**5. Hybrid Benders/Genetic algorithm**

The basic idea of Benders' partitioning algorithm for mixed-integer linear problems is to decompose the original problem into a pure integer master problem and one or more subproblems in the continuous variables, and then to iterate between these two problems. If the objective function value of the optimal solution to the master problem is equal to that of the subproblem, then the algorithm terminates with the optimal solution of the original mixed-integer problem. Otherwise, we add constraints, termed Benders' cuts, one at a time to the master problem, and solve it repeatedly until the termination criteria are met. A major difficulty with this decomposition lies in the solution of the master problem, which is a "hard" problem, costly to compute.

For the addressed CPL problem, however, the constraints are explicit only in the subproblem and the master problem is free of explicit constraints. Thus, the master problem is more amenable to solution by GA.

Lai et al. (2010) introduced a hybrid Benders/Genetic algorithm which is a variation of Benders' algorithm that uses a genetic algorithm to obtain "good" subproblem solutions to the master problem. Lai and Sohn (2011) conducted a study applying the hybrid Benders/Genetic algorithm to the vehicle routing problem. Below is a detailed description of the hybrid algorithm and it is illlustrated in Fig. 1 as well.


Using a Genetic Algorithm to

and potential plant sites (see Fig. 2).

Fig. 2. Fifty Randomly Generated Points

subproblems.

the algorithm was terminated.

Solve the Benders' Master Problem for Capacitated Plant Location 413

randomly generated, and the first 20 of these points were designated as both demand points

The transportation cost between two points is proportional to the Euclidean distance between them. Three variations of Benders' algorithm were applied to this plant location problem: (1) Optimization of master problem using implicit enumeration (BD-Opt); (2) Suboptimization of master problem using implicit enumeration (BD-Subopt); and (3) Suboptimization of master problem using a genetic algorithm (Hybrid BD/GA). In each case, the problem was not solved to completion, but was terminated after solving 50

First, an implicit enumeration algorithm was used to optimize Benders' master problem. Fig. 3 shows the values of the upper and lower bounds, i.e., the solutions of the subproblems and master problems, respectively. The incumbent solution, which was found at iteration #10, is shown in Fig. 4 and requires opening 11 plants with a total cost of 5398, of which 2619, or 48.5%, are fixed costs of the plants and the remaining costs are transportation costs. The greatest lower bound at this stage is 4325, so that the gap is approximately 19.9% when

Secondly, the algorithm was restarted and again 50 iterations were performed, but suboptimizing the master problem using implicit enumeration. Fig. 5 shows the progress of this case. Because the master problem was suboptimized, no lower bound is available. After 50 iterations, the incumbent solution shown in Fig 6, which requires opening seven plants, has a total cost of 5983, of which 1710, or approximately 28.6%, are fixed costs of the plants. It is important to note, of course, that although the quality of the incumbent solution is somewhat inferior to that found by optimizing the master problem, the computational effort

is miniscule compared to that required when the master problem is optimized.

	- **4a.** Initialization. We initialize the variable *Y* as a string of binary bit with the position #*i* corresponding to the plant #*i*. We generate initial population and their fitness function are evluated as well.
	- **4b.** Genetic Operations. We perform a standard single-point crossover approach. The mutation operation to guarantee the diversity of the population is performed as well. The current population is replaced by the new population through the incremental replacement method.
	- **4c.**Termination. We terminate the GA if no improvement within 100 iterations.

Fig. 1. Flowchart of the Hybrid Benders/Genetic Algorithm

This hybrid algorithm would avoid other traditional search methods, i.e., branch-andbound, which were used in the master problem. It will search the solution space in parallel fashion and take advantage of the "easy" evaluation of the fitness function.

