1484 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011 Application of the LP-ELM Model on Transportation System Lifetime Optimization Zhan-Li Sun, Kien Ming Ng, Joanna Soszynska-Budny, and Mohamed Salahuddin Habibullah ´ Abstract—Considering factors such as economic costs and lives, an unreliable transportation system is more likely to cause severe consequences. Therefore, reliability optimization of transportation systems has attracted much attention over the past several decades. The traditional reliability optimization design is usually focused on redundancy allocation or reliability redundancy allocation. In practice, the operation process usually has a significant influence on the transportation system lifetime. By combining linear programming (LP) and extreme learning machine (ELM), a two-stage approach is proposed to optimize the transportation system lifetime, in which a semi-Markov model (SMM) is used to model the operation process. In the proposed method, we first formulate the optimization problem as an LP model, and the LP algorithm is utilized to search for the approximate optimal state probabilities. After data production and sample selection, ELM is trained with the produced training data and used to predict the optimal sojourn time distribution parameters. Applications on three different cases demonstrate that a higher lifetime can be ensured for the transportation system by using the proposed method. Index Terms—Artificial neural network (ANN), extreme learning machine (ELM), lifetime optimization, linear programming (LP), semi-Markov model (SMM), transportation system. ACRONYMS ANN Artificial neural network. ELM Extreme learning machine. LP Linear programming. SLFN Single-hidden-layer feedforward neural network. SM Semi-Markov. SMM Semi-Markov model. SMP Semi-Markov process. Manuscript received March 9, 2010; revised March 19, 2011; accepted June 8, 2011. Date of publication July 14, 2011; date of current version December 5, 2011. This paper is a part of the Singapore-Poland Joint Research Project entitled “Safety and Reliability of Complex Industrial Systems and Process” (Science & Engineering Research Council grant number 072 1340050) granted by Singapore’s Agency for Science, Technology and Research (A*STAR) and Poland’s Ministry of Science and Higher Education (MSHE). The Associate Editor for this paper was L. Li. Z.-L. Sun and K. M. Ng are with the Department of Industrial and Systems Engineering, National University of Singapore, Singapore 117576 (e-mail: [email protected]). J. Soszynska-Budny is with the Department of Mathematics, Gdynia ´ Maritime University, 81-225 Gdynia, Poland. M. S. Habibullah is with the Institute of High Performance Computing, Agency for Science, Technology and Research, Singapore 138632. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TITS.2011.2160053 NOTATION θbl Conditional sojourn time of the system at operation states zb when its next operation state is zl. Hbl Distribution functions of the conditional sojourn time variables θbl. γbl Parameters of the exponential distributions of the variables θbl. Mbl Expectation values of the variables θbl. θb Sojourn time of the system at operation states zb. Mb Mean values of θb. pbl Transition probabilities that the system operation state is transformed from zb into zl. pb Limit values of the transient probabilities pb(t) at the operation states zb. Rn(t) Unconditional reliability function of the system. μ Mean value of the system lifetime. lbi Lower bound for pi. ubi Upper bound for pi. v Number of states of the semi-Markov process. N ˜ Number of hidden neurons of ELM. W Input layer weight of ELM. β Output layer weight of ELM. H Hidden layer output matrix of ELM. H† Moore-Penrose generalized inverse of H. γ Vector (γ11, . . . , γ1v, γ21, . . . , γ2v, . . . , γvv). g(x) Activation function of ELM. I. INTRODUCTION W ITH the rapid development of economy, transportation systems are playing a more and more important role in the society [40]. In the past several decades, many problems of transportation systems have been addressed by researchers, such as traffic monitoring [24], [42], resource allocation [15], [41], route scheme [2], [25], and safety and reliability analysis [1], [29]. Considering factors such as economic costs and lives, an unreliable transportation system is more likely to cause severe consequences. Therefore, reliability optimization of transportation systems has been regarded as one of the most important problems. Due to its importance, reliability has been widely used as an index to assess the performance of engineering systems in various fields, e.g., software systems [6], [7], [12], [30], communications [28], [39], transportation systems [3], [11], [27], and semiconductor systems [16], [44]. With increasingly sophisticated structure and high-tech industrial processes involved, for engineering systems, achieving an optimal systemlevel configuration has become an even greater concern in 1524-9050/$26.00 © 2011 IEEESUN et al.: APPLICATION OF THE LP-ELM MODEL ON TRANSPORTATION