[Retracted] A Hybrid Algorithm of Ant Colony and Benders Decomposition for Large-Scale Mixed Integer Linear Programming

Algorithm 2

Procedures of hybrid algorithm of ant colony algorithm and benders decomposition.

	Initialization: k: = 0, f = 0, o = 0; LB: = −∞, UB: = ∞; set an initial feasible solution , and algorithm stop condition ε = 0.
	Generate the initial solution by ant colony algorithm:
	Let  = 0.7, index = 1, Q = 104, m = 20, NCmax = 30;
	Set : = ones (UBn − LBn + 1, n1 + n2)·(UBn − LBn + 1);
	for i: = 1 ⟶ m do
	Randomly obtained m solutions.
	end for
	Find local minimum solution fmin;
	Update and .
	while index < NCmax − 1 do
	for i: = 1 ⟶ m − 1 do
	Randomly obtained m − 1 new solutions.
	end for
	Add local minimum solution to the new solutions;
	Update fmin, and ;
	index = index + 1.
	end while
	Obtain the initial solution .
	Obtain upper bound by Benders decomposition:
	while UB − LB > ε do
	Solve the PD.
	if Infeasible then
	Exit.
	end if
	if Unbounded then
	Obtain an extreme ray ;
	let f = f + 1;
	let ;
	Add a feasibility cut 0 to PFM.
	end if
	if Bounded then
	Obtain an extreme ray ;
	let o = o + 1;
	let ;
	Add an optimality cut to PFM;
	Obtain a upper bound, UB: = Min{UB, }.
	end if
	Obtain feasible solution and lower bound by feasible adjustment rule:
	Let maxiter: = 25,  = ,  = 1,  = 0.9;
	Set UBFM: = ∞, and set the loop stop condition λ = 10 − 4;
	Solve PFM and obtain the solution (x^∗, θ^∗);
	Let x^∗ = round(x^∗), α^1 = µα^0;
	Update UBFM: = Min{UBFM, ||x^∗ − x^0||2};
	for i: = 1 ⟶ maxiter − 1 do
	if ||x^i + 1 − x^i||2 > λ then
	Let x^i = x^∗;
	Solve PFM;
	Update the solution (x^∗, θ^∗);
	if ||x^i + 1 − x^i||2 < UBFM then
	Let UBFM: = ||x^i + 1 − x^i||2;
	x^∗ = round (x^i + 1);
	end if
	end if
	Update α^i + 1: = µα^i.
	end for
	Update the lower bound, LB: = Max{LB, c};
	k: = k + 1;
	: = ;
	end while
	Return the optimum result