# Suppose that one of the following constraints arises when applying the implicit enumeration algorithm to a 0−1 integer program

Suppose that one of the following constraints arises when applying the implicit enumeration algorithm to a 0−1 integer program:

−2×1 − 3×2 + x3 + 4×4 ≤ −6,                                               (1)

−2×1 − 6×2 + x3 + 4×4 ≤ −5,                                              (2)

−4×1 − 6×2 − x3 + 4×4 ≤ −3.                                                 (3)

In each case, the variables on the lefthand side of the inequalities are free variables and the righthand-side coefficients include the contributions of the fixed variables.

a) Use the feasibility test to show that constraint (1) contains no feasible completion.

b) Show that x2 = 1 and x4 = 0 in any feasible completion to constraint (2). State a general rule that shows when a variable xj, like x2 or x4 in constraint (2), must be either 0 or 1 in any feasible solution. [Hint. Apply the usual feasibility test after setting xj to 1 or 0.]

c) Suppose that the objective function to minimize is:

z = 6×1 + 4×2 + 5×3 + x4 + 10,

and that z = 17 is the value of an integer solution obtained previously. Show that x3 = 0 in a feasible completion to constraint (3) that is a potential improvement upon z with z < z.=”” (note=”” that=”” either=””>1 = 1 or x2 = 1 in any feasible solution to constraint (3) having x3 = 1.)

d) How could the tests described in parts (b) and (c) be used to reduce the size of the enumeration tree encountered when applying implicit enumeration?