Simulation Procedures

Three subprocedures are proposed first since they are frequently called by the simulation procedures.

Subprocedure 4.1 is used to reposition the empty containers and calculate the corresponding operating costs in the simulation. The main part fol-lows from Algorithm 4.1. To generate random number Z, we first generate a pseudorandom number uniformly distributed over (0, 1), then generate Z using the inverse transform method for the discrete uniform distribution and the rejection method for the normal distribution (Ross, 2006). In prac-tice, inventory level i^khas an upper bound M , i.e., 0 6 i^k 6 M for all k and random variables Z₁^k and Z₂^k have an upper bound R, i.e., 0 6 z1^k, z^k₂ 6 R for all k in Subprocedure 4.1.

Subprocedure 4.1. Reposition

1. Set n = 1.

2. If n 6 N, do the following; else stop.

3. For each k within 1 6 k 6 K:

(a) If i^k < A^k,

• Add k to state set Ω^a;

CHAPTER 4. MULTI-PORT CASE 61

• Set u^k= A^k, according to the optimal policy;

• Reset Importing^{T LB}[k] = Importing^{T LB}[k] + c^k_i u^k− i^k
, correspondingly.

(b) If A^k 6 i^k6 S^k,

• Add k to state set Ω^b;

• Set u^k= i^k, according to the optimal policy.

(c) If i^k > S^k,

• Add k to state set Ω^c;

• Set u^k= S^k, according to the optimal policy;

• Reset Exporting^{T LB}[k] = Exporting^{T LB}[k] + c^k_e i^k− u^k
, correspondingly.

4. (a) If Ω^a= ∅and Ω^c= ∅, go to step 8.

(b) If Ω^a6= ∅ and Ω^c6= ∅, go to step 5.

(c) If Ω^a= ∅and Ω^c6= ∅,

If Ω^b 6= ∅, go to step 6; else go to step 8.

(d) If Ω^a6= ∅ and Ω^c= ∅,

If Ω^b 6= ∅, go to step 7; else go to step 8.

5. (a) Find the p with the minimum ∆^k i^k
from k ∈ Ω^a.

CHAPTER 4. MULTI-PORT CASE 62

(b) Fine the q with the minimum ∆^k i^k
from k ∈ Ω^c. (c) Reset i^p = i^p + 1.

(d) Reset i^q= i^q− 1.

(e) Reset Exporting [q] = Exporting [q] + c^q_e, correspondingly.

(f) Reset Importing [p] = Importing [p] + c^p_i, correspondingly.

(g) If i^p = A^p,

• Delete p from Ω^a;

• Add p to Ω^b. (h) If i^q = S^q,

• Delete q from Ω^c;

• Add q to Ω^b. (i) Go to step 4.

6. (a) Find the p with the minimum ∆^k i^k
from k ∈ Ω^c. (b) Find the q with the minimum ∆^k i^k
from k ∈ Ω^b.

(c) If ∆^p(i^p) + ∆^q(i^q) > 0, go to step 8; else

• Reset i^p = i^p− 1.

• Reset i^q = i^q+ 1.

• Reset Exporting [p] = Exporting [p] + c^p_e, correspondingly.

CHAPTER 4. MULTI-PORT CASE 63

• Reset Importing [q] = Importing [q] + c^q_i, correspondingly.

• If i^p = S^p, delete p from Ω^c.

• If i^q = S^q, delete q from Ω^b.

• Go to step 4.

7. (a) Find the p with the minimum ∆^k i^k
from k ∈ Ω^a. (b) Fine the q with the minimum ∆^k i^k
from k ∈ Ω^b.

(c) If ∆^p(i^p) + ∆^q(i^q) > 0, go to step 8; else

• Reset i^p = i^p+ 1.

• Reset i^q = i^q− 1.

• Reset Exporting [q] = Exporting [q] + c^qe, correspondingly.

• Reset Importing [p] = Importing [p] + c^p_i, correspondingly.

• If i^p = A^p, delete p from Ω^a.

• If i^q = A^q, delete q from Ω^b.

• Go to step 4.

8. For each k within 1 6 k 6 K:

(a) Generate a random number Z.