SYSTEM LIFETIME OPTIMIZATION 1485 recent years. The goal of optimal reliability design is to optimize the system configuration by making a tradeoff between system reliability enhancement and the cost in achieving the improvement. In the past several decades, this problem has been addressed by using different system structures, performance measures, optimization techniques, and other options for reliability improvement. In [19] and [20], detailed surveys of system reliability optimization are provided. From the standpoint of problem formulations, optimal reliability design can be concluded as four basic formulations: 1) reliability redundancy allocation [4]; 2) percentile life evaluation [31]; 3) multistate system optimization [23]; and 4) multiobjective optimization [5]. In terms of the optimization techniques, the optimal reliability design can be classified into three typical categories: 1) metaheuristic methods [18], [33]; 2) exact methods [8]; and 3) other optimization techniques [21], [22], [43]. Most of past works focused on redundancy allocation problems or reliability redundancy allocation. In practice, the operation process usually has a significant influence on system reliability. In [13], [14], and [17], the operation processes of transportation systems are modeled with the use of SMMs, and their influence on system reliability is investigated. However, how to further optimize the system reliability is not mentioned. In this paper, we present a system lifetime optimization approach by adjusting the route scheme of transportation systems. After analysis, we first formulate the lifetime optimization problem as an LP model. In this model, the system mean lifetime is adopted as the objective function. Then, the state transient probabilities are chosen as the decision variables. With the LP algorithms, we can obtain the approximate optimal state transient probabilities. In the SMM, the state transient probabilities are determined by the sojourn time distribution. To obtain the corresponding sojourn time distribution parameters, one possible way is to use various evolutionary algorithms [26], [34] to search for the distribution parameters. However, a high level of skill is often required for operators to design an appropriate evolutionary operation scheme, e.g., chromosome coding, gene selection, gene cross, and gene mutation. Moreover, a heavy computation burden is usually encountered in various evolutionary algorithms. Due to random initiation, stability is also a problem for evolutionary algorithms. Given the input and output data, ANN can approximate a complicated map relationship without knowing the detailed process [37], [38]. Therefore, ANN is used here to predict the sojourn time distribution parameters. Recently, a relative novel learning algorithm for SLFNs called ELM has been proposed [10] and widely applied in various fields [32], [35], [36]. In ELM, the input weights and hidden biases are randomly chosen, and the output weights are analytically determined by using Moore–Penrose generalized inverse. ELM not only learns much faster with higher generalization performance than the traditional gradient-based learning algorithms but it also avoids many difficulties that are faced by gradient-based learning methods such as stopping criteria, learning rate, learning epochs, local minima, and overtuning issues. Compared with evolutionary algorithms, ELM has far less of a computational burden and only one parameter to be adjusted. Based on the preceding analysis, we present a two-stage scheme to improve the lifetime of the SMM-based transportation system. In the proposed method, LP is first used to search the approximate optimal state probabilities. After data production and sample selection, ELM is trained with the produced training data. Then, the trained ELM is used to predict the sojourn time distribution parameters corresponding to the optimal state probabilities. Finally, the LP-ELM model is applied on the transportation system and verified that it can efficiently ensure that the transportation system has a high lifetime by adjusting the operation process. The remainder of this paper is organized as follows. Section II gives a brief review of the SMM-based system operation process [13]. The LP-ELM model is presented in Section III. In addition, the applications of the LP-ELM model on three different examples are given in Section IV. Finally, conclusions are made in Section V. II. SEMI-MARKOV MODEL OF THE SYSTEM OPERATION PROCESS Assume that the system operation process Z(t) is an SMP with v states zv(z1, . . . , zv); the conditional sojourn time θbl obeys the exponential distribution with parameters γbl, i.e., Hbl(t) = P(θbl < t) =1 − exp(−γblt), b, l = 1, . . . , v, b = l. (1) For the same operation states, the distributions are set as zeros, i.e., Hbb = 0, b = 1, . . . , v. According to (1), the mean values Mbl of θbl are given by Mbl=E[θbl]= ∞0 tdHbl(t)=1/γbl, b, l=1 · · · , v, b=l. (2) Furthermore, the mean values Mb of θb can be obtained with the following equation: Mb = E[θb] = E l=1 v pblθbl = l=1 v pblE[θbl] = v l =1 pblMbl, b = 1, . . . , v. (3) Then, the limit values of transient probabilities at the operation states zb can be obtained by [9] pb = lim t→∞ pb(t) = πbMb v l=1 πlMl , b = 1, . . . , v (4) where the initial state probabilities πb satisfy the following system of equations: ⎧⎪⎪⎨⎪⎪⎩ ⎡⎣ π1 ... π v ⎤⎦ = [π1, . . . , πv] ⎡⎣ p11 · · · p1v ... · · · ... pv1 · · · pvv ⎤⎦ v l=1 πl = 1. (5) Denoting by T (b) the system lifetimes at operation states zb, the system conditional reliability functions Rb n(t) at operation1486 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011 states zb can be given by Rb n(t) = P T (b) > t|Z(t) = zb , b = 1, . . . , v. (6) With Rb n(t)(b = 1, . . . , v), the system unconditional reliability function Rn(t) can be computed as follows: Rn(t) = P(T > t) ∼ = v i =1 piRi n(t) (7) where T is the unconditional lifetime of the system. Denoting by μb the mean values of the system lifetime at operation states zb, i.e., μb = ∞ 0 Rb n(t)dt, b = 1, . . . , v (8) the mean value of the system lifetime μ can be given by μ = ∞ 0 Rn(t)dt ∼ = v i =1 pi ∞ 0 Ri n(t)dt ∼ = v i =1 piμi. (9) Without considering repair, the safety state of a system should be changed in time only from better to worse. Although the change in the system safety condition is continuous, it is generally discretized into several discrete states for convenience of processing. As the SMP can be used to model the system at each safety state, respectively, lifetime optimization can also be performed at each safety state, respectively, for our proposed model. Therefore, we only consider the safety state in which the system is fully safe in this paper. Denote by μ0 i the mean values of the system lifetime when the system is fully safe, and at operation states zi, the corresponding mean value of the system lifetime μ can be given by μ ∼ = v i =1 piμ0 i . (10) It can be seen from (4) that pb is a limit value when t → ∞. In practice, the system operation time is generally assumed to be large enough so that the operation process is stationary, i.e., a stationary SMM. It can also be seen from (10) that pb is one parameter of μ. As a result, the mean value of the system unconditional lifetime shown in (10) is also an approximate value. III. LINEAR PROGRAMMING–EXTREME LEARNING MACHINE MODEL Given Hbl(t), pbl, and μ0 i , the goal of the LP-ELM model is to search for the optimal state transient probabilities popt i and predict the optimal sojourn time distribution function parameters γbl opt. In the optimization procedure, parameters Mbl, Mb, and πi are computed in terms of the SMP described in Section II. Fig. 1 shows the flowchart of the LP-ELM model. In this figure, the diamond represents the input or output data, whereas the rectangle represents the operation. As shown in Fig. 1, this model consists of three parts. 1) Given μ0 i and constraints, establish the LP model, and use the LP algorithm to obtain the approximate optimal state transient probabilities pLP i . Fig. 1. Flowchart of the LP-ELM model. 2) Prepare for the training data, including data sampling and data selection. 3) Train ELM, and predict γbl opt. Compute popt i and the corresponding μ. A detailed description of these three parts is given here. A. LP-Based System Lifetime Optimization In this section, an LP-based lifetime optimization model is built at first, and then, some illustrations about this model are given. 1) Building LP Model: Higher μ means longer time that the transportation system is in normal operation state. That is to say, the transportation system has higher reliability. As a result, less cost will be spent on repairing or maintaining the system. From (10), it can been seen that μ can be maximized by optimizing parameters pi and μ0 i . In terms of (1)–(4), the state transient probabilities pi are determined by the sojourn time distribution parameters γbl once the exponential distribution function is assumed. Furthermore, we learned that μ0 i are independent of the parameters γbl from [13]. To simplify the optimization problem, we assume that the values of μ0 i are fixed and only the probabilities pi are adjusted. Under this assumption, μ can be maximized by searching for the optimal decision variable values popt i . That is to say, we can optimize the system lifetime by adjusting the system operation scheme, i.e., sojourn time distribution. After optimization, the transportation system can obtain a higher lifetime under the optimized system operation scheme. Based on the preceding analysis, an LP model is built as follows to represent the optimization problem: min p (−pT μ0) s.t. v i =1 pi = 1 lbi ≤ pi ≤ ubi, i = 1, . . . , v (11)SUN et al.: APPLICATION OF THE LP-ELM MODEL ON TRANSPORTATION SYSTEM LIFETIME OPTIMIZATION 1487 Fig. 2. Port oil transportation system [13]. where vector p = [p1, . . . , pv]T , and μ0 = [μ0 1, . . . , μ0 v]T . The variables lbi and ubi define a set of lower and upper bounds on the design variables pi, so that the solution is always in the range [lbi, ubi]. In terms of the physical meaning of pi, lbi and ubi must be selected in the interval [0, 1] and satisfy the constraints lbi ≤ ubi. As a special case, variable pi is equal to the bound value when lbi is equal to ubi. Equality constraint v i=1 pi = 1 means that the sum of the transient probabilities pi in the experiment time should be equal to 1. 