Sample Average Approximation for Stochastic Empty Container Repositioning

Transcription

1 Sample Aerage Approimation for Stochastic Empty Container Repositioning E Peng CHEW a, Loo Hay LEE b, Yin LOG c Department of Industrial & Systems Engineering ational Uniersity of Singapore Singapore a isecep@nus.edu.sg; b iseleelh@nus.edu.sg; c iselongy@nus.edu.sg Abstract To incorporate uncertainties in empty container repositioning problem, we formulate a two-stage stochastic programming model with random demand, supply, ship weight capacity and ship space capacity. The Sample Aerage Approimation (SAA) method is applied to approimate the epected alue function. Seeral non-independent and identically distributed sampling schemes are considered to enhance the performance of the SAA method. umerical eperiments show that near-optimal solutions could be proided by the SAA method with these sampling schemes. Keywords-empty container repositioning; simulation optimization; sample aerage approimation; supersaturated design I. ITRODUCTIO One main issue in the containerized transportation is the imbalance of container flow, which is the result of global trade imbalance between different regions. Thus, maintaining higher operational cost efficiencies in repositioning empty containers becomes a crucial issue in shipping industry. There are increasing studies on empty container flows in recent years [][2]. In the maritime transportation, container operators hae to deal with some uncertain factors lie the real transportation time between two ports/deports, future demand and supply, the in-transit time of returning empty container from customers, and the aailable capacity in essels for empty containers transportation, etc. The uncertain nature of parameters is taen into account in seeral studies [3][4]. A multi-scenario model was proposed in [5] to address the Empty Container Repositioning (ECR) problem in a scheduled maritime system. In their study, opinions of shipping companies were considered to generate scenarios when the distributions of uncertain parameters cannot be estimated through historical data. Our study focuses on deeloping scenario-based model when the distribution of uncertain parameters can be estimated through historical data. Random scenarios could be generated based on these distributions. Howeer, it is difficult to sole the stochastic ECR problem with a large number of scenarios. In this case, we apply the Sample Aerage Approimation (SAA) method to sole the stochastic ECR problem with multiple scenarios. The SAA problem with multiple scenarios is usually difficult to sole due to its large scale. In this study, we try to enhance the SAA method with well-planned samplings which are more representatie. The motiation of this idea is to get acceptable solutions by soling SAA problems with a small number of scenarios. Independent and identically distributed (i.i.d) sampling is well studied for construction approimations [6][7]. On the other hand, SAA with noni.i.d samplings is also studied in recent year [8][9]. Empirically, it was shown that Latin Hypercube (LH) and Antithetic Variates (AV) methods outperform those under i.i.d sampling, with LH outperforming AV [0]. In [], U design was used to further enhance the accuracy of the SAA and their theoretical results showed that the SAA with U designs can significantly outperform those with LH designs. In this study, we try to do as a whole ery few eperiments (een less than the number of degrees of freedom of the system when that is possible) and still get a satisfying approimation. To our nowledge, no eisting study applies the SAA method with non-i.i.d samplings to the stochastic ECR problem where the distributions of uncertain parameters are nown. This study is to fill in this gap. The paper is structured in the following way. Section I concerns introduction; In Section II, we proide the description of a basic deterministic model and our two-stage stochastic model for ECR. Section III shows the soling methodologies to sole our proposed model. The SAA method and the sampling schemes to enhance SAA are eplained. Section IV presents the results of computational studies. Finally, we gie conclusions and outline directions for future research in Section V. II. PROBLE FORULATIO The focus of this study is to mae operational leel maritime ECR decisions for shipping companies. As the global container transportation networ is large and comple, the ocean liners usually diide the global networ into seeral regions and appoint regional operators to manage the container flows for each region. Because of the long lead time of the across-region empty containers, ocean liners usually mae ordering decisions depending on forecasting to mae decisions on ordering. Due to the booing system used in the maritime transportation, demand, supply and the aailable ship capacity in the near future could be forecasted accurately. Howeer, it is difficult to obtain accurate forecasting for more than one or two wees. Currently, container operators mae decisions based on the nominal forecast alue. Because of the differences between the epected alue and the realized alue, inefficient solutions 86

