Accelerating Benders Decomposition

1. Fakhri Ashkan, Ghatee Mehdi,
Fragkogios Antonios, Saharidis K.D. Georgios
UNIVERSITY OF THESSALY
School of Engineering
Department of Mechanical Engineering
Division of Production Management & Industrial Administration
3rd International Conference on Energy, Sustainability and Climate
Change, July 10-16, 2016
With the contribution of the LIFE
programme of the European Union -
LIFE14 ENV/GR/000611

2.  BENDERS METHOD
 PROPOSED ALGORITHM
 APPLICATION-RESULTS
 CONCLUSION

3.  Jacobus Franciscus (Jacques) Βenders: Dutch
Mathematician (1925-today) Emeritus
Professor of Operations Research at the
Eindhoven University of Technology.
 Benders, Jacques F. “Partitioning procedures
for solving mixed-variables programming
problems.” Numerische mathematik 4.1
(1962): 238–252.

4.  Since 1962, more than 5.000 papers have been
published, which modify, extend and
accelerate the method.
 Sections:
 Mixed-Integer Linear Programming,
 Stochastic Programming,
 Multi-Objective Programming,
 Non-Linear Programming
 Etc.

5.  Applications:
 Crew Scheduling,
 Plant Scheduling,
 Supply Chain Network Design,
 Power Systems
 Etc.

6.  Mixed-Integer Problem
 Relaxed Master Problem
(RMP)
 Primal Subproblem
(PSP)
Complicating variables

7. START
Solve Master Problem. Compute
Lower Bound.
Solve SubProblem.
Upper Bound – Lower
Bound >= e
Produce
Benders
Optimality
Cut.
END
YES
NO
SubProblem is
Feasible.
Compute Upper Bound.
NO
YES
Produce
Benders
Feasibility
Cut.
Add Benders Cuts to Master
Problem.

8.  BENDERS METHOD
 PROPOSED ALGORITHM
 APPLICATION-RESULTS
 CONCLUSION

9.  In order to apply Benders decomposition
method, the duality of the subproblem is
necessary (Dual Subproblem exists).
 The Primal Subproblem has to be continuous and no duality
gap exists.
 Application at Mixed-Integer Linear
Problems:
 Master Problem: Integer variables and
 Primal Subproblem: Only continuous variables of the
original problem.

10.  However, the decomposition of the original
problem might lead to a Primal Subproblem,
which includes integer variables.
 The application of the classical Benders
decomposition method is impossible.
 A Branch-and-Cut algorithm is proposed,
which allows the use of Benders method at the
case of integer subproblem.

11.  The integrality constraints of the Primal Subproblem are
relaxed and the Relaxed Subproblem is solved in a Branch-and-
Bound framework.
Integer Primal Subproblem
(PSP)
Relaxed Primal Subproblem
(RPSP)

12. 0
1 2
5
4
6
3
87
109
i
i+1 i+2
Branch-and-Bound

13. 0
1 2
5
4
6
3
87
109
i
i+1 i+2
Branch-and-Cut (with Benders method)
Application of
Benders
method
between RMP
and A-RPSP

14. Augmented Relaxed Primal
Subproblem (A-RPSP(k))
Relaxed Master Problem (RMP)

15. 0
1 2
5
4
6
3
87
109
i
i+1 i+2
Branch-and-Cut (with Benders method)
Application of
Benders
method
between MP
and RPSP
MP0
SP0
MP0
SP1
MP0
SP2
MP0
SP3
MP0
SP4
MP0
SP5
MP0
SP6
MP0
SP7 MP0
SP8
MP0
SP9 MP0
SP10

16. Proof that the Benders cuts, which are produced in
a node of the Branch-and-Bound tree are valid for all
descendant nodes, but not necessarily for the non-
descendant nodes. These cuts, referred to as
LOCAL CUTS, can be used to warm start the
master problem of each descendant node leading
to better initial bounds.
GENERAL CUTS are formed out of the LOCAL
CUTS. This general form enables us to reuse the
generated (local) cuts in the whole tree by
updating some values of the function.

17. Augmented-RMP (A-RMP(k))

18. 0
1 2
5
4
6
3
87
109
Branch-and-Cut (with Benders method+Local Cuts)
MP0
SP0
MP1
SP1
MP1
SP2
MP3
SP3
MP3
SP4
MP5
SP5
MP5
SP6
MP7
SP7 MP7
SP8
MP9
SP9 MP9
SP10
MP1=MP0+BendersCuts0
MP3=MP1+BendersCuts2
MP5=MP3+BendersCuts4
MP7=MP5+BendersCuts5
MP9=MP7+BendersCuts7
“Local Cuts”

19. Generalization of Local Cuts (General Cuts)
if we are going to use the generalized cuts (55) or (57) in another node, say k2, it is sufficient to replace y* by the optimal value of y at node k2

20. Global Cuts
if we are going to use the generalized cuts (55) or (57) in another node, say k2, it is sufficient to replace y* by the optimal value of y at node k2

21.  BENDERS METHOD
 PROPOSED ALGORITHM
 APPLICATION-RESULTS
 CONCLUSION

22.  Capacitated Fixed Charge Multiple Knapsack
Problem- CFCMKP.
 Pure Integer Problem
 Binary variables
 Integer Primal Subproblem
APPLICATION No.1

23.  20 Examples
 Solution with the classical Branch-and-Cut
algorithm with Benders method, but without
use of “Local Cuts” (B&C-NotLC) and with
the proposed algorithm with “Local Cuts”
(B&C-LC).
 Comparison of the results.