#### **6. Example**

To illustrate the hybrid algorithm discussed in the earlier section, we use a randomlygenerated problem with 20 plant sites and 50 customers. Fifty points in a square area were 412 Bio-Inspired Computational Algorithms and Their Applications

**Step 3.** Generation of Benders' Cut. We compute a new linear support using the dual

**Step 4.** Primal Master system by GA. A trial location paln *Y* is to be computed by

implementing a GA whose solution delivers both a feasible investment plan and a

**4a.** Initialization. We initialize the variable *Y* as a string of binary bit with the position #*i* corresponding to the plant #*i*. We generate initial population and their fitness

**4b.** Genetic Operations. We perform a standard single-point crossover approach. The mutation operation to guarantee the diversity of the population is performed as well. The current population is replaced by the new population through the

**4c.**Termination. We terminate the GA if no improvement within 100 iterations.

This hybrid algorithm would avoid other traditional search methods, i.e., branch-andbound, which were used in the master problem. It will search the solution space in parallel

To illustrate the hybrid algorithm discussed in the earlier section, we use a randomlygenerated problem with 20 plant sites and 50 customers. Fifty points in a square area were

fashion and take advantage of the "easy" evaluation of the fitness function.

solution of the transportation subproblem and increment *k* by 1.

lower bound to the minimal cost for the equivalent program.

function are evluated as well.

incremental replacement method.

Fig. 1. Flowchart of the Hybrid Benders/Genetic Algorithm

**6. Example** 

randomly generated, and the first 20 of these points were designated as both demand points and potential plant sites (see Fig. 2).

Fig. 2. Fifty Randomly Generated Points

The transportation cost between two points is proportional to the Euclidean distance between them. Three variations of Benders' algorithm were applied to this plant location problem: (1) Optimization of master problem using implicit enumeration (BD-Opt); (2) Suboptimization of master problem using implicit enumeration (BD-Subopt); and (3) Suboptimization of master problem using a genetic algorithm (Hybrid BD/GA). In each case, the problem was not solved to completion, but was terminated after solving 50 subproblems.

First, an implicit enumeration algorithm was used to optimize Benders' master problem. Fig. 3 shows the values of the upper and lower bounds, i.e., the solutions of the subproblems and master problems, respectively. The incumbent solution, which was found at iteration #10, is shown in Fig. 4 and requires opening 11 plants with a total cost of 5398, of which 2619, or 48.5%, are fixed costs of the plants and the remaining costs are transportation costs. The greatest lower bound at this stage is 4325, so that the gap is approximately 19.9% when the algorithm was terminated.

Secondly, the algorithm was restarted and again 50 iterations were performed, but suboptimizing the master problem using implicit enumeration. Fig. 5 shows the progress of this case. Because the master problem was suboptimized, no lower bound is available. After 50 iterations, the incumbent solution shown in Fig 6, which requires opening seven plants, has a total cost of 5983, of which 1710, or approximately 28.6%, are fixed costs of the plants. It is important to note, of course, that although the quality of the incumbent solution is somewhat inferior to that found by optimizing the master problem, the computational effort is miniscule compared to that required when the master problem is optimized.

Using a Genetic Algorithm to

evaluated by the subproblem.)

Solve the Benders' Master Problem for Capacitated Plant Location 415

generations) at which time all those solutions better than the incumbent were evaluated. (After each subproblem, the trial solutions are re-evaluated, using the updated master problem cost function, ( ) *<sup>T</sup> v Y* , and only those with cost less than the incumbent are

Fig. 5. Subproblem solutions of variation 2 of Benders' algorithm (BD-Subopt).

Fig. 6. Incumbent Solution Found by variation 2 of Benders' algorithm (BD-Subopt).

Fig. 3. Upper and lower bounds provided by Benders' algorithm (BD-Opt).

Fig. 4. Incumbent Solution Found by Benders' algorithm (BD-Opt).

Finally, the algorithm was again restarted, and 50 trial solutions were evaluated by the subproblems, this time using a genetic algorithm, so that the master problem is again suboptimized to generate the trial solutions. Each master problem was terminated after 40 trial solutions better than the incumbent have been found (or after a maximum of 100 414 Bio-Inspired Computational Algorithms and Their Applications

Fig. 3. Upper and lower bounds provided by Benders' algorithm (BD-Opt).

