Computational results

In this section, we discuss the computational results of the branch-and-cut al-gorithm. The algorithm was implemented in C++ (GCC version 4.6.3), using the elements of Standard Template Library (STL) and CPLEX 12.4 framework. As men-tioned in Sec. 2.7, the internal CPLEX cut generation was disabled and, CPLEX was used only to manage the enumeration tree. All the simulations were performed on

Procedure - Greedy Assignment Input: y^∗;

Output: assignments σ, set P of vertices that are spanned by some cycle;

comment: initialization for each i∈ T do σ(i) := −1;

T¯:= T ; comment: customers to be assigned

P := V ; comment: vertices that are spanned by some cycle comment: customer assignment

a Dell Precision T5500 workstation (Intel Xeon E5630 processor @2.53 GHz, 12 GB RAM). The computation times reported were expressed in seconds and we imposed a time limit of 7200 seconds for each run of the algorithm. The performance of the algorithm was tested on different classes of test instances, all generated using the traveling salesman problem library [69].

Instance generation: We generated two classes of test instances (I and II) having the same underlying graph, but with a different assignment cost structure (similar to [4, 40]). For each of the two classes and for each value of |T | ∈ {29, 51, 76, 101}, we generated 12 MDRSP instances using four TSPLIB instances [69] namely, bays29, eil51, eil76 and eil101. We performed a computational study on these instances with

|D| ∈ {3, 4, 5}. The depot locations were randomly generated. The routing costs and assignment costs were generated as follows:

Class I: The routing and assignment cost for a pair of vertices i, j is equal to the Euclidean distance lij between the two vertices.

Class II: For each pair of vertices i, j, the routing cost cij = αlij and the

as-signment cost dij = (10− α)l^ij where α ∈ {3, 5, 7, 9}. We refer to α as the scale factor.

Tables 2.1–2.3 summarize the computational behavior of the branch-and-cut al-gorithm on the two classes of instances. The column headings are defined as follows:

Name: instance name (for Classes I and II);

|D|: number of depots (for Classes I and II);

α: scale factor (for Class II);

%-LB: percentage LB/opt, where LB is the objective value of the LP relaxation computed at the root node of the enumeration tree (for Classes I and II);

%-LB0: percentage LB/opt, where LB is the objective value of the LP relaxation computed at the root node of the enumeration tree without adding the additional valid inequalities for the MDRSP (for Class II);

Pair: number of constraints (2.16) generated (for Classes I and II);

SEC: number of constraints (2.4) with |S| > 2 generated (for Classes I and II);

2mat: number of constraints (2.18) generated (for Classes I and II);

PEC: number of constraints (2.5) and (2.6) generated (for Classes I and II);

Nodes: total number of nodes examined in the enumeration tree (for Classes I and II);

Time: total computation time in seconds (for Classes I and II).

%Ring: total percentage of customers in present in the ring for the optimal MDRSP solution (for Class II)