24. Instances B&C-NotLC B&C-LC
Relative difference B&C-LC over
B&C-NotLC
Α/Α M N
CPU time
(Sec.)
Iterations
CPU time
Iterations CPU time Iterations
(Sec.)
1 5 40 5.54 380 2 59 -63.90% -84.50%
2 5 60 11.15 593 3.27 71 -70.70% -88.00%
3 5 80 3.2 186 1.94 26 -39.40% -86.00%
4 6 80 8.81 375 4.4 77 -50.10% -79.50%
5 7 80 3.72 129 1.75 29 -52.90% -77.50%
6 8 80 31.79 838 0.48 11 -98.50% -98.70%
7 9 80 4.91 185 1.55 24 -68.50% -87.00%
8 10 50 334.29 3431 179.11 638 -46.40% -81.40%
9 10 60 62.64 1642 2.41 53 -96.10% -96.80%
10 10 70 264.7 2582 11.72 175 -95.60% -93.20%
11 10 80 411.44 3477 2.51 34 -99.40% -99.00%
12 10 90 1.23 49 0.76 9 -38.40% -81.60%
13 10 100 25.53 474 4.82 48 -81.10% -89.90%
14 15 50 13.25 391 9.39 92 -29.10% -76.50%
15 15 60 6.23 189 3.59 64 -42.40% -66.10%
16 15 70 3.29 95 2.33 39 -29.20% -58.90%
17 15 80 54.63 892 14.52 91 -73.40% -89.80%
18 15 90 13.01 301 8.39 76 -35.50% -74.80%
19 15 100 1565.1 4404 0.35 7 -99.90% -99.80%
20 15 110 36.78 502 6.5 57 -82.30% -88.60%

25.  Significant reduction of the CPU time (-29.1%
ως -99.9%, Average: -64.7%)
 Large reduction of the total number of
iterations of Benders algorithm inside all nodes
(-58.9% ως-99.8%, Average: -84.9%). The small
number of iterations is due to the better initial
lower bounds. Thus, the MP is solved fewer
times and the proposed algorithm is faster.

26.  Environmental Multi-Modal Journey-
Planning Problem- E-MMJP (Action B.3)
 Pure Integer Problem (Binary variables-Integer Primal
Subproblem)
 Mixed Integer Linear Program (MILP) in order to compute
the optimal journey between the departure and arrival
stops of the public network.
 In-between those two stations, the model prompts the user
to use up to a number of different modes of transport,
depending on his/her input. While in the network, the user
follows an optimal journey that minimizes the travel time
APPLICATION No.2

27.  Decision variables
 Xi,j,k,n Binary Variable used to represent whether a
transfer is made from i to j with mode k and
trip n.
 Si,j,k,n Non-negative general integer variable used
to represent the departure time from i to j
with k and n.

28.  Objective Function
Minimization of 2 criteria, the total environmental cost and
the total travel time of the journey, which is proposed to the
user. Coefficients a and b are predefined by the user.
The Environmental Cost Ci,j,k is pre-computed for each arc i-j
and mode k of the public transportation network using
emission calculation models, that take into consideration
several parameters, such as the type of fuel (gasoline, diesel,
electricity etc.) and the fuel consumption, which concern the
vehicle of the public means of transport. Other parameters
concern the trip, such as the distance and the gradient
between the stops
, , , , ,, ,
1 1 1 1
( * * )*
N N M L
i j k i j k ni j k
i j k n
Min z a bC TT X= = = =
= +∑∑∑∑

29.  Due to the pure integrality of the problem,
when decomposed using Benders
Decomposition method, the Primal
Subproblem will be an integer one.
 Thus, this modelling approach for the
environmental MMJP problem should be
based on the proposed Branch-and-Cut
algorithm using Benders Decomposition with
Local Cuts (B&C-LC)

30.  BENDERS METHOD
 PROPOSED ALGORITHM
 APPLICATION-RESULTS
 CONCLUSION

31.  A novel, better and valid algorithm is
proposed for the solution of problems, where
after application of Benders decomposition,
the Primal Subproblem is integer.
 Acceleration of the classical Branch-and-Cut
algorithm with the application of “Local
Cuts”.

32. Thank you for your attention
With the contribution of the LIFE
programme of the European Union -
LIFE14 ENV/GR/000611