Fig. 4. Incumbent Solution Found by Benders' algorithm (BD-Opt).

Finally, the algorithm was again restarted, and 50 trial solutions were evaluated by the subproblems, this time using a genetic algorithm, so that the master problem is again suboptimized to generate the trial solutions. Each master problem was terminated after 40 trial solutions better than the incumbent have been found (or after a maximum of 100 generations) at which time all those solutions better than the incumbent were evaluated. (After each subproblem, the trial solutions are re-evaluated, using the updated master problem cost function, ( ) *<sup>T</sup> v Y* , and only those with cost less than the incumbent are evaluated by the subproblem.)

Fig. 5. Subproblem solutions of variation 2 of Benders' algorithm (BD-Subopt).

Fig. 6. Incumbent Solution Found by variation 2 of Benders' algorithm (BD-Subopt).

Using a Genetic Algorithm to

by the subproblems.

Benders' algorithm

Hybrid BD/GA, *trial 1* Hybrid BD/GA, *trial 2*

Variation of

BD-Opt BD-Subopt

Solve the Benders' Master Problem for Capacitated Plant Location 417

Fig. 9. Upper bounds provided by Benders' subproblems in variation 3 (Hybrid BD/GA).

Incumbent total cost

Table 1. Summary of results of variations of Benders' algorithm

problem and suboptimizing it by implicit enumeration.

The best of the 50 trial solutions was found at iteration 49, with a total cost of 5303, of which 988 (or approximately 18.6%) were fixed costs. Five plants were opened in this solution (see Fig. 7). Again, because the master problem is being suboptimized, no lower bound is available from the algorithm. Due to the random nature of the genetic algorithm, a second run of this variation was performed and found another incumbent solution (see Fig. 8). Fig. 9 shows the progress of two trials of the hybrid algorithm, i.e., the upper bounds provided

> Fixed costs

As well, Table 1 summarizes the results obtained by these three variations of Benders' algorithm (terminated after 50 subproblems have been solved). Remarkably, in these results we observe no significant degradation of the quality of the solution when the master problem is suboptimized using a genetic algorithm, compared to optimizing the master

% fixed costs

48.5% 28.6% 18.6% 33.8% # plants open

Fig. 7. Incumbent Solution by variation 3 of Benders' algorithm (Hybrid BD/GA) *trial 1*.

Fig. 8. Incumbent Solution by variation 3 of Benders' algorithm (Hybrid BD/GA) *trial 2*.

In this case, it happens that only 7 master problems were required to generate the 50 trial solutions. (A population size of 50 was used, with 75% probability of crossover and 1% probability of mutation.)

416 Bio-Inspired Computational Algorithms and Their Applications

Fig. 7. Incumbent Solution by variation 3 of Benders' algorithm (Hybrid BD/GA) *trial 1*.

Fig. 8. Incumbent Solution by variation 3 of Benders' algorithm (Hybrid BD/GA) *trial 2*.

probability of mutation.)

In this case, it happens that only 7 master problems were required to generate the 50 trial solutions. (A population size of 50 was used, with 75% probability of crossover and 1%

Fig. 9. Upper bounds provided by Benders' subproblems in variation 3 (Hybrid BD/GA).

The best of the 50 trial solutions was found at iteration 49, with a total cost of 5303, of which 988 (or approximately 18.6%) were fixed costs. Five plants were opened in this solution (see Fig. 7). Again, because the master problem is being suboptimized, no lower bound is available from the algorithm. Due to the random nature of the genetic algorithm, a second run of this variation was performed and found another incumbent solution (see Fig. 8). Fig. 9 shows the progress of two trials of the hybrid algorithm, i.e., the upper bounds provided by the subproblems.


Table 1. Summary of results of variations of Benders' algorithm

As well, Table 1 summarizes the results obtained by these three variations of Benders' algorithm (terminated after 50 subproblems have been solved). Remarkably, in these results we observe no significant degradation of the quality of the solution when the master problem is suboptimized using a genetic algorithm, compared to optimizing the master problem and suboptimizing it by implicit enumeration.

