Suppose that one of the following constraints arises when applying the implicit enumeration…

Suppose that one of the following constraints arises when applying the implicit enumeration…
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? Apr 24 2022 07:43 AM

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