Some Studies on the Principles and Mechanisms for Loading and Unloading

Figure ý6. Comparison of makespan for loading three trains

Table ý6. Minimum makespan for loading three trains

STFLTFLBFExistingMakespan599.5 608.5601.5608.5Number of trucks88615Cost for unloading operations

The cost of unloading a containers from trains and transporting it to its location in the storage yard depends upon the travel time of the truck, processing time taken by the crane, and waiting time for the truck. The Eq ý6. is used to compute the cost µ §of unloading a single container.

Figure ý6. gives the comparisons of the cost obtained through three models based on the SFT, LFT, and FAT principles. It can be seen from Figure ý6. that the lowest cost of operation is obtained using the model based on STF principle while the LTF model produces the worst results in terms of cost.

Figure ý6. Comparison of cost for unloading three trains

Table ý6. shows the cost of operation corresponding to minimum makespan obtained through three different models. The existing operation cost is also given in Table ý6..

Table ý6. Operation cost for unloading three trains

STFLTFFATExistingCost (Rupees)6265 7326.5 690810057 Number of trucks58715Cost for loading operations

The cost of loading a container onto a train after bringing it from its location in the storage yard depends upon the travel time of the truck, processing time taken by the crane, and waiting time for the truck. The Eq ý6. is used to compute the cost µ §of loading a single container

Figure ý6. gives the comparisons of the cost obtained through three models based on the SFT, LFT, and LBT principles. It can be seen from Figure ý6. that the lowest cost of operation is obtained using the model based on LBT principle while the LTF model produces the worst results in terms of cost.

Figure ý6. Comparison of cost for loading three trains

Table ý6. shows the cost of operation corresponding to minimum makespan obtained through three different models. The existing operation cost is also given in Table ý6.. it is found that STF gave the minimum cost while LTF gave the worst results.

Table ý6. Operation Cost for loading three trains

STFLTFLBTExistingCost (Rupees)535655104746.58209.5 Number of trucks88715Waiting time for unloading operations

The waiting time of truck to unload a container from a train after return back to the rail yard depends upon the arrival time and departure time of a truck. The Eq ý6. has been used to compute the waiting time µ § of unloading a single container

Figure ý6. gives the comparisons of the waiting time obtained through three models based on the SFT, LFT, and FAT principles. It can be seen from Figure ý6. that the lowest waiting time of operation is obtained using the model based on STF principle while the LTF model produces the worst results in terms of waiting time.

Figure ý6. Comparison of waiting time for unloading three trains

Table ý6. shows the waiting time of operation corresponding to minimum makespan obtained through three different models. The existing operation cost is also given in Table ý6.

Table ý6. Waiting time for unloading three trains

STFLTFFATExistingWaiting time (minutes)1139 29082225.58209.5Number of trucks58715Waiting time for loading operations

The waiting time of truck to load a container to a train after coming back to the storage yard depends upon the arrival time and departure time of a truck. The Eq ý6. is used to compute the waiting time µ §of loading a single container.

Figure ý6. gives the comparisons of the waiting time obtained through three models based on the SFT, LFT, and LBT principles. It can be seen from Figure ý6. that the lowest waiting time of operation is obtained using the model based on LBT principle while the LTF model produces the worst results in terms of waiting time

Figure ý6. Comparison of waiting time for loading of three trains

Table ý6. shows the waiting time of operation corresponding to minimum makespan obtained through three different models. The existing operation cost is also given in Table ý6.

Table ý6. Waiting Time for loading three trains

STFLTFLBTExistingWaiting time(minutes)2818.5 2969.516647093Number of trucks88615Case study 4

This section describes the results of application of greedy algorithm model based on FAT principle, reverse greedy algorithm model based on LBT principle, greedy algorithm model based on STF principle, and greedy algorithm model based on LTF principle. In this case, as given in Table ý6. there are four trains, 18 trucks and 270 jobs to be processed. The processing time (s) and travel time (2d) for each job for case study 3 is given in Table ý6.