2) Illustration About the LP Model: Here, we want to get the maximum of pT μ0. However, the target of LP algorithms is to find the minimum of the objective function. Minimizing −pT μ0 is equivalent to maximizing pT μ0. Thus, the objective function is −pT μ0 instead of pT μ0. Another problem should be addressed for the bound constraints. If μ0 b is the maximum of μ0 i and pb is the corresponding state transient probability, we generally can get a high μ value if we set pb to be 1 and other pi to be 0. However, this case is not reasonable and infeasible. In this case, other subsystems will be not used, except for the subsystem corresponding to the operation state zb. In fact, other operation states are sometimes necessary. In addition, the transportation cost may be greatly increased if all things are moved to one place by the tools on land. For example, Fig. 2 shows a port oil transportation system. The system is composed of three terminal parts (A, B, and C) and three subsystems (S1, S2, and S3). The pier PIRS is used to unload the tankers. Assume that factory A is an oil factory, the oil can be transported to terminal part A by the truck and then from A to C by the oil transportation system, or transported to terminal part B and then from B to C by the oil transportation system. The operation cost per unit time, i.e., the rate of all costs (including the equipment cost, the cost spend on transportation, etc.) and the operation time before failure, is a good index to evaluate different routines. If the operation cost of the first routine (oil factory → A → C) is less than that of the second routine (oil factory → B → C), we should increase the operation time from A to C, which means increasing the pi value corresponding to the state from A to C. In practice, there are usually bound constraints for pb. For example, if every operation state is required with at least 1% probability, we can set the lower bounds to be 0.01, i.e., pi ≥ 0.01 for all i. According to the size of the LP problem, the LP algorithms can be classified into simplex, medium-scale, and large-scale TABLE I INPUT AND OUTPUT FOR EXTREME LEARNING MACHINE Fig. 3. Multiple-input–multiple-output ELM regression model. algorithms. Here, we only focus on the proposed model. The details about the specific algorithms of LP can be found in [45]. As shown in Part 1 of Fig. 1, we can get pLP by the LP algorithm with the given μ0 i , equality constraint, and bound constraints. B. Training Data Production To train ELM shown in Fig. 3, the training data must be produced at first. As shown in Table I and Part 2 of Fig. 1, we first randomly select N sets of parameters γi(i = 1, . . . , N) from a given interval. In terms of the SMP described in Section II, we can get the corresponding N sets of state transient probability vectors pi(i = 1, . . . , N) with γi. To ensure that the predicted results satisfy the constraints given in (11), we should remove the data pairs (pi, γi) that do not satisfy these constraints. After deletion, the remained data pairs (pi, γi) are used as the training data of ELM. It should be pointed out that we want to predict the optimal parameter vector γopt by using ELM. Thus, the inputs and outputs of ELM are pi and γi, respectively. As shown in Table I, it is an oppositional process compared with the training data creation procedure. Although the evolutionary algorithms can also be directly used here to select an optimal γi, as we analyzed in Section I, they usually suffer from a high computation burden and a complicated operation. Moreover, the global optimization solution sometimes cannot be obtained when the parameters are not correctly selected. C. Computing Parameters γbl Through ELM For the N distinct samples (pi, γi), the standard SLFN with N ˜ hidden nodes and activation function g(x) can approximate1488 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011 these N samples with zero error, i.e., there exist βi = (βi1, . . . , βiv)T , wi = (wi1, . . . , wiv)T , and bi such that N i =1 βig(wipj + bi) = γj. (12) The preceding N equations can be compactly written as Hβ = γ (13) where the hidden-layer output matrix H(w1, . . . , wN ˜ , b1, . . . , bN ˜ , p1, . . . , pN) = ⎡⎢⎣ g(w1 · p1 + b1) · · · g(wN ˜ · p1 + bN ˜) ... · · · ... g(w1 · pN + b1) · · · g(wN ˜ · pN + bN ˜) ⎤⎥⎦ (14) β = ⎡⎢⎣ β1 T ... βN T ˜ ⎤⎥⎦ N ˜×v and γ = ⎡⎢⎣ (γ1)T ... (γN)T ⎤⎥⎦ N×v . (15) The smallest norm least-square solution [10] of the preceding linear system is β = H†γ. (16) With the pLP obtained by the LP algorithm, the optimal parameter vector γopt can be computed through (13), i.e., γopt = (hopt)T β (17) where hopt =[ g(w1 · popt+b1) · · · g(wN ˜ · popt+bN ˜) ]T . (18) To select an optimal hidden node number N ˜, we should