Numerical Experiments

To examine the performance, the proposed heuristic framework is evaluated by a suite of ran-domly generated problems, which contains 7 instance sets of dierent problem size with 30 instances each. Table 5.1 summarizes the details for the 7 instance sets. For a Task i (i ∈ Ω), the handling time pik, ∀ k ∈ Qa follows a discrete uniform distribution in the range [30, 100];

while pik, ∀ k ∈ Qb is derived by adding pik, ∀ k ∈ Qa a random integer variable generated from a discrete uniform distribution in the range [−15, 15]. The length of a vessel measured by the number of bays is equal to the size of the tasks, n = |Ω|. To construct an instance, rst of all, the n tasks are randomly distributed to the n bays one by one. Next for each bay, if the number of tasks that assigned to the bay is greater than 2, then each task in that particular bay is tagged to be loading or discharging task randomly with equal possibility. The information obtained from this part can be used to construct the precedence constrained pair set, Φ. Finally, let the safety distance δ equal to one ship bay and ˆt= 0.1.

In this research, two criteria are used to assess the eectiveness of the proposed heuristic framework: computation time and solution quality. Because it is challenging to nd the optimal solutions in reasonable time for each instance, a compromise strategy is to compare the approx-imate solutions that the proposed heuristic framework generates to lower bounds. Three kinds of lower bound from unrelated machine scheduling problem are calculated for each instance:

• LB1 =

∑

i∈Ωmink∈Qpik

|Q| . LB1 distributes the sum of the minimal handling times among the quay cranes.

• LB2 = max_i∈Ω{mink∈Qp_ik}.

• The third lower bound LB3, is found by solving the surrogate dual problem of problem P. Let λ = (λ1, . . . , λ_m) be a vector of non-negative multipliers, the surrogate relaxation problem Sλ is obtained by dualizing constraints (5.24) in P:

[Sλ]min

∑_m

k=1

∑_n

j=1λ_k· pjk· ukj

∑_m

k=1λ_k (5.29)

s.t.

∑m k=1

u_kj = 1, j = 1, . . . , n (5.30)

u_kj ∈ {0, 1}, k = 1, . . . , m; j = 1, . . . , n (5.31)

Note that problem Sλ is solvable in O(mn) (Ghirardi and Potts, 2005). Let S(λ) be the optimal solution of Sλ, which is obviously a lower bound for problem P. Then the surrogate dual problem D for problem P is:

[D] max S(λ) (5.32)

s.t. λ≥ 0 (5.33)

Problem D can be solved by ascent direction algorithm proposed in Van de Velde (1993).

Let LB3 be the optimal solution of problem D.

After calculating the above three lower bounds, the lower bound for each instance LB is equal to max{LB1, LB₂, LB₃}.

Table 5.1: Detail of the 7 instance sets

Instance set A B C D E F G

Tasks: |Ω| 10 15 20 25 30 35 40

Quay cranes: |Qa| 1 1 2 2 2 3 3 Quay cranes: |Qb| 2 2 3 3 3 4 4

The proposed heuristic framework is coded in Matlab and ran in a PC with 2.4 GHz CPU

and 2.99 GB of RAM. There are 4 solution methods in the framework to nd the approximate solutions for the QCSP at indented berth: the combination of ASH1 and SSH (denoted as ASH1+SSH), the combination of ASH2 and SSH (denoted as ASH2+SSH), Tabu1, and Tabu2.

The rst two solution methods are deterministic heuristics while the latter two are stochastic ones. For Tabu1 and Tabu2, the maximum iteration number L1 and the number to test the convergence L2 are set to 200 and 10, respectively. Besides, for each instance, both Tabu search algorithms are executed 3 times and the information of the best solution found in the 3 runs for each algorithm is recorded for evaluation. For a particular instance, let µ be the makespan for the solution found by one of the four solution methods. Then the eectiveness of the corresponding solution method can be measured by:

gap = µ− LB

LB × 100% (5.34)

To select the suitable size of neighborhood solutions searched in each iteration (denoted as L3), Tabu1 with dierent values of L3 is executed to solve the 30 instances in Set A. Figure 5.11 depicts the performance of Tabu1 to solve instance Set A with dierent value of L3ranging from 20 to 160 with step size 20. In Figure 5.11, the horizontal axis measures the eectiveness of the algorithm while the vertical axis is the average CPU time consumed by the algorithm to solve the 30 instances. As can be observed, except L3 = 20 and L3 = 60 which lies on the pareto front, other options are dominated. By putting more weight on eectiveness, 20 can be a suitable value for the parameter L3. For simplicity, in the rest of numerical experiments, the value of L3 is xed to 20.

Table 5.2 summarizes the average computational time, the means and standard deviations of gaps for each solution method under dierent instance set. From the computational results shown in Table 5.2, several facts can be observed: 1) On average, the solutions generated by the two deterministic heuristics (i.e., ASH1+SSH and ASH2+SSH) are not good enough since the means of gap are ranging from 36% to 100%. 2) The dierence between the two deterministic heuristics in terms of solution quality is marginal. 3) The improvement of solution by Tabu search is signicant especially for Tabu2. Compared with Tabu1, Tabu2 not only can nd better

solutions but also achieves that in a faster way. To explain this phenomenon, it is worth noting that the exact size of searching space for Tabu1 is (|Q|)^|Ω|while for Tabu2 is less 2^|Ω|(recall that the tasks in the same ship bay are assigned to the same quay crane). Since in practice |Q| ≥ 2, the exact searching space for Tabu1 is remarkably larger than the one for Tabu2. Besides, the task assignment plan generated by Tabu2 has taken the Non-crossing Constraints among quay cranes into account which will help to reduce the number of disjunctive arcs in graph G. In light of the above facts, Tabu2 should be essentially more ecient than Tabu1.

7.5% 8.0% 8.5% 9.0% 9.5% 10.0% 10.5% 11.0%

Efficiency (the average CPU time in second)

L3= 20

Figure 5.11: The selection of L3

5.5 Summary

This paper discusses the QCSP at indented berth, an extension to the current QCSP in the

eld of container terminal operation. An MIP model by considering the unique features of the QCSP at indented berth is formulated. For solution, decomposition heuristic framework is developed and enhanced by Tabu search. To evaluate the performance of the proposed heuristic

Table 5.2: The performance of the proposed heuristic framework

Instance set A B C D E F G

CPU^a 0.09 0.21 0.41 0.79 1.49 2.32 3.85 ASH1+SSH Mean^b 48.16 36.28 79.43 68.55 62.48 100.47 95.65

STD^c 18.32 18.96 18.86 18.44 23.47 19.03 16.66 CPU 0.07 0.21 0.44 0.82 1.52 2.38 3.95 ASH2+SSH Mean 41.43 37.6 79.42 68.55 62.48 100.47 95.65

STD 16.94 18.39 18.86 18.44 23.47 19.03 16.66 CPU 9.58 40.75 82.38 227.99 453.99 915.13 1434.12 Tabu1 Mean 7.79 4.22 16.52 11.7 12.38 28.44 29.27

STD 3.35 1.87 6.77 4.45 4.96 7.24 8.17 CPU 4.38 20.46 22.61 59.79 101 100.41 192.61 Tabu2 Mean 6.11 3.31 8.02 5.88 4.59 8.29 6.93

STD 2.55 1.33 2.51 1.78 1.16 1.75 1.49 a: the average computational time in seconds

b: the mean of gaps (%)

c: the standard deviation of gaps (%)

framework, a comprehensive numerical test is carried out and its results show the good quality of the proposed heuristic framework.

A Combinatorial Benders' Cuts