Non-vehicle related FSMPLIRP model parameters. Vehicles’ parameters. The above operators are embedded in two shaking procedures (the first does not include the S;). Their pseudocodes are provided in Algorithms 3 and 4. The most commonly used diversification method within VNS is the intensified shaking, which randomly selects a shaking operator and applies it k times, where k denotes the intense of diversifica- tion and it is 1<k <kmax, with kmax being the shaking strength. Additional to the intensified shaking, this work proposes two adap- tive shaking procedures. Initially, the five shaking operators are or- dered in a set. According to that initial order, two adaptive shaking procedures are formed. In the first procedure the initial order of operators is based on their computational complexity, while in the second one their ordering is performed randomly. However, both of them are executed similarly. More specifically, in each GVNS it- eration and for a specific k value, the shaking operators are ex- ecuted sequentially (shaking operator - pVND - solution renewal Algorithm 8 General VNS 3. docodes, the re-ordering mechanism of shaking operators is pro- vided in Algorithm 5. The ShakingOrder is the ordered set of shak- The pseudocodes of the first three GVNS schemes are provided in Algorithms 6-8. However, before the presentation of these pseu- Fig. 1. Flowchart of the proposed solution method. kmax analysis on the GVNS_1 performance on 30 GLIRP . Table 6 GVNS_2 vs GVNS_2p. Average and best values of GVNS_1, GVNS_2 and GVNS_3. Compare the results achieved by GVNS_2 and CPLEX (using light-duty vehicles). Compare the results achieved by GVNS_2 and CPLEX (using medium-duty vehicles) . Furthermore, it is interesting to examine the reduced case of GVNS_5. Table 12 contains the average and the best found solu- tions of GVNS_5p and their gap (%) from the corresponding GVNS_5 solutions. weight) which is obviously increased in the case of medium-duty fleet of vehicles. Fig. 3. The average fuel consumption (L) in cases of light- and medium-duty vehicles. Fig. 2. The average fuel consumption cost in cases of light- and medium-duty vehicles. Average and best values of GVNS_4, GVNS_5 and GVNS_6. sity. However, in order to control this diversity, operator Ng is also used. The numerical results of GVNS_5* and GVNS_5} are given in Table 13. shaking iterations which enables the improvement phase to be executed more times. Furthermore, during the experiments with GVNS_5 and GVNS_5p, it is noticed that the solutions with an in- crease in vehicle mixing, are found to be the best. In this direc- tion, an alternative of the GVNS_5 and GVNS_5p schemes (GVNS_5* and GVNS_5z respectively), which use the Shake instead of Shake, are tested. Due to the fact that a local search operator is more complex than a shaking operator, the shaking operator Ss is se- lected to be used as the expedient on increasing the fleet diver- Fig. 5. The average fuel consumption cost in cases of medium-duty vehicles and mixed-fleet. Fig. 4. The average CO, emissions (kg) in cases of light- and medium-duty vehicles. Compare the results achieved by GVNS_5* and CPLEX . The average and best found results of GVNS_5* and GVNS_5}. Compare the results achieved by GVNS_5* using different initialization meth- ods . Number of opened depots per instance. Fig. 6. The average fuel consumption (L) in cases of medium-duty vehicles and mixed-fleet. Fig. 7. The average CO2 emissions (kg) in cases of medium-duty vehicles and mixed-fleet. The opened depots and their capacity levels. Fig. 9. The average fuel consumption cost using different initialization rules. Fig. 10. The average fuel consumption cost of ten small-sized instances using different initialization rules. Solution by the proposed GVNS-based heuristic algorithms lead to opening the minimum required number of depots. As shown in Table 16, the proposed GVNS algorithms managed to open equal or less depots than the CPLEX solver for the case of small-sized instances. In all problem cases the same depots are selected to be opened. The reason behind this fact, is that in all these cases the structure of locations are kept unmodified. A more detailed infor- mation about the opened depots and their planned capacity levels is given in Table 17. Table 18 provides the fleet composition for the mixed-fleet problem case as it has been obtained by CPLEX solver for some of the small-sized instances and the GVNS_5;.... for all problem in- stances. The fleet composition decided by GVNS_5;,,,., corresponds to the best found solution for each problem instance. The letter “L” means light-duty vehicle and the letter “M” is used for medium- duty vehicles. Fig. 11. The average fuel consumption (L) using different initialization rules. The fleet composition by each method. Table 18