Table ý6. Input data of case study 4

jobs(min)d (min)job.s(min)d (min)job.s (min)d (min)14.002.50911.002.501814.002.5022.003.50923.002.001822.002.0032.004.00932.002.501833.003.5042.002.00943.003.501843.002.50 52.503.00953.006.001852.005.00 62.002.50963.001.501865.003.50 72.003.00974.004.001873.002.50 82.002.50983.00 3.501884.002.00 92.005.00993.005.001891.503.00103.002.501003.003.501902.002.50113.003.001014.002.501912.003.50125.003.501022.001.501925.003.50134.001.501032.002.501934.001.50143.003.501042.003.501943.003.50153.003.001052.002.001953.003.00162.002.501063.002.501962.002.50172.002.001075.003.501972.002.00183.004.001083.002.001983.004.00192.002.501093.00 4.501992.002.50203.002.001102.003.502001.504.50213.503.001112.005.002012.005.00223.003.501123.002.502022.002.50232.504.001132.001.502032.503.00242.002.501143.002.002042.003.50253.001.501154.002.502052.002.50262.002.501163.003.502063.003.50273.004.001172.001.502072.001.50281.502.001182.006.002083.004.50293.003.501192.002.502093.004.00302.002.501202.003.502102.002.50314.003.501214.001.502113.002.00323.002.001224.004.502122.002.00332.002.501232.002.502134.004.50343.004.001244.006.002146.002.50352.002.001254.002.502153.003.00363.003.501263.003.502163.002.50372.002.001272.002.502175.006.00383.005.001283.003.002181.503.00393.502.001292.002.002193.003.50402.002.501304.002.502205.004.00413.003.001314.003.502213.002.00424.002.501324.001.502225.003.50434.004.001335.003.502234.004.50443.003.501343.005.002241.503.00452.002.001353.502.002252.003.50462.005.001364.002.502262.002.50472.002.501372.003.502272.004.00483.002.001382.004.002282.503.50493.004.501392.002.002296.002.50502.002.501402.503.002305.003.50513.003.001412.002.502312.003.00522.002.501422.003.002324.003.50532.002.001432.002.502332.004.00543.003.501442.005.002344.002.50554.002.501453.002.502354.003.50562.002.001463.003.002362.002.50573.003.501473.503.002373.002.00583.002.501483.003.502385.003.00592.005.001492.504.002392.002.50605.003.501502.002.502402.003.00613.002.501513.001.502413.003.50624.002.001522.002.502424.004.00631.503.001533.004.002432.002.50642.002.501541.502.002444.003.00652.003.501553.003.502452.004.50663.004.001562.002.502461.502.50674.002.501574.003.502471.501.50684.003.501583.002.002482.002.50693.005.001592.002.502492.503.00702.002.501603.004.002505.003.50713.003.501612.002.002513.002.00722.001.501623.003.502526.002.50733.002.501632.002.002533.004.00743.004.501643.005.002543.001.50752.003.001653.502.002554.003.00763.003.501662.002.502564.002.50774.002.501673.003.002573.003.00784.001.501684.002.502582.003.50793.002.001694.004.002593.003.50802.002.501703.003.502603.001.50813.003.501712.002.002613.002.00822.004.501722.005.002622.002.50833.002.501732.002.502632.503.00843.002.001743.002.002642.003.50852.002.501753.004.502652.003.50862.002.501762.002.502662.002.50873.001.501773.003.002672.501.50882.002.501782.002.502683.003.50893.004.501792.002.002692.002.50901.003.501803.003.502702.002.00Makespan for unloading operations

A Comparison of makespan obtained using the three models based on STF, LTF, and FAT principles is presented in Figure ý6. It can be observed from Figure ý6. that the minimum makespan of 760 minutes is achieved using the model based on FAT. The number of trucks obtained corresponding to the minimum makespan of 760 is 7. For the existing situation, the minimum makespan with 18 trucks is 774 minutes. Therefore, there is a saving of 14 minutes and 11 trucks when the model based on FAT principle is used. The minimum makespan and the corresponding number of trucks for the three models and for the existing situation are shown in . In terms of makespan, the worst performance was obtained by using the STF model.

Figure ý6. Comparison of makespan for unloading four trains

Table ý6. Minimum makespan for unloading four trains

STFLTFFATExistingMakespan (minutes)771 760.5760774Number of trucks88718Makespan for loading operations

A comparison of the makespan obtained using the three models based on STF, LTF, and LBT principles is presented in Figure ý6.. It can be observed from Figure ý6. that the minimum makespan of 761.5 minutes is achieved using the model based on STF. The number of trucks obtained corresponding to the minimum makespan of 761.5 is 8. For the existing situation, the minimum makespan with 18 trucks is 770.5 minutes. Therefore, there is a saving of 9 minutes and 11 trucks when the model based on STF principle is used. In terms of makespan, the worst performance was obtained by using the LTF model. The minimum makespan and the corresponding number of trucks for the three models and for the existing situation are shown in Table ý6..

