Due to the computational effort required for the iterative algorithm and the qual-ity of the solutions in Section 4.1, we next develop a heuristic based approach.
In the heuristic approach, we remove the integrality constraints (4.6) and replace them by non-negativity constraints. We let u be the dual variable associated with constraint (4.4). The dual of the problem (4.1) - (4.6) is given as
min X
The algorithm that we use to solve the gateway capacity expansion problem is presented Algorithm 4. We start by relaxing M ILP formulation and initializing restricted master problem. To iterate while loop, we first have to solve the restricted master problem to obtain corresponding dual variables. Then, we use a topological algorithm to determine the longest paths for each OD pair. The pricing sub-problem is solved according to the longest path information and we get the reduced cost corresponding to the longest path of OD pairs. Then, we add the column(s) that have to enter the restricted master problem and we solve the updated restricted master problem to check whether there is left any column having positive reduced cost. The algorithm iterates until we reach the dual feasibility. At the end of the algorithm, we solve M ILP formulation corresponding to RM P obtained at the end of column generation.
By solving the gateway capacity expansion problem by column generation, we ob-tain an optimal solution to the LP relaxation of the problem (4.1) - (4.6). However, the solution obtained by this approach does not necessarily satisfies the integrality constraints (4.6). For this reason, after obtaining an optimal LP relaxation, we add
Algorithm 4: Algorithm for the Gateway Network Design Problem
1 Obtain LP relaxation of (4.1) - (4.6)
2 Initialize RM P for the Gateway Capacity Expansion Problem
3 Set f ound = true
4 while f ound do
5 solve RMP
6 obtain dual variables of u, ti and wg
7 forall i ∈ I
8 solve the pricing sub-problem for finding the longest path
9 add the column with maximum positive reduced cost to the RMP
10 end forall
11 set f ound = f alse if no column with positive reduced cost is found
12 end while Solve M ILP corresponding to RM P
integrality constraints for variables zg and solve the M ILP problem formulation to optimality. The approach does not guarantee optimality. However, in our experi-ments, it performed better than the iterative approach in both computational time and quality of solutions.
4.3 Computational Results
In the following part, we present a computational study related to the gateway capacity expansion problem. Due to the mixed integer linear programming formu-lation, we obtain the optimal solution of capacity expansion problem through an integer programming solver. A proposed column generation algorithm solves the ca-pacity expansion problem iteratively. While solving the resulting column generation algorithm, we realize that the performance of column generation majorly depends on skillful selection of entering column(s) during the pricing sub-problem process.
Therefore, we utilize the shadow price information of dual variable wg by sorting in non-increasing order. Then, we set a constant on the number of columns to be added at each pricing sub-problem iteration instead of directly adding all of those columns having positive reduced cost. This approach allows us to reach a good quality solution easily even though it takes longer to reach an optimal solution. For this reason we stop column generation when the improvement between two successive
column generation iterations is less than 1.0%. We illustrate this in Table 4.1, where we study two different column selection criteria to solve an iteration of the gateway capacity expansion problem. CG stands for adding one column for each OD pair having positive reduced cost to the restricted master problem, whereas Best5000 stands for adding 5000 columns to the restricted master problem at each pricing sub-problem iteration. Additionally, those best 5000 columns are selected among the columns having positive reduced cost by considering weightsortedgateway. It[CG]
and It[Best5000] represent the required number of iterations to reach the termination, i.e. all promising columns are already added to the restricted master problem. We observe that the total number of columns added to the restricted master problem significantly reduces by Best5000 selection criterion, even though computation time is slightly longer.
C It[CG] It[Best5000] Time[CG] Time[Best5000] Nb Columns[CG] Nb Columns[Best5000]
10000 5 11 103.7 172.7 89791 36718
50000 5 11 102.7 161.4 87205 41973
Table 4.1: Results for the gateway capacity expansion problem with respect to different column selection criteria
Figure 4.1: The objective function value results for gateway capacity expansion problem with respect to different column selection criteria
In Figure 4.1, we see that adding all promising columns at each pricing sub-problem yields better results in terms of computation time. However, Best5000 selection criterion reaches good quality solutions in a few steps.
In the following discussion, we present the results obtained by computational exper-iments on the gateway capacity expansion problem with respect to different solution procedures in Table 4.2. First three columns refers to initial capacity of gateways, container capacity as to be assigned an extra capacity and number of containers available in the model, respectively. OFV[IP] stands for the optimal solution of the problem; whereas OFV[BC] refers to adding fixed number of columns at each itera-tion (at most 5000). At the end of column generaitera-tion, we add gateway capacities to gateways for which the corresponding dual variable is positive. OFV[One] refers to assigning one container at each iteration (and solving column generation to optimal-ity). OFV[CER] stands for the solutions obtained by relaxation heuristic. OFV[BC], OFV[One] and OFV[CER] are solved through utilizing column generation method.