2 may be produced. These decisions hae to be recoered at real-time operations. A. The Deterministic odel If all parameters in the planning horizon are nown, the deterministic time space model for ECR could be formulated as follows, where the actual serice schedule and most port requirements are considered. Details of this model could be found in [2]. t in TC Subject to t u TC ct, i, u K ip ( s, ) {( s, ) V, ss, p, s i, ta, s } w t, i, K ip ( s, ) {( s, ) V, ss, p, s i, td, s } ip K c t,2,...,t ( c y c z ) y z t, i, t, i, t, i, t, i, K ( s, ) {( s, ) V, ss, td, s } ( g t, s,, ) t, s, (, s, t) {(, s, t) V, s S, t D, s} () K ( h t, s,, ) t, s, (, s, t) {(, s, t) V, s S, t D, s} (2) K u w tb, s, s,, t, s,, t d, s, s,, t d, s, s,, u tbs,, s,, t, s,, (, s, t) {(, s, t) V, s S, t D } The objectie function is to minimize the total operation t cost in the planning horizon. TC is the total operational cost at time t. The operational costs include the handling cost (unloading cost and loading cost), the holding cost, the penalty cost and the transportation cost. u, w,, y i, and z i are decision ariables, where u is the number of empty containers of size unloaded at stop s from serice at time t, w is the number of empty containers of size loaded from stop s onto serice at time t, is the number of empty containers of size transported from stop s to net stop on serice leaing stop s at time t, y i is the number of empty containers of size c w, s K, (, s, t) {(, s, t) V, s S, t A } y u z t, i, t, i, t, i, t, i, ( s, ) {( s, ) V, ss, p, s i, ta, s } ( s, ) {( s, ) V, ss, p, s i, td, s } w y t, i,, s K, i P, t,2,...,t u, w,, y, z 0 (6) t, i, t, i, (3) (4) (5) stored at port i at time t, and z i is the number of empty containers of size that cannot be satisfied by the empty u w y containers stored at port i at time t. c i, c i, c, c i, z and c i are the corresponding cost parameters of unloading, loading, transportation, storing and penalty respectiely. V is the set of serices. P is the set of ports. Q is the set of regions. K is the set of container sizes. And S is the set of stops on serice. D s, is the set of periods in which serice departs from its stop s. A s, is the set of periods in which serice arries at its stop s. Constraint () and constraint (2) are ship capacity constraints. t, s, is residual space capacity on serice when it leaes stop s at time t, and t, s, is residual weight capacity on serice when it leaes stop s at time t. g and h are olume and weight of one container of size. These two constraints should be considered when there is a serice leaing the port. Constraint (3) guarantees the balance of the container flows at each serice. bs, is the transportation time from stop s to net stop on the serice. d s, is the number of days that the serice stays at stop s. Constraint (4) ensures that the number of empty containers unloaded from a essel should not eceed the total number of empty containers in the essel. These two constraints should be considered when there is a serice arriing at a port. Constraint (5) considers the balance of the container flows at each port at each time. ti,, is the supply of empty containers of size in port i at time t, and i is the demand of empty containers of size in port i at time t. ps, is the port corresponding to the stop s on serice. Constraint (6) ensures that all ariables are non-negatie. B. The Stochastic odel In this study, we deelop a stochastic programming model which taes account into four uncertain parameters, i.e. the demand (the empty containers that piced up by the customers to load cargos), the supply (the empty containers that returned by the customers), the aailable ship space capacity and aailable ship weight capacity for empty containers. Other uncertain factors lie the transportation time between two ports are not considered. We also do not consider container substitution in this study. We assume that serice schedule is gien and fied in the planning horizon. This assumption is alid as the planning horizon of our operation model is short (seeral wees), and the serice schedule is not changed frequently. In order to incorporate the deterministic information and the uncertain information, a two-stage stochastic programming is deeloped. This model is run in a rolling horizon manner. ECR decisions are made at the beginning of stage and will be made again when new information is collected. Let denotes a scenario that is unnown when decisions at stage are made, but that is nown when the 87