(b) If i^k+ Z > 0,

Reset Holding [k] = Holding [k] + c^k_hmin{i^k+ Z, M };

CHAPTER 4. MULTI-PORT CASE 64

Else

Reset Stockout [k] = Stockout [k] − c^k_s i^k+ Z
.

(c) If u^k+ Z > 0,

Reset Holding^{T LB}[k] = Holding^{T LB}[k] + c^k_hmin{u^k+ Z, M };

Else

Reset Stockout^{T LB}[k] = Stockout^{T LB}[k] − c^k_s u^k+ Z
.

(d) Reset i^k= minn

i^k+ Z
+

, Mo .

9. Clear Ω^a, Ω^b, Ω^c.

10. Reset n = n + 1, go to step 2.

Lemma 4.2. U⁰ = a + (b − a) U ∼ U (a, b)if U ∼ U (0, 1).

Proof. The result is self-evident.

Lemma 4.2 is a well-known result, which is usually used to generate the desired uniform variable from the standard uniform distribution in simu-lation.

Subprocedure 4.2 is used to initialize the problem instance, which is de-termined by the combination of 8 parameters ci, ce, ch, cs, i1, M , N and

CHAPTER 4. MULTI-PORT CASE 65

R, in which ci, ce, ch, cs and i1 are all vectors. The elements of each vec-tor are all randomly generated from a given uniform distribution in order to better test the effectiveness of Algorithm 4.1. The supports of the uni-form distributions are defined by two preassigned values, respectively, i.e., (c^min_i , c^max_i ), (c^min_e , c^max_e ), (c^min_h , c^max_h ), (c^min_s , c^max_s ) and (i^min, i^max). Lemma 4.2 is applied to conduct the transformation of uniform variables.

Subprocedure 4.2. Initialization

For each k within 1 6 k 6 K:

1. (a) Generate a pseudorandom number U uniformly distributed over (0, 1).

(b) Set c^k_i = c^min_i + (c^max_i − c^min_i ) U to make c^k_i uniformly distributed over (c^min_i , c^max_i ).

2. (a) Generate a pseudorandom number U uniformly distributed over (0, 1).

(b) Set c^k_e = c^min_e + (c^max_e − c^min_e ) U to make c^k_e uniformly distributed over (c^min_e , c^max_e ).

3. (a) Generate a pseudorandom number U uniformly distributed over (0, 1).

CHAPTER 4. MULTI-PORT CASE 66

(b) Set c^k_h = c^min_h + (c^max_h − c^min_h ) U to make c^k_huniformly distributed over (c^min_h , c^max_h ).

4. (a) Generate a pseudorandom number U uniformly distributed over (0, 1).

(b) Set c^k_s = c^min_s + (c^max_s − c^min_s ) U to make c^k_s uniformly distributed over (c^mins , c^max_s ).

5. (a) Generate a pseudorandom number U uniformly distributed over (0, 1).

(b) Set i^k = i^min + (i^max− i^min) U to make i^k uniformly distributed over (i^min, i^max).

Subprocedure 4.3 is proposed to calculate the RE-TLB, which is used to evaluate the performance of Algorithm 4.1.

Subprocedure 4.3. RE-TLB

1. Calculate the cost given by the heuristic algorithm by adding up Importing [k], Exporting [k], Holding [k] and Stockout [k] for all 1 6 k 6 K, denoted as Sum^Algorithm.

2. Calculate the TLB by adding up Importing^{T LB}[k], Exporting^{T LB}[k],

CHAPTER 4. MULTI-PORT CASE 67

Holding^{T LB}[k] and Stockout^{T LB}[k] for all 1 6 k 6 K, denoted as Sum^{T LB}.

3. Calculate the RE-TLB as:

RE-T LB = Sum^Algorithm/Sum^{T LB} − 1. (4.19)

The following simulation procedures are used to simulate the empty con-tainer repositioning process between multi-ports over multi-periods and then calculate the average relative error with respect to the tight lower bound (AVG-RE-TLB) to evaluate the performance of Algorithm 4.1. We run this simulation for numerous ports, ranging from M in-K to M ax-K.

