultes are incorporated to demonstrate the validity and applicability of the approach. Comparisons with a number of conventional methods are made in order to verify the nature of accuracy and the applicability of the proposed approach. Keywords: Systems of nonlinear Volterra integro-differential equations; The Bernstein polyno- mials and series; Collocation points. 2010 AMS Subject Classication: 34A12, 34A34, 45D05, 45G15, 45J05, 65R20. Manuscript profile

‎In this paper we propose a numerical scheme to solve the one dimensional nonlinear Klein-Gorden equation‎. ‎We describe the mathematical formulation procedure in details‎. ‎The scheme is three level explicit and based on nonstandard finite difference‎. ‎It has nonlinear denominator function of the step sizes‎. ‎Stability analysis of the method has been given and we prove that the proposed method when applied to one dimensional nonlinear Klein-Gorden equation‎, ‎is unconditionally stable‎. ‎We illustrate the usefulness of the proposed method by applying it on two examples. Manuscript profile