3 decisions at stage 2 are made, where Ω is the set of all scenarios. A two-stage stochastic model for ECR is formulated as follows. Stage min g( ) c E [ Q(, ( ))] (7) : Decisions at stage 2 (ω): Decisions for scenario ω at stage 2 gien c, c 2 : The cost ector at stage and stage 2 respectiely A, B, A 2, B 2 : The coefficient matrices of or 2 a, a 2 (ω): The RHS of constraint (8) and constraint (2) respectiely : The ector of end container states of stage. It is the empty container inentory at each port and at each essel at the end of stage. {, 2,..., K } ( ): The ector of initial container states of stage 2 gien The objectie function of stage is to minimize the total operational cost in the planning horizon. c is the operation cost at stage. Ep [ Q (, ( ))] is the epected cost at stage 2, where p is the probability distribution of uncertain parameters. We assume that the probability distribution p on Ω is nown in the stage. Q (, ( )) is the objectie function of stage 2, which is the operational cost at stage 2 gien and scenario ω. Constraint (8) and constraint (2) includes the typical constraints of ECR problem in the deterministic model, i.e. ship capacity constraint, serice flow constraint, and port flow constraint. Constraint (9) is to set the end container states of stage, which are also the initial container states of stage 2. Constraint (3) is to set the initial container states of stage 2. III. p subject to A a (8) B (9) 0 (0) Stage 2: For a realized scenario, we hae Q[, ( )] min c ( ) () 2 2 subject to A2 2 ( ) a2 ( ) (2) B ( ) ( ) (3) ( ) 0 (4) SOLVIG ETHODOLOGY Our stochastic problem is hard to sole as it is difficult to ealuate the epected cost of stage 2 for a gien, i.e. Ep [ Q (, ( ))]. It requires the solutions of a large number of stage 2 optimization problems. In this study, we consider applying the SAA method to sole the stochastic ECR problem. The basic idea of SAA method is that the epected objectie function of the stochastic problem is approimated by a sample aerage estimate deried from a random sample and the resulting SAA problem could then be soled by deterministic optimization techniques [6]. A. The SAA ethod 2 A sample with scenarios {,,, } is generated according to the probability distribution p. The SAA problem is formulated as follows. n gˆ min c [ c ( )] (5) 2 2 n Subject to (8), (9); and (2), (3) for n=,2,, 0, 0, ( n ) 0 for n=,2,, (6) 2 The SAA problem can then be soled by deterministic optimization methods. The optimal solution ˆ and the optimal alue g ˆ of the SAA problem could be obtained. ˆ is a candidate solution of the true problem. ote that when this sample is an i.i.d random sample of the random ector, gˆ ( ) is called a (standard) onte Carlo estimator of g. ( ) To ealuate the objectie alue gien the candidate solution ˆ, we consider generating an independent sample with scenarios, where is much larger than. Let gˆ ( ˆ ) is defined to estimate the objectie alue g ( ˆ) of an ' feasible solution ˆ. In order to get a better solution, we can generate independent samples equally with scenarios. By soling the corresponding independent SAA problems, we can get candidate solutions ˆ, 2 ˆ,, ˆ and the objectie 2 alues g ˆ, g ˆ,, ˆ * g. It is natural to tae ˆ as one of the optimal solutions of these SAA problems which proides the smallest estimated objectie alue, To estimate the performance of SAA method, we need to calculate the optimality gap, i.e. the difference between the lower bound and the upper bound. This gap can be used to * ealuate the quality of the solution. As ˆ is a feasible solution of the stochastic ECR problem, ' g ( ) gies an estimate of the upper bound of the true optimal objectie alue of the true problem. On the other hand, as realized scenarios are considered in the SAA problems, the objectie alue of the SAA problem g ˆ has a negatie bias. Let g denotes the aerage objectie alue of the SAA problems, g proides a statistical estimate for a lower bound of the true optimal alue of the true problem. The optimality gap could be estimated as ' n gˆ ( ˆ ) min c ' ˆ Q[ ˆ, ( )] ' (7) n * 2 ' ˆ argmin{ gˆ ( ˆ ) : ˆ { ˆ, ˆ,..., ˆ }} (8) g m m gˆ (9) 88

4 ' g ( ) g (20) B. on-i.i.d Samplings Due to its large scale, the SAA problem (5)-(6) for the real scale ECR is difficult to sole. In this case, the sampling should be well-planned. We try to generate samplings with a small number of scenarios (the number of scenarios is een less than the random ariables of the stochastic ECR problem) and still get acceptable solutions. In this study, three sampling schemes are considered to enhance the performance of the SAA method for the stochastic ECR problem. ) Latin Hypercube (LH) Sampling: In computer eperiments, it is well now that LH design achiees maimum stratification in one-dimensional projections. The idea is to partition the sample space, and the number of sample points on each region should be proportional to the probability of that region. This way we ensure that the number of sampled points on each region will be approimately equal to the epected number of points to fall in that region. 