For each number of ports, we run N o. of Instances problem instances, and in each instance we run this simulation N o. of Iterations times and then calculate the average values to obtain AVG-RE-TLB, making the results more reliable.

Simulation Procedures

For each K within M in-K 6 K 6 Max-K:

1. Set number = 1.

2. If number 6 No. of Instances, do the following; else stop.

CHAPTER 4. MULTI-PORT CASE 68

3. Randomly generate c^k_i, c^k_e, c^k_h, c^k_s and initial i^kfor all 1 6 k 6 K by Subprocedure 4.2.

4. Calculate A^k_n, S_n^kand V_n^k(i)for all 1 6 n 6 N, 1 6 k 6 K and 0 6 i 6 M + R by Algorithm 3.1.

5. Set i^k₁ = i^kto copy initial i^k to i^k₁ for all 1 6 k 6 K.

6. Set Sum^{RE-T LB} = 0.

7. Set Sum-Importing [k], Sum-Exporting [k], Sum-Holding [k] and Sum-Stockout [k]; Sum-Importing^{T LB}[k], Sum-Exporting^{T LB}[k], Sum-Holding^{T LB}[k]and Sum-Stockout^{T LB}[k]equal to 0 for all 1 6 k 6 K.

8. Do this step N o. of Iterations times, in each times of execution:

(a) Set Importing [k], Exporting [k], Holding [k] and Stockout [k];

Importing^{T LB}[k], Exporting^{T LB}[k], Holding^{T LB}[k]

and Stockout^{T LB}[k]equal to 0 for all 1 6 k 6 K.

(b) Reposition the empty containers by Subprocedure 4.1.

(c) Calculate the corresponding RE-T LB by Subprocedure 4.3.

(d) Reset Sum^{RE-T LB} = Sum^{RE-T LB}+ RE-T LB.

CHAPTER 4. MULTI-PORT CASE 69

(e) Add the values of Importing [k], Exporting [k], Holding [k]

and Stockout [k]; Importing^{T LB}[k], Exporting^{T LB}[k], Holding^{T LB}[k]and Stockout^{T LB}[k]

to Sum-Importing [k], Sum-Exporting [k], Sum-Holding [k]

and Sum-Stockout [k]; Sum-Importing^{T LB}[k], Sum-Exporting^{T LB}[k], Sum-Holding^{T LB}[k]and Sum-Stockout^{T LB}[k], respectively, for all 1 6 k 6 K.

(f) Reset i^k= i^k₁ to restore initial i^kfor all 1 6 k 6 K.

9. (a) Calculate the average values of Importing [k], Exporting [k], Holding [k]and Stockout [k]; Importing^{T LB}[k],

Exporting^{T LB}[k], Holding^{T LB}[k]and Stockout^{T LB}[k]

as Sum-Importing [k], Sum-Exporting [k], Sum-Holding [k]

and Sum-Stockout [k]; Sum-Importing^{T LB}[k], Sum-Exporting^{T LB}[k], Sum-Holding^{T LB}[k]and Sum-Stockout^{T LB}[k]divided by N o. of Iterations, respectively, for all 1 6 k 6 K.

(b) Reset the values of Importing [k], Exporting [k], Holding [k]

and Stockout [k]; Importing^{T LB}[k], Exporting^{T LB}[k],

Holding^{T LB}[k]and Stockout^{T LB}[k]with their average values, respectively, for all 1 6 k 6 K.

CHAPTER 4. MULTI-PORT CASE 70

10. Calculate the AVG-RE-TLB:

AV G-RE-T LB = Sum^{RE-T LB}/N o. of Iterations. (4.20)

11. Output the following values:

(a) c^k_i, c^k_e, c^k_h, c^k_s and initial i^kfor all 1 6 k 6 K.

(b) A^k_nand S_n^k for all 1 6 n 6 N and 1 6 k 6 K.

(c) Importing [k], Exporting [k], Holding [k] and Stockout [k]

for all 1 6 k 6 K.

(d) Importing^{T LB}[k], Exporting^{T LB}[k], Holding^{T LB}[k]and Stockout^{T LB}[k]for all 1 6 k 6 K.

(e) AV G-RE-T LB.