On the other hand, Time[IP], Time[BC], Time[One] and Time[CER] represent the required computational time (in seconds) for respective solution procedures.
In Table 4.2, we conduct several computational experiments using the parameters of gateway capacity, container capacity and number of available containers where IP solutions correspond to the optimal solution of resulting problem cases. Assigning container capacities one by one to gateways is computationally expensive, it promises near optimal solution. When we consider adding capacities to a set of promising gateways, the computation time is reduced drastically and results in almost 10 times faster computationally inexpensive solutions against one by one gateway capacity expansion method. However, the quality of solutions decreases. Relaxation heuristic is the best approach in terms of computation time. In our experiments, it solves each instance to optimality, even though this is not guaranteed.
C K N OFV[IP] OFV[BC] OFV[One] OFV[CER] Time[IP] Time[BC] Time[One] Time[CER]
10000 1000 100 196762 186436 196700 196762 993.7 208.8 2466.8 131.7 1000 378356 301559 371521 378356 985.7 507.4 24241.9 66.4 10000 10 196762 193724 196469 196762 991.7 134.1 277.5 131.7
100 378356 327055 362676 378356 987.1 203.7 2468.7 66.8 30000 1000 100 382449 369369 382449 382449 984.7 199.2 2454.8 109.7
1000 582619 486987 581690 582619 989.7 620.0 24412.7 66.3 10000 10 382449 378096 382449 382449 989.3 124.4 268.2 109.2
100 582619 513062 575551 582619 988.7 194.0 2452.6 67.5 50000 1000 100 522617 509584 522617 522617 983.5 169.9 2426.9 109.8
1000 735650 632499 735650 735650 987.8 692.0 24328.6 69.1 10000 10 522617 518596 522617 522617 987.7 96.5 240.6 108.8
100 735650 656000 732187 735650 988.2 166.9 2436.1 67.9
Table 4.2: Results for the gateway capacity expansion problem with respect to different solution procedures
We also present optimality gaps for different approaches in Table 4.3. The perfor-mance of the column generation algorithm employing adding capacities to a set of promising gateways fluctuates depending on different model parameters. Generally, it performs well under limited number of available container instances; whereas the gap become wider as we have a plenty of available containers.
C K N GAP[BC] GAP[One] GAP[CER]
10000 1000 100 5.25% 0.03% 0%
1000 20.30% 1.81% 0%
10000 10 1.54% 0.15% 0%
100 13.56% 4.14% 0%
30000 1000 100 3.42% 0.00% 0%
1000 16.41% 0.16% 0%
10000 10 1.14% 0.00% 0%
100 11.94% 1.21% 0%
50000 1000 100 2.49% 0.00% 0%
1000 14.02% 0.00% 0%
10000 10 0.77% 0.00% 0%
100 10.83% 0.47% 0%
Table 4.3: Optimality gap for different solution procedures with respect to the IP solution
Chapter 5
GATEWAY NETWORK DESIGN PROBLEM
In gateway network design problem, the location of gateways and the OD pair de-mands are known in advance. However, the gateways are not capable of consolida-tion operaconsolida-tion due to lack of consolidaconsolida-tion capacity. Skillful assignment of containers to the gateways is the primary objective of gateway network design problem. There-fore, the gateways that are assigned container capacity can perform consolidation operation.
Since the initial design of the network does not contain any gateway with the consol-idation capability, we seek for those, which are located critically on the flight route network. Accompanied by offering limited consolidation capacity, i.e. containers for the sake of our context, the solution of gateway network design problem plays an important role in maintaining maximum possible saving through the use of limited capacity. After identifying the critical gateways, we need to allocate the desirable amount of consolidation capacity by utilizing the sub-route information.
To solve the gateway network design problem, we use the M ILP formulation (4.1) - (4.6) by setting C = 0. In the modified constraint, we do not have any initial capacities but use capacity expansion variables to assign consolidation capacity to gateways. Thus, the container capacity K with a multiplier of associated positive integer decision variable zg will be the only consolidation capacity provided for gateways.
Solution Approach and Computational Study
To solve the gateway network design problem, we use the relaxation based heuristic presented in Section 4.2 and conduct several experiments. In order to compare the computational experiments implemented for gateway capacity expansion with the gateway network design problem, we start with modifying the capacity, container capacity and number of available container parameters in a way that the total avail-able capacity is equal with all problem instances as presented in Tavail-able 4.2.
