Kriging with maximization of a multi-point expected improvement presents an interesting direction in the parallel budgeted optimization. However, this statistical criterion is a difficult-to-evaluate integral, and all the presently available methods are either extremely slow or not accurate enough. We introduce an extremely simple and fast integration method which is also accurate when the covariance matrices of kriging responses are diagonally-dominant, a frequently occuring case in practice. In addition, we state preliminary results with a novel importance sampling method which is developed as a more accurate alternative to theMonte Carlo based methods. Our tests are carefully designed to represent early and late stages of a budgeted optimization. We emphasize that integrating 4-variate functions can be extremely challenging.