12. Reset number = number + 1, go to step 2.

Due to the same reason stated in Section 3.3 for the single-port case, we continue to use the normal distribution and uniform distribution to con-duct the simulation, and the planning horizon also contains 12 consecutive decision periods. The following parameters are used in the simulation.

M = 1000, R = 50, N = 12, α = 0.99, M in-K = 5, M ax-K = 50, N o. of Instances = 10, N o. of Iterations = 100, (c^min_i , c^max_i ) = (140, 160),

CHAPTER 4. MULTI-PORT CASE 71

(c^min_e , c^max_e ) = (140, 160), (c^min_h , c^max_h ) = (160, 200), (c^min_s , c^max_s ) = (900, 1100), (i^min, i^max) = (0, 40).

We summarize the results, i.e., AVG-RE-TLB, in Tables A.1 and A.2 for the normal distribution and discrete uniform distribution, respectively. We next compute the maximum value, minimum value, average value and standard deviation of the 10 problem instances for each number of ports, abbreviated as Max, Min, Avg and SD in Tables A.1 and A.2 for the normal distribution and discrete uniform distribution, respectively.

The data corresponding to Max, Min and Avg for all the ports are plotted in Figures 4.1 and 4.3 for the normal distribution and discrete uniform distribution, respectively. The data corresponding to SD for all the ports are plotted in Figures 4.2 and 4.4 for the normal distribution and discrete uniform distribution, respectively.

The simulation is conducted by running a C++ program on a PC with Pentium 4 CPU 3GHz and 504MB of RAM.

4.3.2 Case I: Normal Distribution

Suppose random variables Z₁^k and Z₂^k for all k follow the same normal distribution with µ_Z^k

1 = µ_Z^k

2 and σ²_Zk

1 = σ_Z²k

2 = 50. Since Z₁^k and Z₂^k are

CHAPTER 4. MULTI-PORT CASE 72 The results are shown in Figures 4.1 and 4.2.

0

Figure 4.1: AVG-RE-TLB under the Normal Distribution

Figure 4.1 shows that the maximum values for all the cases are less than 7 per cent, and the average values appear to be stable at 3 per cent level, which means that Algorithm 4.1 is very effective under the normal distri-bution.

CHAPTER 4. MULTI-PORT CASE 73

0.002 0.004 0.006 0.008 0.01 0.012 0.014

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 No. of Ports

Standard Deviation

Figure 4.2: Standard Deviation under the Normal Distribution

Figure 4.2 shows that the standard deviation approaches 0 with the num-ber of ports increasing, which is due to the fact that the AV G-RE-T LB ap-proaches a constant as the number of ports increases. Therefore, the range in Figure 4.1 also approaches 0, namely, two dashed lines converge to the middle real line with the number of ports increasing, which means that the stability of Algorithm 4.1 improves as the number of ports increases under the normal distribution.

4.3.3 Case II: Discrete Uniform Distribution

Suppose random variables Z₁^k and Z₂^k for all k follow the same discrete uniform distribution in the interval [0, R]. Thus random variable Z^k = Z₁^k− Z₂^k −R 6 z^k6 R
follows the discrete triangular distribution

estab-CHAPTER 4. MULTI-PORT CASE 74

lished in Lemma 3.1. The results are shown in Figures 4.3 and 4.4.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 No. of Ports

AVG-RE-TLB

Max Avg Min

Figure 4.3: AVG-RE-TLB under the Discrete Uniform Distribution

Figure 4.3 shows that the maximum values for all the cases are less than 4 per cent and the average values approach 0 as the number of ports in-creases, which means that Algorithm 4.1 is very effective under the dis-crete uniform distribution.

CHAPTER 4. MULTI-PORT CASE 75

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 No. of Ports

Standard Deviation

Figure 4.4: Standard Deviation under the Discrete Uniform Distribution

Figure 4.4 shows that the standard deviation approaches 0 with the num-ber of ports increasing, which is due to the fact that the AV G-RE-T LB ap-proaches a constant as the number of ports increases. Therefore, the range in Figure 4.3 also approaches 0, namely, two dashed lines converge to the middle real line with the number of ports increasing, which means that the stability of Algorithm 4.1 improves as the number of ports increases under the discrete uniform distribution.

