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

Algorithm 1

Procedures of benders decomposition.

	Initialization: k: = 0, f = 0, o = 0; LB: = −∞, UB: = ∞; set an initial feasible solution , and algorithm stop condition ε = 0.
	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 PM.
	end if
	if Bounded then
	Obtain an extreme ray ;
	let o = o + 1;
	let ;
	Add an optimality cut to PM;
	Obtain a upper bound, .
	end if
	Solve PM.
	Obtain a solution ;
	Update the lower bound, ;
	k: = k + 1;
	: = ;
	end while
	Return the optimum result.