choose an objection function. Since we do not know the actual γopt corresponding to pLP, it is impossible to use the error between the predicted value and the actual value. The solution method we present is to compute pi and μ using the γopt in terms of SMP. When the number of hidden neurons (N ˜) is increased from N ˜min to the maximum (N ˜max), we can get N ˜max − N ˜min + 1 sets of pi and μ. The N ˜ corresponding to the maximum μ is selected as the optimal hidden node number. The specific steps of ELM training and prediction can be summarized here. Step 1: Given the interval of parameter N ˜ ([N ˜min, N ˜max], set N ˜ to be N ˜min at first. Step 2: With the input pi and the output γ, compute the matrix β according to (16). TABLE II πi VALUES OF EXAMPLE 1 TABLE III μ0 i VALUES OF EXAMPLE 1 TABLE IV pLP i AND μ OBTAINED VIA THE LP ALGORITHM IN EXAMPLE 1 Step 3: With pLP and β, compute the corresponding output γopt by (17). Step 4: In terms of SMP, compute μ using the γopt. Step 5: Increase N ˜ from N ˜min + 1 to N ˜max, and repeat Steps 2–4, respectively; select the maximum μ and the corresponding parameters N, ˜ popt. IV. APPLICATIONS Three different examples are presented here to demonstrate how to apply the proposed LP-ELM model for analyzing the transportation systems. The port oil transportation system with five and eight operation states is adopted as the first and second examples [13], [14], respectively. The third example is the port bulk cargo transportation system [17]. The sojourn time distribution function of the third example is different from that of the first two examples. A. Application on Port Oil Transportation System With Five Operation States A detailed description of the analysis procedure is given for the first example to serve as an illustration of each step of the method described in this paper. The port oil transportation system shown in Fig. 2 is composed of three terminal parts A, B, and C and three piping transportation subsystems S1–S3. S1 consists of two identical pipelines. There are 178 pipe segments of length 12 m in each pipeline. S2 also consists of two identical pipelines, but each pipeline includes 717 pipe segments. There are three different pipelines in S3. Each pipeline is composed of 360 pipe segments with lengths of either 10 or 7.5 m. Five operation states (Z = {z1, . . . , zv}, v = 5) are defined in terms of the operation process of the transportation system [13]. Equation (19), shown at the bottom of the page, is the matrix of sojourn time distribution functions, in which the parameter [Hbl(t)] = ⎡⎢⎢⎢⎣ 0 0 0 1 − e−217.4t 0 0 0 1 − e−156.3t 0 0 1 − e−434.8t 1 − e−370.4t 0 0 0 0 1 − e−1111.1t 0 0 1 − e−1111.1t 0 0 0 1 − e−6.083t 0 ⎤⎥⎥⎥⎦ (19)SUN et al.: APPLICATION OF THE LP-ELM MODEL ON TRANSPORTATION SYSTEM LIFETIME OPTIMIZATION 1489 TABLE V RANDOMLY PRODUCED γi VALUES AND THE CORRESPONDING pi VALUES IN EXAMPLE 1 values of γbl are evaluated by the experts. The state transition probability matrix is given as follows: [pbl] = ⎡⎢⎢⎢⎣ 0 0 0 1 0 0 0 1 0 0 0.11 0.89 0 0 0 0 0.5 0 0 0.5 0 0 0 1 0 ⎤⎥⎥⎥⎦ . (20) In terms of the SMP described in Section II, we know that pbl are independent of γbl and that the πi, (i = 1, . . . , v) values are determined by pbl. Therefore, the πi values in Table II are not changed in the optimization process. Table III shows the parameter values of μ0 i . It should be pointed out that the parameter values of pbl, πl, and μ0 i are all provided by [13]. With μ0 i and the constraints, we first search for the approximate optimal parameters pLP i via the LP algorithm. As a result, pLP i and the corresponding μ are given in Table IV. It can be seen from (19) that only part of the transition processes existed in practice. Correspondingly, only part of the parameters γbl(γ14, γ23, γ31, γ32, γ42, γ45, γ54) are needed to be optimized. To get the training data of ELM, 8000 sets of parameter γbl k (k = 1, . . . , 8000) values are selected from the interval [0 2500]. Then, the corresponding pk i (i = 1, . . . , v, k = 1, . . . , 8000) values are computed according to the SMP described in Section II. As an example, the first five sets of samples are shown in Table V. In simulation, we assume that the lower and upper bounds of pi are 0.01 and 0.95, respectively. We can see from Table V that the first four samples do not satisfy the bound requirements. Therefore, they are removed from the training data set. As a result, only 2233 samples are reserved after filtering and used as the final training data of ELM. The maximum number of hidden neurons is set to be 50 for ELM, which has been verified to be enough to process most problems [10], [35]. The activation function used here is the sigmoidal function g(x) = 1 1 + e−x (21) which has been demonstrated to be an efficient activation function of ELM. In the training process, we increase the number of hidden neurons (N ˜) from 1 to 50. After the training, the optimal parameters γbl opt are predicted via ELM by using the pLP i values as the inputs, which are obtained through the LP algorithm, as