After third iteration of the modified lower bound estimate algorithm (i.e., deleting rows corresponding to 1,3 and 5) , we have the following matrix

After eliminating minterm7, we exhaust all rows, but need not add 1 to the cost because there are two rows PI7 and PI8, which have all 1s.

As discussed in Section 4.4, we start exploring the solution space by selecting PI1 (i.e., PI1=1). With PI1 being taken, the following happens in the matrix

•  rows minterm1, minterm4, and minterm112 are covered,

•  columns PI2 (by PI3) and PI4 (by PI3) are dominated

•  column PI3 becomes essential.

After reduction (i.e., taking PI1, PI3 and eliminating PI2, PI3) we get the following matrix. As of now, the cost of the solution is 2 (PI1, PI3).

In this matrix Lower Bound = 2 comprising {5,7}; this is same, using the original lower bound computation algorithm as well as the modified one. So if we explore on this matrix the solution cost lower bound is 2+2=4. As this value is equal to that of the Lower Bound on the initial matrix, we explore on the search space. Let us now consider column PI5.

With PI5 being taken, the following happens in the matrix

•  rows minterm9, minterm10, and minterm11 are covered,

•  columns PI9 (by PI6), PI10 (PI6) and PI11 (by PI6) are dominated

After reduction (i.e., taking PI5 and eliminating PI92, PI10, PI1) we get the following matrix. As of now the cost of the solution is 3 (PI1, PI3,PI5).

In this matrix Lower Bound = 1 comprising {6}; this is the value obtained using the using the original lower bound computation algorithm of Section 4.3. Using the modified algorithm the Lower Bound = 2 (in this case 1 is added to the cost because there is no column with all 1s.) Therefore, if we explore on this matrix the solution cost lower bound is 3+2=5. As this value is higher than the Lower Bound on the initial matrix, we do not explore on the search space. As shown in Section 4.4, exploring in path PI1=1, PI5=1….., would have the cost of 5. As this value is available without actually exploring the path, when we use the modified bound computation algorithm, we save computation time in the branch and bound algorithm.