4.4 Summary

We have extended the single-port case to multi-ports in this chapter. We have mathematically formulated the multi-port empty container

reposi-CHAPTER 4. MULTI-PORT CASE 76

tioning problem with stochastic demand and lost sales. After determining a tight lower bound on the cost function, we have introduced the concept of relative error with respect to the tight lower bound, which can be used to measure the performance of Algorithm 4.1 in accordance with Lemma 4.1. Based on the two-threshold optimal policy established for a single port in Theorem 3.1, we have developed a polynomial time Algorithm 4.1 to find an approximate repositioning policy for multi-ports and then use simulation method to test its performance. Simulation results show that the average relative error with respect to the tight lower bound is within 5 per cent under the normal distribution and uniform distribution, respec-tively. Thus the underlying true relative error must be within 5 per cent or even smaller according to Lemma 4.1, which indicates Algorithm 4.1 per-forms very effectively for the multi-port empty container repositioning.

Furthermore, Algorithm 4.1 performs very efficiently due to its polyno-mial running time. The stability of Algorithm 4.1 improves as the number of ports increases. More importantly, Algorithm 4.1 is easy to understand and implement from a practical perspective because of its simplicity.

Chapter 5

Concluding Remarks

In this study, we have analyzed the multi-period empty container repo-sitioning problem with stochastic demand and lost sales. Maritime con-tainer shipping is a highly competitive industry. Therefore, we assume that unsatisfied customer demand due to the unavailability of empty con-tainers will be lost forever, and will incur a stockout cost, i.e., we assume lost sales scenario in our model. We do not consider leasing policy as an option to supply empty containers in our model based on the reasonable justifications. We aim to establish an effective empty container reposition-ing policy with the objective to minimize the total operatreposition-ing cost, i.e., con-tainer holding cost, stockout cost, importing cost and exporting cost.

77

CHAPTER 5. CONCLUDING REMARKS 78

First, we have mathematically formulated the single-port case as an inven-tory problem over a finite horizon with stochastic import and export of empty containers. We have analytically established the two-threshold op-timal policy for a single port, i.e., for period n: importing empty containers up to An when the number of empty containers in the port is fewer than A_n; exporting empty containers down to Sn when the number of empty containers in the port is more than Sn; and doing nothing, otherwise. We have also developed a polynomial time algorithm to numerically calcu-late the two thresholds Anand Snfor each period. We have provided two examples to illustrate the solution procedures based on the normal distri-bution and uniform distridistri-bution, respectively. The results show that the proposed algorithm performs highly effectively and efficiently.

Next, we have extended the single-port case to multi-ports. We have also mathematically formulated the multi-port problem and determined a tight lower bound on the cost function. We then introduce the concept of rela-tive error with respect to the tight lower bound, which is used to measure the performance of the proposed algorithm. Based on the two-threshold optimal policy established for a single port, we have developed a polyno-mial time algorithm to find an approximate repositioning policy for

multi-CHAPTER 5. CONCLUDING REMARKS 79

ports and then use simulation approach to test its performance. The sim-ulation results show that the proposed approximate repositioning algo-rithm performs very effectively since the calculated average relative error with respect to the tight lower bound is within 5 per cent under the nor-mal distribution and uniform distribution, respectively. Furthermore, the algorithm performs very efficiently as a result of its polynomial running time. The stability of the proposed algorithm improves as the number of ports increases. More importantly, the proposed approximate reposition-ing algorithm features bereposition-ing easy to understand and implement from a practical perspective. In reality, a liner shipping company manager can first calculate the two thresholds for all ports in all periods at the begin-ning of the whole planbegin-ning horizon; then at the beginbegin-ning of each decision period, the manager can apply the repositioning algorithm to determine the specific repositioning policy for this period.

There are several promising directions for future research. For example, it can examine the effect of shipping capacity on empty container reposi-tioning, i.e., it means that there is an upper bound on the number of repo-sitioning empty containers, and thus the recursive relation is minimized under the constraint, which may change the structure of the optimal

pol-CHAPTER 5. CONCLUDING REMARKS 80

icy. Furthermore, this study can be extended by analyzing the empty con-tainer repositioning problem over an infinite planning horizon, e.g., one can discuss the convergence of two thresholds in an infinite setting.