shown in Table IV. Given the optimal γbl opt, we can compute the corresponding pi values and μ in terms of TABLE VI OPTIMAL PARAMETERS γbl opt PREDICTED VIA ELM IN EXAMPLE 1 TABLE VII p opt i AND μ OBTAINED ACCORDING TO THE SMP AND THE γbl opt OF TABLE VI TABLE VIII pi AND μ COMPUTED ACCORDING TO THE PARAMETER VALUES SUGGESTED BY THE EXPERTS IN EXAMPLE 1 the SMP described in Section II. The value of N ˜ that gives the highest μ is selected as the final number of hidden neurons of ELM. Due to the random initiation of ELM, the final results are changed when the experiments are repeated. Here, 20 trials are carried out to investigate the robustness of the proposed model. The best results are given in Tables VI and VII. Table VI shows the optimal γbl opt values whereas Table VII presents the corresponding popt i and μ computed in terms of the SMP. The optimal number of hidden neuron of this trial is 28, which corresponds to the highest μ value. As the popt i and μ of Table VII are computed according to the SMP, they satisfy the constraints in (11) and the SMP. In [13], the pi values are computed according to the parameter values suggested by the experts, as shown in Table VIII. With these pi values and the μ0 i values, we can get the μ value according to (10), as shown in Table VIII. We can see that the μ value (0.5779) in Table VII is higher than the result (0.308) in Table VIII. The result indicates that a higher lifetime can be ensured for the transportation system by adjusting the routine scheme according to the optimization results. B. Application on the Port Oil Transportation System With Eight Operation States In [14], eight operation states are defined for the port oil transportation system shown in Fig. 2. Different from [13], due to lack of statistical data, it is impossible to verify hypotheses on the distributions of the sojourn times. In this case, the1490 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011 TABLE IX Mbl VALUES IN EXAMPLE 2 TABLE X πi VALUES IN EXAMPLE 2 TABLE XI μ0 i VALUES IN EXAMPLE 2 TABLE XII pLP i AND μ OBTAINED VIA THE LP ALGORITHM IN EXAMPLE 2 TABLE XIII OPTIMAL PARAMETERS Mopt bl (×107) PREDICTED VIA ELM IN EXAMPLE 2 approximate empirical values of Mbl are directly evaluated by the experts, as shown in Table IX. A similar optimization process can also be performed on this case. However, the parameters to be optimized are Mbl instead of γbl. Therefore, the computation of (2) is not necessary during sampling and optimization. The state transition probability matrix is given by [pbl] = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 0 0 0 0.06 0.06 0.86 0.02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.125 0 0 0 0 0.125 0.687 0.063 0.4 0 0 0 0.6 0 0 0 0.82 0 0 0 0.16 0 0 0.02 0.67 0 0 0 0 0 0.33 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ . (22) It can been seen from (22) that rows 2–4 are all zeros. The reason is that transitions between these operation states did not happen during the observation time. However, it does not mean that these transitions are not possible. They may happen TABLE XIV p opt i AND μ OBTAINED ACCORDING TO THE SMP AND THE Mopt bl OF TABLE XIII TABLE XV pi AND μ COMPUTED ACCORDING TO THE PARAMETER VALUES SUGGESTED BY THE EXPERTS IN EXAMPLE 2 TABLE XVI πi VALUES IN EXAMPLE 3 TABLE XVII μ0 i VALUES IN EXAMPLE 3 after the observation time, and then, the evaluations of the probabilities will be different from zero [14]. Parameters πi and μ0 i are given in Tables X and XI, respectively. As the transition probabilities of operation states z2−z4 are zeros, the corresponding lower bounds for these three operation states are set as zeros in this case. Except for these three operations, the lower bounds of other operation states are still set as 0.01. As done in example 1, the upper bounds are also 0.95. The pLP i and μ obtained by the LP algorithm, the optimal M opt bl predicted by ELM, and the corresponding popt i and μ computed in terms of the SMP are given in Tables XII–XIV, respectively. It can be seen that the μ value of Table XIV is quite different from that of Table XII. The reason is that the SMP is not considered in the LP model given in (11). As a result, the pLP i values of operation states z2−z4 are not fit for the practical condition for this specific case, as shown in Table XII. Therefore, further correction is needed for the results obtained via the LP algorithm. As the popt i values and μ are computed according to the SMP, the results satisfy the constraints in (11) and the SMP. In [14], the pi values are computed according to the parameter values suggested by the experts, as shown in Table XV. With the pi values and the μ0 i values given in Table XI, we can get the μ value according to (10), as shown in Table XV. We can see that the μ value (0.2726) in Table XIV is higher than the result (0.173) in Table XV. The result verifies again that a higher lifetime can be ensured for the transportation system via the proposed optimization model.SUN et al.: APPLICATION OF THE LP-ELM MODEL ON TRANSPORTATION SYSTEM