Both the proposed models fall in the SMIP with random recourse property. As a solution approach, both the L-shaped method and the L2 algorithm are efficiently applied. Laporte and Louveaux derive the L2 optimality cut for the piecewise linear approximation of the expected value function [29]. That L2 optimality cut requires lower bound of the recourse function. Thus, LP-relaxation of the proposed model is required to get the lower bound, and L-shaped method is utilized to get the lower bound value. Tighter lower bound value is essential to make the algorithm con- verged faster. Once the lower bound value is obtained by the L-shaped method, the L2 algorithm is utilized to solve the proposed models. To make the L2 algorithm converged faster in solving large-scale problems, the Benders’ cuts that are used in L-shaped method are applied in L2 algorithm. We call the proposed algorithm ’L2 with Benders’ Cuts Algorithm’. This approach provides a very tighter initial cut as
well as generates two strong valid inequalities that are the L2 optimality cut and the Benders cut in every iteration which significantly reduce the computational time in solving large-scale SMIPs. Before stating the algorithm, some preliminaries are introduced as follows.
• Original problem is given as below.
Min c>x + q>y (5.1)
s.t. Ax ≥ b (5.2)
T x + W y ≥ r (5.3)
x : binary, y : mixed integer (5.4)
q, W, r may have randomness, and let eω denote random, and ω denote any realiza- tion of eω variable. Then it can be decomposed as below where first-stage only have deterministic data, and second-stage may have stochastic and deterministic data.
1st Stage : Min c>x + E
e
ω[f (x, eω)] (5.5)
s.t. Ax ≥ b (5.6)
x : binary (5.7)
where for any realization ω of eω we have
2nd Stage : f (x, ω) = Min q(ω)>y (5.8)
s.t. W (ω) ≥ r(ω) − T (ω)x (5.9)
L-shaped Method
The L-shaped method is a efficient algorithm in solving two-stage recourse model if all the decision variables are continuous. Although the proposed models have mixed integer variables, the L-shaped method is utilized in solving the proposed models by applying LP-relaxation. This give the lower bound value L that is used in generating the L2 optimality cut in the L2 algorithm. Thus the following L-shaped method pro- cedure is applied to get the lower bound L. For notational conveniences, it is assumed that there are only equality constraints.
Step[0] : Initialization Let x0 be given.
Set ² ≥ 0, LB = −∞, U B = ∞, k ← 0
Step[1] : Solve Subproblem
Let s be scenario index, then for s = 1,· · · ,S Solve fk
s = Min qs>y
s.t. Wsy = rs − Tsxk
y ≥ 0
If infeasible for some s: Generate feasibility cut -Get dual extreme ray: µk
s -Compute αk = µks > rs βk = µks > Ts -Go to Step[2]
Else if feasible ∀ s: Generate optimality cut - Get πk s - Compute αk = P spsπks > rs
βk =
P
Spsπsk >
TS where ps: probability of scenario s
Upper Bounding Compute UB:
Vk = c>xk + P spSfsk
UB = min{Vk, UB}
If UB is updated, set incumbent solution to x∗ = xk
Go to Step[2]
Step[2] : Add Cut to Master Problem and Solve If some subproblem was infeasible
Add β>
kx ≥ αk to master problem
Else
Add β>
kx + η ≥ αk
Solve the master problem to get {xk+1, ηk+1} and Vk+1 as the master problem’s ob-
jective value Lower Bounding LB = max{Vk+1, LB} Master Problem Vk+1 = Min c>+ η s.t. Ax = b β> t x + η ≥ αt, t ∈ θk β> t x ≥ αt, t ∈ θk x ≥ 0
Where θk denotes iteration index set at which an optimality cut is generated,
and θk denotes iteration index set at which a feasibility cut is generated
Step[3] : Termination If UB − LB ≤ ²|UB| Stop ELSE k ← k + 1 Return to Step[1]
L2 with Benders’ Cuts Algorithm
Once the lower bound L of the recourse function is obtained by the L-shaped method, the L2 with Benders’ Cuts algorithm is applied to solve the proposed models. The detailed steps of the algorithm are as follows.
Step[0] : Initialization
Let ² ≥ 0, x1 ∈ Ax ≥ b, x ∈ {0, 1}, be given. Also let L be obtained by L-shaped method.
Set v1 ← −∞, V 1 ← ∞, k ← 1
Step[1] : Solve subproblem for all ω ∈ Ω Step[1-1] : Solve LP-relaxation of the subproblem:
Apply Step[1] in L-shaped method by applying LP-relaxation to solve subproblem. (Not to apply ’Upper Bounding’ part in Step[1])
Get the dual solutions of the subproblem, then create Benders cut. Step[1-2] : Solve the mixed-integer subproblem:
Set Vk+1 ← min{c>xk + E[xk, eω], Vk} This gives upper bound
Set F (xk) ← E[xk, eω]
Step[2]: Update and solve master problem
Step[2-1] : Append Benders cut to the master problem: If some subproblem was infeasible
Add β>
kx ≥ αk to master problem
Else
Add β>
kx + η ≥ αk to the master problem
Step[2-2] : Derive L2 Cut, then append the cut to the master Problem: Using L, xk, F (xk), derive the L2 optimality cut.
The cut that is a valid inequality for E[xk, eω] is defined as below.
η ≥ (F (xk) − L)(P j∈Skxj − P j∈ ¯Skxj − |Sk| + 1) + L, where Sk = {j|xk j = 1}
Append the cut to the master problem
Step[2-3] : Solve the master problem and get xk+1:
Master Problem vk+1 = Min c>+ η s.t. Ax = b β> t x + η ≥ αt, t ∈ θk β> t x ≥ αt, t ∈ θk βk t + η ≥ αt, t = 1, · · · , k x ∈ {0, 1}
Where θk ≡ denotes iteration index set at which a Benders optimality cut is gener-
ated, θk denotes iteration index set at which a Benders feasibility cut is generated
Let xk+1 be the solution of the master problem. Step[3]: Termination If Vk+1 - vk+1 ≥ ², Stop ELSE k ← k + 1 Return to Step[1]
• Note that this L2 with Benders’ Cuts algorithm can be a L2 algorithm if step[1-1] and step[2-1] are omitted from the procedures.