Figure ý6. Comparison of makespan for loading three trains

It can be seen from Figure ý6. that there is no reduction in the makespan even when the number of trucks are increased beyond 7. This results shows that no further savings in terms of makespan could be achieved by increasing the number of trucks beyond 7 with either of the three models considered here. However, the optimal number of trucks in this case study is 5.

Table ý6. Minimum makespan for loading four trains

STFLTFLBFExistingMakespan( minutes)761.5 770.5763.5770.5Number of trucks87518Cost for unloading operation

The cost of unloading a containers from trains and transporting it to its location in the storage yard depends upon the travel time of the truck, processing time taken by the crane, and waiting time for the truck. The Eq ý6. is used to compute the cost µ §of unloading a single container. The comparisons of the cost obtained through three models based on the STF, LTF, and FAT principles is presented in Figure ý6.. It can be seen from Table ý6. that the lowest cost of operation is obtained using the model based on FAT principle while the LTF model produces the worst results in terms of cost.

Figure ý6. Comparison of cost of unloading four trains

Table ý6. shows the cost of operation corresponding to minimum makespan obtained through three different models. The existing operation cost is also given in Table ý6..

Table ý6. Comparison of operation cost for unloading four trains

STFLTFFATExistingCost (Rupees.)912492788784.513974Number of Trucks88718Cost for loading operation

The cost of loading a container onto a train after bringing it from its location in the storage yard depends upon the travel time of the truck, processing time taken by the crane, and waiting time for the truck. The Eq ý6. is used to compute the cost µ §of loading a single container.

Figure ý6.gives the comparison of cost obtained through three models based on the SFT, LFT, and LBT principles. It can be seen from Figure ý6. that the lowest cost of operation is obtained using the model based on LBT principle while the STF model produces the worst results in terms of cost.

Figure ý6. Comparison of cost for loading four trains

Table ý6. shows the cost of operation corresponding to minimum makespan obtained through three different models. The existing operation cost is also given in Table ý6..

Table ý6. Operation cost of for loading four trains

STFLTFLBTExistingCost (Rupees)6780.5 6488.5 560411621 Number of trucks87518Waiting time for unloading operation

The waiting time of truck to unload a container from a train after return back to the rail yard depends upon the arrival time and departure time of a truck. The Eq ý6. has been used to compute the waiting time µ § of unloading a single container

Figure ý6. gives the comparison of the waiting time obtained through three models based on the SFT, LFT, and FAT principles. It can be seen from Figure ý6. that the lowest waiting time of operation is obtained using the model based on FAT principle while the LTF model produces the worst results in terms of waiting time.

Figure ý6.Comparison of waiting time for unloading four trains

Table ý6. shows the waiting time of operation corresponding to minimum makespan obtained through three different models. The existing operation cost is also given in Table ý6.

Table ý6. Waiting time for unloading four trains

STFLTFFATExistingWaiting time(minutes)3645.5 3694.5286311287.5Number of trucks88718Waiting time of loading operation

The waiting time of truck to load a container to a train after coming back to the storage yard depends upon the arrival time and departure time of a truck. The Eq ý6. is used to compute the waiting time µ §of loading a single container.

Figure ý6. gives the comparisons of the waiting time obtained through three models based on the STF, LTF, and LBT principles. It can be seen from Figure ý6. that the lowest waiting time of operation is obtained using the model based on LBT principle while the STF model produces the worst results in terms of waiting time

Figure ý6. Comparison of waiting time for loading four trains

Table ý6. shows the waiting time of operation corresponding to minimum makespan obtained through three different models. The existing operation cost is also given in Table ý6..