LIFETIME OPTIMIZATION 1491 TABLE XVIII pLP i AND μ OBTAINED VIA THE LP ALGORITHM IN EXAMPLE 3 TABLE XIX OPTIMAL PARAMETERS γbl opt PREDICTED VIA ELM IN EXAMPLE 3 TABLE XX p opt i AND μ OBTAINED ACCORDING TO THE SMP AND THE γbl opt OF TABLE XIX TABLE XXI pi AND μ COMPUTED ACCORDING TO THE PARAMETER VALUES SUGGESTED BY THE EXPERTS IN EXAMPLE 3 C. Application on the Port Bulk Cargo Transportation System The bulk conveyor system is a part of the Baltic Bulk Terminal of the Port of Gdynia, which is used to load ships with bulk cargo from the Terminal Storage. The transportation system is composed of nine subsystems. In terms of the transportation scheme, three operation states are defined in [17]. The matrix of sojourn time distributions is given as follows: [Hbl(t)]= ⎡⎣ 0 1−e−0.01935t2 1−e−0.00882t2 1−e−0.08190t2 0 0 1−e−0.03697t2 0 0 ⎤⎦ . (23) As the distribution functions in (23) are non-integrable, the mean values Mbl are approximately estimated via the following formula: Mbl = γbl −βblΓ 1 + β1 bl (24) where βbl = 2, Γ(u) = 0∞ tu−1e−tdt, and u > 0 is the gamma function.1 [pbl] = ⎡⎣ 0 0.37 0.63 1 0 0 1 0 0 ⎤⎦ . (25) Parameters pbl, πi, and μ0 i are given in (25), Tables XVI; and XVII, respectively. The pLP i and μ obtained by the LP algorithm, the γbl opt predicted via ELM, and the corresponding popt i and μ computed in terms of the SMP are given in Tables XVIII–XX, respectively. In [17], the pi values are computed according to the parameter values suggested by the experts, as shown in Table XXI. With the pi values and the μ0 i values given in Table XVII, we 1Part values of the gamma function are given in Table XXIII in the Appendix. TABLE XXII MEAN STD AND CV FOR THE μ VALUES OF 20 TRIALS can get the μ value according to (10), as shown in Table XXI. We can see that the μ value (0.0248) in Table XX is higher than the result (0.0219) in Table XXI. The result verifies on the port bulk cargo transportation system, a higher lifetime can be achieved for the transportation system with the proposed LP-ELM model. D. Discussions A brief discussion for five problems is given here. 1) When the simplex, medium-, and large-scale algorithms are used in the LP algorithm, respectively, the obtained pLP i are all the same. Therefore, the changes in different LP algorithms have no influence on the optimization results. 2) Fig. 4 shows the unconditional lifetime μ of 20 trials for three examples. In addition, the mean, standard deviation (std), and the coefficients of variation (i.e., the ratio between the std and mean) are given in Table XXII. From Fig. 4 and Table XXII, it can be seen that the fluctuations of the final results are relatively small. 3) The optimization problem addressed in this paper can also be formulated as nonlinear optimization problems. Nonlinear optimization techniques may be used in two ways. i) Substitute ELM by nonlinear optimization techniques within the proposed two-stage framework. The approximate optimal transient probabilities pLP i are first obtained through the LP algorithm in terms of the model given in (11). Denoting by pi a function of γbl, i.e., pi(γbl), the γbl opt can be found via nonlinear optimization techniques by minimizing the following objective function: min γbl v i =1 pi(γbl) − popt i 2 . (26) The preceding optimization problem can be regarded as a multiple-input–multiple-output regression problem. As it has been demonstrated that ELM is superior to other optimization techniques in the regression or classification problem in many literatures,2 we present no more discussions here for this case. ii) In terms of the SMP described in Section II, we can also formulate the optimization problem as the following nonlinear optimization model: max γbl f(γbl) = max γbl v i =1 pi(γbl)μ0 i (27) 2http://www.ntu.edu.sg/home/egbhuang/1492 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011 where pi are regarded as the functions of γbl. Take the first application for example; the nonlinear optimization problem can be represented as follows: max γbl v i =1 pi(γbl)μ0 i = min γbl − v i =1 pi(γbl)μ0 i = min γbl − v i =1 πiMi(γbl) v l=1 πlMl(γbl)μ0 i (28) where function Mi(γbl) can be derived from (2) and (3). Two problems arise when we use the preceding model to find the optimal transient probabilities pi. On the one hand, it can be seen from (27) that an explicit expression of the objective function f(γbl) is generally necessary for the preceding optimization model. Although the relationships between γbl and Mbl are very simple in this case, as shown in (2), the final objective function is still fair complicated. On the other hand, how to enforce constraints on the variables γbl to ensure that pi is meaningful is still a difficult problem to handle. Compared with the nonlinear optimization model (27), we decompose the optimization problem in two steps in the proposed LP-ELM model. An explicit expression of the objective function is not necessary. Applications on three different cases demonstrated that a higher lifetime can be ensured for the transportation system, whereas the obtained optimal transient probabilities popt i still satisfy the requirements given in (11). 