An efficient memetic algorithm for the uncapacitated single allocation hub location problem

Abstract

In this paper, we present a memetic algorithm (MA) for solving the uncapacitated single allocation hub location problem (USAHLP). Two efficient local search heuristics are designed and implemented in the frame of an evolutionary algorithm in order to improve both the location and allocation part of the problem. Computational experiments, conducted on standard CAB/AP hub data sets (Beasley in J Global Optim 8:429–433, 1996) and modified AP data set with reduced fixed costs (Silva and Cunha in Computer Oper Res 36:3152–3165, 2009), show that the MA approach is superior over existing heuristic approaches for the USAHLP. For several large-scale AP instances up to 200 nodes, the MA improved the best-known solutions from the literature until now. Numerical results on instances with 300 and 400 nodes introduced in Silva and Cunha (Computer Oper Res 36:3152–3165, 2009) show significant improvements in the sense of both solution quality and CPU time. The robustness of the MA was additionally tested on a challenging set of newly generated large-scale instances with 520–900 nodes. To the best of our knowledge, these are the largest USAHLP problem dimensions solved in the literature until now. In addition, in this paper, we report for the first time optimal solutions for 30 AP and modified AP instances.

Appendices

Appendix 1

In Tables 14 and 15 we present detailed results of the MA method on CAB instances. The largest CAB instance contains 25 nodes, while the smaller ones with 10, 15 and 20 nodes are obtained as its subsets. For each CAB instance, we considered four values of fixed costs: 100, 150, 200 and 250.

Table 14 Results of the MA on CAB instances (n = 10, 15)

Table 15 Results of the MA on CAB instances (n = 20, 25)

Stopping criteria imposed on the MA for the CAB data set are: G _max = 50 and R _max = 20.

From the results presented in Tables 14 and  15, it is obvious that the MA approach reaches all known optimal solutions in extremely short CPU times. Other heuristic methods, the HubTS, MSTS-3 and GA, have similar performance: all three of them are executed very quickly providing optimal solutions on all CAB instances. The optimal solutions for CAB data set and AP instances with 10 ≤ n ≤ 50 and n = 100 are obtained by CPLEX 9.1 solver and taken from Silva and Cunha (2009), together with corresponding CPU times.

Appendix 2

The new large-scale instances including n = 520, 600, 720, 800, 900 nodes with tight and loose fixed costs are created in the following way:

• We use two constants d _min = 500 and d _max = 2,000, taken from Silva and Cunha (2009). These values are chosen based on the average and maximum distances between two nodes in the original AP data set with 200 nodes;

• In order to obtain a new instance with $2\times n$ nodes from an original AP instance with n nodes we use the approach described in Silva and Cunha (2009). We will refer to this approach as Method 1

• To generate new instance with $3\times n$ nodes from an original AP instance with n nodes, for each node i in the original AP data set with n nodes located at (X _i , Y _i ), we generate three new nodes (X ^k _i , Y ^k _i ), k = 0, 1, 2, whose coordinates are calculated as:

  $$ X_i^{k} = X_i + (d_{\rm{min}} + r_i\times(d_{\rm{max}}-d_{\rm{min}}))\times {\rm{cos}}(\varphi+2k\times\pi/3), k=0,1,2 $$

  and

  $$ Y_i^{k} = Y_i + (d_{\rm{min}} + r_i\times(d_{\rm{max}}-d_{\rm{min}}))\times sin(\varphi+2k\times\pi/3), k=0,1,2, $$

  where r _i is a random number within the [0,1] interval and $\varphi$ is a random angle within the [0, π/3] interval. The new instance contains only newly generated nodes $(X_i^{k}, Y_i^{k}), k=0,1,2; i=1,\ldots,n.$ For any given pair of nodes (i, j) in the original AP data set, the corresponding amount of flow W _ij is equally distributed among the nine newly generated pairs of nodes, that is: $W_{ i^{k}j^{l}} = 1/9W_{ij}, k,l=0,1,2.$ The fixed costs for the new nodes are different and given by $f_i^{k} = (1- r_i^{k})\times f_i, k=0,1,2,$ where r ^k _i , k = 0, 1, 2 are three random numbers within the [0,0.3] interval. We will denote this approach for generating three times larger instance as Method 2;

• Instance with 600 nodes is obtained from the original AP instance with 200 nodes, by applying the Method 2;

• Instance of dimension n = 520 is created from the AP instance with n = 130 nodes, by applying the Method 1, which gives an instance with $2\times 130=260$ nodes. The Method 1 is then applied again, producing an instance with 2 × 260 = 520 nodes;

• The instance with n = 720 nodes is obtained from the original AP instance with n = 120. The Method 2 is first applied, creating an instance with $3\times 120=360$ nodes. The Method 1 is then used to generate an instance of dimension 2 × 360 = 520;

• Instance with n = 800 nodes is created from the AP instance with n = 200 in the same manner as the new instance with 520 nodes (i.e., Method 1 is applied two times);

• By applying Method 2 on AP instance with 100 nodes, we obtain a instance of dimension n = 300, which is used to create the instance with n = 900 by using Method 2 again.