Using a Genetic Algorithm to

225.

pp. 175-192.

Solve the Benders' Master Problem for Capacitated Plant Location 419

Geoffrion, A. M. & Graves, G.W. (1974). Multicommodity distribution system design by Benders decomposition. *Management science*, Vol. 20, No. 5, pp. 822-844. Goldberg, D. E. (1989). *Genetic Algorithms in Search, Optimization, and Machine Learning*, Addison-Wesley Professional, ISBN 978-0201157673, Boston, MA, USA. Goldberg, D. E. & Lingle, R. J. (1985). Alleles, loci and the traveling salesman problem,

He, S.; Chaudhry, S. & Chaudhry, P. (2003). Solving a class of facility location problems

Heragu, S. S. & Chen, J. (1998). Optimal solution of cellular manufacturing system design:

Holland, J. H. (1992). *Adaptation in Natural and Artificial Systems: An Introductory Analysis* 

Holmberg, K.. & Ling, J. (1997). A Lagrangean heuristic for the facility location problem

Kennington, J. L. & Whitler J. E. (1999). An efficient decomposition algorithm to optimize

Kershenbaum, A. (1997). When genetic algorithms work best. *INFORMS Journal on* 

Kim, H.; Sohn, H. & Bricker, D.L. (2011). Generation expansion planning using Benders'

Kratica, J.; Tosic, D.; Filipovic, V. & Ljubic, I. (2001). Solving the simple plant location problem by genetic algorithm. *RAIRO Operations Research*, Vol. 35, pp. 127-142. Levine, D. (1997). Genetic algorithms: a practitioner's view. *INFORMS Journal on computing*,

Lai, M.; Sohn, H.; Tseng, T. & Chiang, C. (2010). A Hybrid Algorithm for Capacitated Plant Location Problems. *Expert Systems with Applications*, Vol. 37, pp. 8599-8605. Lai, M. & Sohn, H. A Hybrid Algorithm for Vehicle Routing Problems. (2011) working

Liepins, G. E. & Hilliard, M. R. (1989). Genetic algorithms: foundations and applications.

Michalewicz, Z. (1998). *Genetic Algorithms + Data Structures = Evolution Programs*, Springer,

Oliver, I. M.; Smith, D. J. & Holland, J. R. (1987). A study of permutation crossover

Randazzo, C.; Luna, H. & Mahey, P. (2001). Benders decomposition for local access network

operators on the traveling salesman problem. *Proceedings of the 2nd International Conference on Genetic Algorithms*, pp. 224-230, ISBN 0-8058-0158-8, Hillsdale, NJ,

design with two technologies. *Discrete mathematics and theoretical computer science*,

using genetic algorithms. *Expert Systems*, Vol. 20, No. 2, pp. 86-91.

0-8058-0426-9, Pittsburgh, PA, USA, July, 1985.

978-0262581110, Cambridge, MA, USA

*computing*, Vol. 9, No. 3, pp. 254-255.

*Engineering*, Vol. 18, No. 1, pp. 25-39.

*Annals of operations research*, Vol. 21, pp. 31-58.

ISBN 978-3540606765, New York, USA

Vol. 11, No. 2, pp. 149-160.

Vol. 9, No. 3, pp. 256-259.

paper.

USA, July, 1985.

Vol. 4, pp. 235-246.

*Canadian Journal of Operational Research and Information Processing*, Vol. 37, pp. 194-

*Proceedings of the 1st International Conference on Genetic Algorithms*, pp. 154-159, ISBN

Benders' decomposition approach. *European journal of operational research*, Vol. 107,

*with Applications to Biology, Control, and Artificial Intelligence*, A Bradford Book, ISBN

with staircase costs. *European journal of operational research*, Vol. 97, No. 1, pp. 63-74.

spare capacity in a telecommunications network. *INFORMS Journal on computing*,

decomposition and generalized networks. *International Journal of Industrial* 