The computational results are given in the following Table 5.1. Total_Cap represents the total available consolidation capacity for each instance. OFV[GD] stands for the objective value of gateway network design problem, whereas OFV[CER] stands for the solutions obtained by relaxation heuristic for the gateway capacity expansion problem. Ratio[GD vs CER] refers to ratio between the objective function values of GD and CER. We might think of the Ratio[GD vs CER] column as representing how much we can improve the gateway capacity expansion problem, if we distribute the initial consolidation capacities efficiently. Time[GD] and Time[CER] refer to the required computational time (in seconds) to solve corresponding instances.
Total_Cap K OFV[GD] OFV[CER] Ratio[GD vs CER] Time[GD] Time[CER]
2920000 1000 515366 196762 38.2% 62.9 131.7
3820000 672864 378356 56.2% 55.1 66.4
2920000 10000 513627 196762 38.3% 83.5 131.7
3820000 671126 378356 56.4% 78.9 66.8
8560000 1000 1491020 382449 25.7% 51.5 109.7
9460000 1641010 582619 35.5% 51.2 66.3
8560000 10000 1480890 382449 25.8% 51.9 109.2
9460000 1630910 582619 35.7% 52.6 67.5
14600000 1000 2430840 522617 21.5% 52.4 109.8
15500000 2647670 735650 27.8% 50.9 69.1
14600000 10000 2420970 522617 21.6% 49.4 108.8
15500000 2637550 735650 27.9% 48.8 67.9
Table 5.1: Results for the gateway network design problem vs corresponding gate-way capacity expansion problem
According to the Table 5.1, we observe that OFV[GD] reduces as container capac-ity increases for the equal amount of total available consolidation capaccapac-ity as the problem becomes more restricted with increasing container capacity provided that the total capacity remains the same.
Due to the pre-determined consolidation capacity for the gateway capacity expansion problem, we are not allowed to utilize selected gateway capacities subject to the total amount of container capacity. This causes a significant loss in profits from consolidation operations. Hence, we observe that the performance gap between GD and CER gets wider as the total available container capacity decreases. Solving the gateway network design problem is computationally less expensive.
Chapter 6
CONCLUSION
We study freight consolidation and routing plans of an air cargo carrier aiming to maximize the savings due to less handling effort at capacitated gateways. The operational problem constructs network-wide consolidation and routing plans given the capacities of gateways and the flight network information. The tactical level problem integrates the capacity expansion decision process based on initial capacity levels. The gateway network design problem determines the hub locations and how the available container capacity would be allocated to those selected hubs without initial capacities.
To solve the freight consolidation and routing problem, we develop a set-covering type linear programming problem formulation where a decision variable represent the routing and consolidation scheme of an OD pair. We propose a column genera-tion algorithm due to computagenera-tional complexity of enumerating all possible decision variables. According to our computational study, we conclude that column gener-ation approach is almost 9 times faster than solving linear programming problem with a commercial solver.
We extend the set-covering type formulation for the gateway capacity expansion problem; the problem turns out to be a M ILP problem. We develop an optimization-based heuristic, which solves the capacity expansion problem in an iterative manner, utilizing the column generation approach as a sub-routine. Different column selec-tion criteria are also studied for the performance of pricing sub-problem process;
we conclude that selecting a subset of promising columns entering in the restricted
master problem is superior to adding all possible promising columns at each pric-ing iteration. However, the Relaxation heuristic is the best approach to solve the capacity expansion problem.
Finally, we study the gateway network design problem to determine the hub loca-tions over the flight network. For this purpose, we further extend the set-covering type M ILP formulation. We develop a column generation-based algorithm to solve the gateway network design problem iteratively. We investigate the computational results by comparing the gateway network design and capacity expansion problems using variations of the real-life problem.
In air cargo transportation literature, air cargo scheduling and routing and strate-gical level hub location problems have already been studied individually. However, our study is the first to focus on both cargo consolidation and hub network de-sign problems simultaneously. We also contribute to the literature by focusing on tactical level capacity expansion problem. Our major assumption in this work is the aggregation of time dimension considering that the demand and the availability of resources during a finite planning horizon can be aggregated. We also want to avoid this assumption and consider the demand, the flight network information and the availability of resources over the planning horizon. This is particularly essential for the operational level plans. However, adding the time component will critically change the problem formulations, and therefore the solution procedures.
As the future work, we intend to investigate the performance of a dual ascent pro-cedure for the hub network design problem.
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