Appendix A

Simulated Data

81

APPENDIX A. SIMULATED DATA 82

TableA.1:SimulatedDataforMulti-portsundertheNormalDistribution No. of PortsCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9Case 10MaxMinAvgSD 50.02640.03900.03580.04070.02980.03000.02880.01800.03220.03120.04070.01800.03120.0065 60.03810.05590.03330.04070.02300.03650.04130.03790.01810.02300.05590.01810.03480.0111 70.03600.02490.04360.03560.00310.01900.02190.01950.03540.01210.04360.00310.02510.0125 80.01860.02520.03370.03590.02450.06460.03230.02990.02220.02590.06460.01860.03130.0129 90.03060.03880.02770.03940.03070.02500.03070.04130.03560.02590.04130.02500.03260.0058 100.04810.03060.01820.02070.03800.02030.02550.03790.04880.04870.04880.01820.03370.0123 110.02650.03340.01930.03330.03290.03740.03130.02160.04650.02610.04650.01930.03080.0079 120.02750.03350.05310.04090.03050.02670.04180.02990.03350.04290.05310.02670.03600.0084 130.03470.03700.02310.03210.03050.02300.04090.03230.03690.03300.04090.02300.03240.0057 140.04390.03050.03250.03070.02790.03580.03260.02850.03240.01760.04390.01760.03130.0066 150.03330.02730.02430.03820.03080.04120.03840.02540.02950.03070.04120.02430.03190.0058 160.04550.03570.03400.02180.02720.03270.02980.03240.04340.05340.05340.02180.03560.0094 170.03330.05610.03070.03920.03900.03430.02990.03470.02460.02550.05610.02460.03470.0090 180.03660.02520.05760.03820.03400.05420.03000.03970.03940.03320.05760.02520.03880.0101 190.03270.03450.04400.03270.03250.03830.03530.03940.02550.03280.04400.02550.03480.0050 200.03260.02570.03790.03330.04070.03880.04200.03480.04200.04350.04350.02570.03710.0055 Note:TableA.1continuesonthenextpage.

APPENDIX A. SIMULATED DATA 83

No. of PortsCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9Case 10MaxMinAvgSD 210.02970.03290.02830.02820.03210.03180.04350.03080.02840.02960.04350.02820.03150.0045 220.03760.04090.02560.03140.04590.02920.03810.02920.02850.03390.04590.02560.03400.0064 230.02170.03810.03160.03150.03410.03720.03610.02560.03430.03660.03810.02170.03270.0053 240.02940.02700.04070.03580.02770.03760.04100.04390.03700.02880.04390.02700.03490.0062 250.04270.02860.03550.03490.02740.03310.03810.03490.04090.04520.04520.02740.03610.0057 260.04540.03650.04040.03680.03840.03280.03840.03550.03850.04240.04540.03280.03850.0036 270.03680.03350.03730.03800.02840.03030.03960.02730.03150.03500.03960.02730.03380.0042 280.03220.02980.03530.02730.03890.03500.02620.03570.04240.04180.04240.02620.03440.0056 290.04610.03360.04180.02570.03180.03960.03990.04330.03680.04310.04610.02570.03820.0062 300.02860.03320.03390.04360.04500.04460.03130.04120.03780.03720.04500.02860.03760.0058 310.04170.02910.03490.03720.03870.03570.03640.02800.04310.03680.04310.02800.03620.0048 320.04530.04010.03240.03670.04690.03990.02860.03480.03160.03640.04690.02860.03730.0059 330.03650.04400.03420.03240.03190.03820.03500.03350.03930.02860.04400.02860.03540.0044 340.04160.04350.04040.02790.02580.03080.03760.03860.03720.04460.04460.02580.03680.0065 350.03360.03310.02450.03910.03680.03340.03520.03870.03500.03450.03910.02450.03440.0041 360.03910.03580.03610.04160.03670.03280.03760.04270.03170.03110.04270.03110.03650.0040 Note:TableA.1continuesonthenextpage.

