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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