4) In the proposed model, the training data of ELM are sampled around the parameter values evaluated by the experts. Therefore, the final results are influenced by the empirical values to some extent. Moreover, the final results are also affected by the random initiation of ELM more or less, as shown in Fig. 4. How to reduce these influences is an issue to be investigated in our future work. 5) The discussion about the proposed model has been kept simple for ease of understanding, with the assumptions used being similar to those existing literatures. For problems that are more complex, the proposed model can sometimes be modified to be applicable. For example, when the objective function in (10) is more complicated, e.g., nonlinear, or more constraints are enforced, instead of the LP algorithm, other optimization methods, such as various nonlinear optimization approaches, can be used to find the approximate optimal pi, whereas other procedures (e.g., the prediction of γbl opt and the computation of p opt i ) are not changed. V. CONCLUSION In this paper, a two-stage LP-ELM model has been proposed to optimize the transportation system lifetime, in which SMM has been used to model the operation process. We have first formulated the lifetime optimization problem as an LP model. Through this model, the approximate optimal transient probabilities can be obtained via the LP algorithm. Then, Fig. 4. Unconditional lifetime μ of 20 trials for three examples. TABLE XXIII PART VALUES OF GAMMA FUNCTION the relationship between the transient probabilities and the sojourn time distribution parameters is modeled as a multipleinput–multiple-output regression problem. With data sampling and sample selection, the corresponding optimal distribution parameters can be predicted by ELM. Finally, the optimal transient probabilities are computed in terms of the SMP and the optimal distribution parameters. Applications on three examples have demonstrated that a higher lifetime can be ensured for the transportation system by the LP-ELM model. Although different LP algorithms can be chosen, the performance of the proposed model is not sensitive to the specific algorithm. The results on multiple repeated trials have indicated that the LP-ELM model is robust to the random initiation of ELM. APPENDIX Table XXIII contains the part values of the gamma function. ACKNOWLEDGMENT The authors would like to thank the editors and the anonymous reviewers, who have given many helpful comments and constructive suggestions, and Prof. M. Xie (Singapore principal investigator (PI) of this project) and Prof. K. Kolowrocki (Poland PI of this project) for their support on this research and strong encouragement for our collaboration.SUN et al.: APPLICATION OF THE LP-ELM MODEL ON TRANSPORTATION SYSTEM LIFETIME OPTIMIZATION 1493 REFERENCES [1] A. Alonso-Ayuso, L. F. Escudero, and F. J. 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Zhan-Li Sun received the Ph.D. degree from the University of Science and Technology of China, Hefei, China, in 2005. Since 2006, he has been with Hong Kong Polytechnic University, Kowloon, Hong Kong, Nanyang Technological University, Singapore, and National University of Singapore, Singapore. His research interests include machine learning and image and signal processing.1494 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011 Kien Ming Ng received the B.S. degree in mathematics from the National University of Singapore and the M.S. degree in engineering-economic systems and operations research and the Ph.D. degree in management science and engineering from Stanford University, Stanford, CA. He is currently an Assistant Professor with the Department of Industrial and Systems Engineering, National University of Singapore. His research interests include nonlinear and integer optimization algorithms, scheduling, and routing. Joanna Soszynska-Budny ´ received the M.Sc. degree in physics and mathematics from the University of Gdansk, Gdansk, Poland, and the Ph.D. degree in automatics and robotics from Polish Academy of Science, Warsaw, Poland. She is currently an Assistant Professor with the Department of Mathematics, Gdynia Maritime University, Gdynia, Poland. She has published more than 80 papers in scientific journals and conference proceedings. Her research interests are mathematical modeling of the safety and reliability of complex systems under variable operation conditions. Mohamed Salahuddin Habibullah received the B.S. degree in mechanical engineering (first-class honors) and the Ph.D. degree from the University of Leicester, Leicester, U.K., in 1999 and 2004, respectively. He is currently a Scientist with the Institute of High Performance Computing (IHPC), Agency for Science, Technology, and Research, Singapore. At IHPC, he works on many research and development projects in diverse computational engineering fields. His current research interests include optimization, safety and reliability, and system-level integration.