  In this paper, we deal with sequential testing of multiple hypotheses. In the
general scheme of construction of optimal tests based on the backward
induction, we propose a modification which provides a simplified (generally
speaking, suboptimal) version of the optimal test, for any particular criterion
of optimization. We call this DBC version (the one with Dropped Backward
Control) of the optimal test. In particular, for the case of two simple
hypotheses, dropping backward control in the Bayesian test produces the
classical sequential probability ratio test (SPRT). Similarly, dropping
backward control in the modified Kiefer-Weiss solutions produces Lorden's
2-SPRTs .
  In the case of more than two hypotheses, we obtain in this way new classes of
sequential multi-hypothesis tests, and investigate their properties. The
efficiency of the DBC-tests is evaluated with respect to the optimal Bayesian
multi-hypothesis test and with respect to the matrix sequential probability
ratio test (MSPRT) by Armitage. In a multihypothesis variant of the
Kiefer-Weiss problem for binomial proportions the performance of the DBC-test
is numerically compared with that of the exact solution. In a model of normal
observations with a linear trend, the performance of of the DBC-test is
numerically compared with that of the MSPRT. Some other numerical examples are
presented.
  In all the cases the proposed tests exhibit a very high efficiency with
respect to the optimal tests (more than 99.3\% when sampling from Bernoulli
populations) and/or with respect to the MSPRT (even outperforming the latter in
some scenarios).