APPENDIX A. SIMULATED DATA 84

No. of PortsCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9Case 10MaxMinAvgSD 370.03410.03550.02510.03710.03680.04170.04090.03530.03630.04020.04170.02510.03630.0047 380.03960.03560.03560.04670.04380.03280.03260.03250.04230.03680.04670.03250.03780.0051 390.04110.04450.03960.03320.04110.04210.03860.03200.03830.03270.04450.03200.03830.0043 400.03020.03760.03380.04380.05040.04130.03890.03620.03790.03730.05040.03020.03880.0056 410.03270.03470.03640.04130.03710.03110.04540.03270.03600.03370.04540.03110.03610.0044 420.03270.04050.03950.03800.03820.03910.03670.02880.04080.04040.04080.02880.03750.0039 430.03040.04180.04120.03240.03760.02460.04100.04300.03640.03900.04300.02460.03670.0059 440.03770.03250.03240.03650.03790.03700.04070.04390.04340.03000.04390.03000.03720.0046 450.03700.03900.02900.03370.03450.03970.03900.03850.03090.03690.03970.02900.03580.0037 460.03650.03360.03780.04290.04060.03080.03720.04350.03140.03390.04350.03080.03680.0045 470.03730.04100.03450.03780.04110.03810.03600.02950.03520.03620.04110.02950.03670.0033 480.04710.03840.03640.03780.03420.03370.03260.03570.02910.03530.04710.02910.03600.0047 490.03220.02890.03910.04210.03830.03300.03610.03840.03990.03540.04210.02890.03630.0040 500.04040.03830.03740.03380.03850.03000.03660.03410.04420.03910.04420.03000.03720.0039 Note:Caseninthistablerepresentsthen-thprobleminstance.

APPENDIX A. SIMULATED DATA 85

TableA.2:SimulatedDataforMulti-portsundertheDiscreteUniformDistribution No. of PortsCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9Case 10MaxMinAvgSD 50.01590.03410.02240.02720.01100.03020.02520.02580.02320.03080.03410.01100.02460.0069 60.02450.01210.01890.02770.03170.02280.01460.02150.03130.02080.03170.01210.02260.0065 70.02250.01130.02530.02020.01930.00720.03100.01970.02020.02160.03100.00720.01980.0067 80.01640.01930.02040.02320.01720.01620.01090.01630.01650.02260.02320.01090.01790.0036 90.01330.01480.01720.01770.01890.01250.01540.01260.01140.01230.01890.01140.01460.0026 100.01080.01910.01900.01050.01590.01580.01220.01100.01160.01310.01910.01050.01390.0033 110.01350.01200.00940.01240.01290.01260.00940.01420.01110.01780.01780.00940.01250.0024 120.01210.01050.01860.01360.00830.01330.00490.01340.01620.01720.01860.00490.01280.0041 130.00710.01450.01540.01300.00900.01270.01240.01350.00990.01540.01540.00710.01230.0028 140.01140.01270.01620.00530.00970.00940.00770.00970.00570.00890.01620.00530.00970.0032 150.01040.01210.00390.00950.00990.00720.00880.01050.00740.00800.01210.00390.00880.0023 160.01470.00550.01260.01090.00420.00660.01030.00520.01060.01000.01470.00420.00910.0035 170.00670.00550.00860.00920.00840.00910.00970.01100.00950.00810.01100.00550.00860.0015 180.01050.00750.00830.01010.01070.00650.00910.01360.01010.00810.01360.00650.00950.0020 190.01040.00830.00680.00690.00960.01050.00870.01190.00930.00660.01190.00660.00890.0018 200.00820.00580.00950.00650.00600.00770.00330.00540.00730.00580.00950.00330.00660.0017 Note:TableA.2continuesonthenextpage.