Name |D| %-LB Pair SEC 2mat PEC Nodes Time

bays29 3 94.81 133 2939 17 618 119 4.52

bays29 4 99.30 46 676 8 1107 21 5.46

bays29 5 100.00 42 374 1 282 0 0.69

eil51 3 100.00 76 739 5 24 0 1.59

eil51 4 100.00 74 1182 6 83 0 6.76

eil51 5 100.00 78 1251 2 614 0 10.18

eil76 3 99.83 129 2615 23 1519 44 105.19

eil76 4 99.74 130 2483 10 2835 34 39.04

eil76 5 99.54 148 3738 70 7182 353 260.42

eil101 3 99.93 176 5441 8 1328 5 261.57

eil101 4 99.92 178 4551 9 1954 4 252.69

eil101 5 99.96 174 4118 8 3135 3 277.35

Averages 99.42 121.88 2508.92 13.92 1723.42 48.85 102.12

Table 2.1: Computational results for Class I instances

Table2.2:ComputationalresultsforClassIIinstances(bays29andeil51) Name|D|α%-LB%-LB0PairSEC2matPECNodesTime%-Ring bays293398.0897.45427622447354.29100.00 bays294398.0897.45219121473454.48100.00 bays295398.3197.918293255721284.3100.00 bays293598.9998.991262939176181194.5965.52 bays294599.3098.994667681116215.5265.52 bays2955100.0099.9342374130100.7265.52 bays293799.8099.75258309046903.9741.38 bays294799.7799.72221323029102.1641.38 bays295797.3297.271945122215635238.2227.59 bays2939100.0099.84257510000.090.00 bays2949100.0099.8821960000.060.00 bays2959100.0099.9017760000.060.00 eil513398.9598.791322801877452577266.88100.00 eil514398.7398.7018175416112715881451.58100.00 eil515398.6998.338144211991601274387.43100.00 eil5135100.0099.757673952401.5966.67 eil5145100.0099.7674118268306.7564.71 eil515599.9799.977812512614110.1954.90 eil513799.9099.87424796020284123.1129.41 eil514797.9197.91394276251948085272567.9925.49 eil515797.8597.483873692191908978412680.5523.53 eil5139100.0099.97402200000.240.00 eil5149100.00100.00355200000.250.00 eil5159100.00100.00353200000.250.00

Table2.3:ComputationalresultsforClassIIinstances(eil76andeil101) Name|D|α%-LB%-LB0PairSEC2matPECNodesTime%-Ring eil763399.7999.52126805010754138.13100.00 eil764399.7599.26219294323227444.1100.00 eil765399.6899.26108511912843522.19100.00 eil763599.8399.72129261523151944105.373.68 eil764599.7499.7313124071128353539.1973.68 eil765599.5699.561392899456075317195.5871.05 eil763799.3899.5214101564030966236.4338.16 eil764799.2299.19139115920199913280.3132.89 eil765798.7598.649777627841309816371928.6130.26 eil763999.8399.9814171105023353223.095.26 eil7649100.00100.0013881016017870139.855.26 eil7659100.0099.49943450040703.880.00 eil1013399.4699.241268050107541211.37100.00 eil1014399.4299.172192943232274131.63100.00 eil1015399.6299.371085119128435103.39100.00 eil1013599.9399.83129261523151944258.5172.28 eil1014599.9299.74131240711283535253.1674.26 eil1015599.9699.881392899456075317273.271.29 eil1013799.7899.7814101564030966724.7635.64 eil1014799.9099.88139115920199913562.434.65 eil1015799.5599.559777627841309816372938.6634.65 eil10139100.0099.5714171105023353449.836.93 eil10149100.0099.5013881016017870266.986.93 eil10159100.0099.2994345004070199.86.93 Averages:399.0598.7111.58929.6763.253348.42270.00139.15100.00 599.7799.66103.331916.9216.421967.8377.7596.1968.26 799.2699.04786.172496.6716.176658.251669.671007.2632.92 999.9999.78771.58438.750.00754.830.50107.032.61

The results tabulated in Tables 2.1–2.3 indicate that the proposed branch-and-cut algorithm can solve instances involving up to 101 customers with modest computa-tion times. All the instances were solved by the branch-and-cut algorithm within an hour. For a scale factor value of 3, we observe that the MDTSP solution is the optimal solution to the MDRSP. As the scale factor value is increased, this is clearly not the case because the percentage of customers present in the cycles decreases considerably. Furthermore, we observe that the Class II instances are more difficult, on an average, especially for a scale factor equal to 7. For the scale factor value of 7, the average percentage of customers present in the cycle in the optimal solution is 68%. These are the instances that take the maximum average computation time of 1007 seconds. Hence, the difficult instances tend to be those with relatively few assignment edges in the optimal solution. This is in contrast to the RSP [40], where the difficult instances tend to be those where the optimal cycle consists of about 20% of the customers. This major variation in the trade-off between the cycle costs and the assignment costs is due to the presence of the path elimination constraints in the MDRSP and the inherent challenges involved in solving multiple depot vari-ants. The %-LB column in both the tables indicate that the lower bound obtained at the root node of the enumeration tree is very tight, typically within 0.5% of the optimum. The %-LB0 column in the Tables 2.2 and 2.3 is the ratio of the lower bound obtained at the root node of the enumeration tree to the optimal solution;

here the lower bound is obtained by not using any of the additional valid inequalities developed for the MDRSP. This average %-LB0 is observed to be within 1.2% of the optimal solution for all the instances in Class II. Hence, we conclude that proposed mixed-integer linear programming formulation for the MDRSP is by itself very tight.

But a numerically observed advantage of the depot-2-matching inequalities was that for the instances where the number of violated depot-2-matching inequalities were

large, the number of path-elimination constraints added to the enumeration tree was reduced leading to an overall reduction in the computation time. This is be-cause these inequalities can themselves eliminate depot to depot paths. Overall, we were able to solve all the 60 test instances within an hour, with the largest instance involving 101 customers and 5 depots.