Table ý6. Waiting time for loading four trains

STFLTFLBTExistingWaiting time (minutes)3605 29951396.511232.5Number of trucks87518Summary of loading and unloading results

In this section, summary of results of minimum makespan and related number of trucks, cost and waiting time is presented in Table ý6., Table ý6., Table ý6.. The results presented here have obtained from models based on STF, LTF, FAT, and LBT scheduling mechanisms as discussed in the previous sections. The number of jobs considered in case studies 1 to 4 ranges from 70 to 270. The number of trucks varies from 9 to 18, and upto four trains have been considered in the four case studies discussed in the previous sections. Summary of minimum makespan with related number of trucks obtained with four models based on different scheduling algorithms for loading unloading operations with different number of trains are given in Table ý6.. The summary of cost related to the minimum makespan with number of trucks is given in Table ý6.. In same way, the summary of waiting time is given in Table ý6..

Table ý6. Summary of minimum makespan for loading-unloading operations

Scheduling MechanismsSTFLTFLBTFATOne Train

(loading operations)Makespan(min)204.5213.5208-Trucks675-One Train

(unloading operations)Makespan(min)214.5204.5-201.5Trucks59-5Two Trains

(loading operations)Makespan(min)423.5432.5427.7-trucks786-Two Trains

(unloading operations)Makespan(min)434422.5-421trucks56-7Three Trains

(loading operations)Makspan(min)599.5608.5601.5-trucks886-Three Trains

(unloading operations)Makespan(min)610.5602.5-601trucks55-7Four Trains

(loading operations)Makespan(min)761.5770.5763.5-trucks875-Four Trains

(unloading operations)Makspan(min)771760.5-760trucks88-7

Table ý6. Summary of cost corresponding to the minimum makespan for loading and unloading operations

Scheduling mechanismSTFLTFLBTFATOne Train

(loading operations)Cost (Rs)168418691563-trucks675-One Train

(unloading operations)Cost(Rs)21462180-2164.5trucks55-5Two Trains

(loading operations)Cost(Rs)35743935.53384-trucks786-Two Trains

(unloading operations)Cost(Rs)44004695-4861trucks56-7Three Trains

(loading operations)Cost(Rs)535655104746.5trucks886-Three Trains

(unloading operations)Cost (Rs)62656319-6908trucks587Four Trains

(loading operations)Cost(Rs)6780.56488.55604-trucks875-Four Trains

(unloading operations)Cost(Rs)91249278-8784.5trucks88-7

Table ý6. Summary of waiting time corresponding to the minimum makespan for loading and unloading operations

Scheduling mechanismSTFLTFLBTFATOne Train

(loading operations)Waiting time

(min)562.5838333-trucks675-One Train

(unloading operations)Waiting time

(min)397391-322trucks55-5Two Trains

(loading operations)Waiting time

(min)1576.521171151trucks786-Two Trains

(unloading operations)Waiting time

(min)8141234-1525.5trucks56-7Three Trains

(loading operations)Waiting time

(min)2818.52969.51664trucks886-Three Trains

(unloading operations)Waiting time

(min)11392908-2225.5trucks58-7Four Trains

(loading operations)Waiting time

(min)360529951396.5-trucks875-Four Trains

(unloading operations)Waiting time

(min)3645.53694.5-2863trucks88-7

Analysis of models results

Optimal number of trucks

In this section, analysis of results obtained from models based on STF, LTF, FAT, and LBT scheduling mechanisms has been carried out. The results have been presented for optimal number of trucks, makespan, cost, and waiting time for different number of trains. In each case study, the results have presented for loading, unloading and combined loading-unloading operations. The optimal number of trucks required to unload different number of trains is presented in Figure ý6.. It can be seen from Figure ý6. that for unloading one to three trains, the best results in terms of minimum number of trucks required is obtained using STF model. For unloading four trains, the best results were obtained using FAT model. Results obtained by FAT model are as good as those obtained by STF model for unloading one train. The results produced by LTF model are inferior to STF and FAT models for unloading any number of trains.

Figure ý6. shows optimal number of trucks required for loading a given number of trains using STF, LTF, and LBT models. The optimal solution obtained using LBT model is superior to those produced by other models for loading operations. For loading one train, the STF model suggests that six numbers of trucks must be used for loading one train. However, a better optimal solution (five trucks) is obtained using LBT model. Similarly, LBT model produces the best solution for loading two to four trains. The worst results are produced by the LTF model.