APPENDIX A. SIMULATED DATA 86

No. of PortsCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9Case 10MaxMinAvgSD 210.00830.00650.00710.00710.00780.01320.00600.00790.00750.00570.01320.00570.00770.0021 220.00570.00730.00780.01140.00790.00430.00720.00850.00410.00700.01140.00410.00710.0021 230.00720.00700.00800.00940.00860.00420.00740.00730.00460.00580.00940.00420.00700.0017 240.00680.00710.00970.00640.00640.00410.00390.00680.00570.00670.00970.00390.00630.0016 250.00470.01010.00630.00690.00890.00890.01020.00780.00890.00780.01020.00470.00800.0017 260.00760.00620.00540.00910.00380.00940.00910.00560.00980.00430.00980.00380.00700.0023 270.00480.00790.00700.00420.00700.00940.00680.00540.00390.00690.00940.00390.00630.0017 280.00730.00500.00470.01000.00630.00470.00610.00720.00590.00600.01000.00470.00630.0016 290.00780.00290.00610.00780.00500.00660.00730.00540.00680.00550.00780.00290.00610.0015 300.00600.00880.00590.00350.00350.00790.00550.00520.00930.00660.00930.00350.00620.0020 310.00520.00690.00580.00700.00640.00730.00820.00550.00570.00480.00820.00480.00630.0011 320.00340.00140.00440.00380.00340.00450.00510.00550.00520.00800.00800.00140.00450.0017 330.00430.00750.00250.00390.00480.00490.00780.00590.00540.00410.00780.00250.00510.0016 340.00190.00760.00700.00210.00680.00560.00510.00550.00460.00650.00760.00190.00530.0020 350.00670.00530.00400.00440.00320.00310.00490.00550.00620.00630.00670.00310.00490.0013 360.00680.00180.00690.00530.00400.00810.00310.00660.00300.00360.00810.00180.00490.0021 Note:TableA.2continuesonthenextpage.

APPENDIX A. SIMULATED DATA 87

No. of PortsCase 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9Case 10MaxMinAvgSD 370.00310.00530.00560.00230.00500.00570.00420.00630.00540.00290.00630.00230.00460.0014 380.00750.00490.00440.00870.00490.00620.00390.00510.00360.00480.00870.00360.00540.0016 390.00200.00560.00330.00420.00420.00750.00650.00800.00390.00380.00800.00200.00490.0019 400.00600.00590.00260.00660.00500.00500.00460.00470.00310.00300.00660.00260.00460.0013 410.00550.00220.00660.00350.00400.00700.00340.00440.00370.00470.00700.00220.00450.0015 420.00330.00380.00190.00620.00360.00340.00480.00560.00540.00370.00620.00190.00420.0013 430.00460.00490.00410.00440.00560.00370.00390.00670.00510.00300.00670.00300.00460.0011 440.00360.00320.00450.00240.00260.00500.00650.00500.00330.00430.00650.00240.00400.0013 450.00560.00410.00460.00390.00360.00280.00190.00580.00340.00810.00810.00190.00440.0018 460.00430.00290.00420.00470.00340.00410.00740.00320.00540.00640.00740.00290.00460.0014 470.00290.00420.00450.00330.00400.00520.00450.00120.00380.00430.00520.00120.00380.0011 480.00320.00440.00280.00670.00380.00490.00130.00230.00550.00440.00670.00130.00390.0016 490.00400.00210.00400.00400.00520.00140.00360.00520.00390.00530.00530.00140.00390.0013 500.00480.00240.00460.00470.00490.00230.00410.00480.00370.00350.00490.00230.00400.0010 Note:Caseninthistablerepresentsthen-thprobleminstance.

Appendix B

Papers Arising from the Thesis

• Zhang, B., C.T. Ng, T.C.E. Cheng. A stochastic dynamic model for empty container management in a single port. Under 2nd round review in Journal of the Operational Research Society.

• Zhang, B., C.T. Ng. A threshold control based heuristic algorithm for empty container repositioning between multi-ports with stochastic demand and lost sales. Working Paper.

88

Appendix C

Conference Presentations

• Zhang, B., C.T. Ng, T.C.E. Cheng. Empty container management with lost sales in a single port. The Second POMS-HK International Conference, Hong Kong, January 6-7, 2011.

• Zhang, B., C.T. Ng. Empty container allocation between multi-ports with lost sales. The 22nd Annual POM Conference, Reno, Nevada, USA, April 29 - May 2, 2011.

89

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