LOGISTICS
5. EXPERIMENTAL ANALYSIS
In order to identify the potentiality of proposed MRCVRP model in the improvement of the cold logistics, computational experiments on testing the performance of MRCVRP are reported. First, a set of MRCVRP instances are created in Section 5.1; then, a three-phase experiment are designed in Section 5.2 for testing; and finally, the results of the testing are summarized and analyzed in Section 5.3.
5.1 Bank of Testing Instances
As a new type of vehicle routing related problem, MRCVRP is still unavailable with a bank of benchmark instances or with a real-world case of cold logistics in any literature. Therefore, we generated a set of sixty MRCVRP instances based on the classical VRPTW instances. A set of 56 VRPTW instances which were originally generated by Solomon (1983), are classified as three types of geographical scatter of customers, i.e., C for clustering, R for randomizing, and RC for mixture of randomizing and clustering. Figure 2 demonstrates the configurations of these three patterns of customers spread. All of the VRPTW instances comprise 100 customer nodes with corresponding (x,
y) coordinates, volume of demand, and lower bound and upper bound of required time window.
(a) Randomized scatter (b) Clustered scatter (c) Mixed scatter Figure 2. Three Geographical Scatter of Customers for VRPTW Instances
We chose three VRPTW instances, C101 from type C, R101 from type R, and RC101 from type RC, as the bases to generate customer’s coordinates and demands for MRCVRP instances. Due to the cold logistics, we set three levels of commodities’temperature, i.e., cold (0oC ~ 5oC), refrigerant (-18oC ~ -5oC) and frozen (-30oC below). Moreover, four scenarios were hypothesized for the demands of three-temperature goods as follows.
●Scenario 1: Demands of goods on three temperature levels for all customers are identical.
We established five types of demands for every customer and every temperature level, 10, 20, 30, 40, 50, which adopt the coordinates of customers from C101, R101 or RC101 correspondingly. For example, Instance 1_C-10 means that this instance belongs to scenario 1, the coordinates of customers come from C101, and the demands of goods on three temperature levels for all customers are equal to 10 units. There are total of 15 instances in Scenario 1.
●Scenario 2: Demands of goods for customers are not identical; demands on three temperature levels for some customer are identical.
The coordinates of customers and the demands of goods on every temperature level for customers adopt C101, R101 or RC101 respectively. For example, Instance 2_R-C means that this instance belongs to scenario 2, the coordinates of customers come from R101, the demands of goods for customers come from C101, and the demands of goods on three temperature levels for some customer are identical. There are total of 9 instances in Scenario 2.
●Scenario 3: Demands of goods for customers are not identical; demands on three temperature levels for some customer are different and independent.
The coordinates of customers and the demands of goods on the first temperature level (i.e., cold) for customers adopt C101, R101 or RC101 respectively. To generate independent demands of goods between three temperature levels, we utilize systematic manner to assign three demands of every customer. For example, Instance 3_R-C-11-22 means that this instance belongs to scenario 3, the coordinates of customers come from R101, and the demands of goods for customers on the first temperature level come from C101. In this case, the demand of goods on the second temperature level (i.e., refrigerant) for some customer i is equal to the demand of cold goods for customer i + 11. Similarly, the demand of goods on the third temperature level (i.e., frozen) for customer i is equal to the demand of cold goods for customer i + 22. We design three systematic lags for refrigerant goods (11, 33 and 66) and frozen goods (22, 55 and 88).
Therefore, there are total of 27 instances in Scenario 3.
●Scenario 4: Demands of goods for customers are not identical; demands on three temperature levels for some customer are different and dependent.
Similar to Scenario 3, the coordinates of customers and the demands of goods on the first temperature level for customers adopt C101, R101 or RC101 respectively. In order to generate dependent demands of goods among three temperature levels, we assume that the demands of refrigerant and frozen goods for some customer i are separately equal to reduce and to add the demand of cold goods for customer i by 5 units. For example, Instance 4_R-C means that this instance belongs to scenario 4, the coordinates of customers come from R101, and the demands
of cold goods for customers come from C101. Let the demand of cold goods for some customer be 20 units, therefore the demands of refrigerant and frozen goods for this customer are 15 and 25 respectively. There are total of 9 instances in Scenario 4. Generally, the demands in Scenarios 3 and 4 are considered closer to the real-world cases than that in Scenario 1 and 2.
We suppose that the demand of goods on every temperature level for every customer is dividable to load on different refrigerated containers with the same temperature level in the identical vehicle.
Additionally, the capacity of vehicles is equal to 400 units and the number of customers is 100 for all sixty instances. The time windows corresponding to every customer originally set in VRPTW are ignored.
5.2 Design of Experiments
This research designs three phases of experiments to analyze the performance and potential of the proposed MRCVRP model and heuristic. The first experiment aims to decide an efficient size of the refrigerated containers, the second experiment tests the effect of the heuristic method proposed in Section 4, and the third experiment compare the MRCVRP with traditional VRP according to their performance on solving instances. Because that the practical cost of trucks and refrigerated containers are unavailable, all experiments consider the quality of solutions at two criteria, the total distance of routes and the total number of vehicles. The details of three-phase experiments explain as follows.
●Phase I: The size of the refrigerated container means the capacity of the container, q. Due that the capacity of a vehicle is 400 units, which is equal to the product of q by p (the maximal number of refrigerated containers that a vehicle can load), we set four combinations of (p, q), i.e., (5, 80), (8, 50), (10, 40) and (20, 20). Four combinations are tested on all sixty instances and solved by six savings algorithms, i.e., sequential SA0 (SSA0), sequential SA1 (SSA1), sequential SA2 (SSA2), parallel SA0 (PSA0), parallel SA1 (PSA1) and parallel SA2 (PSA2).
Some combination with the best performance is selected for Phase II.
●Phase II: First, six savings algorithms are tested on all instances. The best savings algorithm is chosen as following initial solution construction module. Then, four interchange-based heuristics, i.e., 2_OPT, (1_0), (1_1) and (2_1), are tested respectively. Finally, various combinations of sequential execution these interchange heuristics are tested.
●Phase III: Instances of Scenario 3 and 4 are used to test the MRCVRP and VRP. In order that VRP can not deliver goods on three temperature levels at one time, we must divide every MRCVRP instance into three VRP instances. Among the three VRP instances, the coordinates of customers are identical, but demands of goods are corresponding to those of different
temperature levels. On the other hand, for the consistent purpose, we adopt the SSA0 to construct initial solutions of MRCVRP and VRPs.
5.3 Computational Results
Results of the three-phase experiments are reported and analyzed in this Subsection. As previously mentioned, the total distance of routes (routing distance) and the total number of vehicles (fleet size) are compared respectively.
The results of Phase I testing present that the capacity of refrigerated container greatly influences upon the routing cost and the fleet size. As shown in Table 2, the combination (20, 20) performs well than other combinations. Judging from the averages of the six savings algorithms, the routing distance under the combinations (10, 40) and (20, 20) are smaller than that under combinations (5, 80) and (8, 50) in all four scenarios, and the routing distance under the combination (10, 40) is very close to that under the combination (20, 20) which makes the smallest routing distance, 1530.67. In terms of fleet size, the combination (20, 20) still demands a minimal fleet size of 17.44. It appears that small size of refrigerated container makes full use of the capacity. To simplify the following tests, we only adopt the combination (20, 20) in terms of the refrigerated container’s capacity and the maximum number of refrigerated containers packed in each vehicle.
Table 2. Results of Test on the Combination of Parameters p and q