In the paper two new feasibility tests are given. The first test gives the sufficient condition of feasibility of a design. The second test gives a sufficient condition of infeasibility of a design. The values of the tests give the upper and lower bounds of the value of the feasibility test introduced by Halemane and Grossmann (AlChE J. 29(3), 425-433, 1983). The procedure for determination of the value of the feasibility test of Halemane and Grossmann based on the use of the "branch and bound" method is suggested. The approaches to solving a one-stage programming problem and a two-stage programming problem are considered. Two problems are introduced and that permits to obtain upper and lower bounds of the criterion in the two stage programming problem. Application of the suggested approaches is illustrated by three examples.