Figure ý6. Number of trucks versus number of trains in unloading process

Figure ý6. Number of trucks versus number of trains in loading process

Figure ý6. Number of trucks versus number of trains in loading-unloading process

The optimal results obtained by using different models for combined loading and unloading operations are presented in Figure ý6.. The STF and LTF model have been applied to both loading and unloading operations. The LBT model is used for loading operations only while FAT model can be used for unloading operations. It can be seen from Figure ý6. that ten trucks are required for loading and unloading a single train. For processing both loading and unloading jobs for two and three trains, the best results are obtained using STF model. For four trains, the best results (12 trucks) are obtained by using FAT and LBT models. The worst results for combined operations are produced by the LTF model.

Optimal makespan

Figure ý6. shows optimal makespan for unloading different number of trains using STF,LTF, and FAT models while Figure ý6. shows optimal makespan for loading different number of trains using STF, LTF, and LBT model. Figure ý6. shows optimal makespan for combined loading and unloading number of trains using STF, LTF, LBT, andFAT algorithms. The optimal makespan for combined laoding and unloading process is shown in Figure ý6..

Figure ý6. Makespan versus number of trains in an unloading process

Figure ý6. Makespan versus number of trains in a loading process

Figure ý6. Makespan versus number of trains in loading and unloading process

It can be seen from Figure ý6. that the FAT model produces the best results for unloading one, two and four number of trains. For unloading three trains, LTF model produces the best results. For unloading three trains, the results produced by FAT are only marginally lower than the LTF model. Therefore, FAT model can be considered as the best model for unloading processes in terms of makespan, This result is consistent with the theorem 1 which states that greedy algorithm based on FAT principle produces optimal results for unloading processes. The worst results are produced by the STF model for unloading any number of trains. For loading jobs, STF model produces the best results for any number of trains considered. The worst results in case of loading jobs were produced by LTF as can be seen from Figure ý6.. For combined loading and unloading process, FAT and LBT models together produce the best results. The worst results for combined loading and unloading of one, three or four number of trains are produced by the STF model. However, for combined loading and unloading of a single train, the worst results are obtained by LTF model.

Optimal cost

Figure ý6. shows optimal cost for unloading different number of trains using STF, LTF, and FAT algorithms while Figure ý6. shows optimal cost for loading different number of trains using STF, LTF, and LBT models. For combined loading and unloading operations, the results are shown in Figure ý6..

Figure ý6. Cost versus number of trains in unloading process

Figure ý6. Cost versus number of trains in loading process

Figure ý6. Cost versus number of trains in combined loading-unloading process

It can be seen from Figure ý6. that the STF model produces best results for unloading one, two or three number of trains. For unloading trains, FAT model produces the best results. The LTF model produces worst results for most cases for unloading operations. For loading operations, LBT produces the best results for all cases of number of trains. The worst results for loading operations are produced by LTF for one, two and three number of trains. For loading four number of trains, STF produces the worst results as can be seen from Figure ý6.. For combined loading and unloading operations, the FAT and LBT model produces the best results in terms of cost of operation for one and four number of trains. For combined loading and unloading operations of two and three trains, STF model produces the best results. The LTF model produces the worst results for most of the cases.

Optimal waiting time

Figure ý6. shows optimal waiting time for unloading different number of trains using STF, LTF, and FAT models. Figure ý6. shows optimal waiting time for unloading different number of trains using STF, LTF, and LBT models. Figure ý6. shows optimal waiting time for combined loading and unloading operations for different number of trains.

Figure ý6. Waiting time versus number of trains in an unloading process

Figure ý6. Waiting time versus number of trains in loading Process

Figure ý6. Waiting time versus number of trains in combined loading-unloading process

From the Figure ý6., it can be seen that the best results are obtained for two and three number of trains using STF model. For one and four number of trains the best results are obtained by FAT model. The FAT model produces worst results for two and three number of trains while for one and four number of trains the LTF model produces result that are substantially inferior to that produced by FAT. For unloading a single train, the results produced by LTF model are 3 times inferior to those produced by FAT model. For loading operations, LBT produces far better results that are produced by other two models (Figure ý6.). It can be seen from Figure ý6. that the worst results for loading operations are produced by LTF models for one, two, or three number of trains. For four number of trains, STF produces the worst results. For combined loading-unloading operations, the best results for one, three and four trains are obtained using the LBT and FAT models together. For combined loading-unloading of two trains, the best results are obtained using STF. The worst results for all the cases except for four trains are obtained by LTF. When the numbers of trains are four, the STF model produces the worst results.