2) AG Design: AG deign is a supersaturated design which is introduced in [3]. One good property of the AG design is that the saturation increase rather fast with the number of scenarios. Besides, for a two-leel design with m scenarios and n factors ( m n), each column has the same number of - s and s in an een case. This property is necessary for a stable regression analysis as each ariable has to be ealuated fairly from its smallest alues to its highest alues. 3) AGLH Design: We also propose a superstaturated design which combines the AG deisn and LH sampling, e.g., a two-leel case, we can generate the AGLH design as follows, a) Generate a AG design, B b) Randomly permute the rows, columns and symbols of B ( m n) c) In each columns of B, replace the m/2 0s by a uniform random permutation of,,m/2. The m/2 s by a uniform random permutation of m/2+,,m. d) Coupling B with U[0,] random ariables and we can get our desire design, C. IV. UERICAL STUDY To ealuate the performance of the SAA method, we first generate an ECR transportation networ as shown in Fig.. Fie ports, three serices, and one type of container (twentyfoot standard container) are considered. The planning horizon is three wees, and we define the first wee as stage and the second and the third wees are stage 2. All information in the first wee is nown when decisions in stage are made, while some parameters in stage 2 are unnown when decisions in stage are made. These parameters are nown when decisions in stage 2 are made. The lead time of across-region empty containers is one wee. The serice schedules are gien in Fig.. Figure. A networ with three serices and fie ports We apply the SAA method to sole the two-stage stochastic problem (with i.i.d sampling). We can sole the SAA problem directly by using CPLEX.2 when the sample size is not too large ( < 000). We set the sample size as 00. The number of scenarios to ealuate the solution is set to be 000. Replication number is set to be 20. The performance of the SAA method ( = 00) for the small scale case is eamined with the ey results shown in Table I. As shown in Table I, the estimated objectie alue of the true problem gˆ ( ˆ* ) is , and a statistic lower ' bound for the objectie alue ' g is The ' optimality gap g ( ) g is (0.73% of g ( )) which is quite small. The small optimality gap implies that the SAA method can proide solutions with good quality. In the case study aboe, samples are independently and identically distributed. We also consider applying the supersaturated design to generate samples. The results of the SAA method with AG design are shown in Table II. The optimal estimated objectie alue we can obtain with =0 (note that the number of sample scenarios is smaller that the number of random ariables of the ECR problem, i.e. 56) is 336.8, which is quite close to the optimal estimated objectie alue in Table I, i.e , with =00. It indicates that SAA method based on supersaturated design TABLE I. RESULTS OF THE SAA ETHOD (=00) Estimate Value g ' g ( ) ' g ( ) g 24.52(0.73%) 89

5 Probability Cost can proide good solutions with a small number of sample scenarios. The performances of the SAA method with i.i.d sampling, LH sampling, AG design, and AGLH design are compared in Fig. 2 and Fig. 3 (all with =0). The replication number is 000 and we can obtain 000 feasible solutions for each sampling method. In Fig. 2, the mean and confidence interal of the estimated objectie alue gˆ ( ˆ) of each sampling method are analyzed. We find that all the three non-i.i.d samplings can reduce the aerage estimated objectie alue and the ariance of the estimated objectie alue. Fig. 3 is the probability plot. Based on Fig. 3, we find that the non-i.i.d samplings are less liely to proide bad solutions compared with the i.i.d sampling. We also find that TABLE II ' RESULTS OF THE SAA ETHOD (=0, AG DESIG) Estimate Value g ' g ( ) ' g ( ) g 86.0(2.56%) iid sampling LH sampling AG design LH_AG sampling Figure 2. Epect cost estimates (α=0.05) AG design AGLH sampling LH sampling iid sampling Cost Figure 3. Probability plot of the objectie estimates the SAA method with AGLH design has the smallest probability to proide bad solutions. V. COCLUSIO AD FUTURE STUDY In this study, we deeloped an operational model to sole the ECR problem. In order to incorporate uncertainties, we built a two-stage stochastic model with uncertain demand, supply, residual ship space capacity, and residual ship weight capacity. The distributions of these parameters can be estimated based on historical data. We applied the SAA method to sole this stochastic problem. In the future, we will consider applying the SAA method to real-scale problems. Based on the results in numerical study section, we found that using LH design, AG design, and the combination of AG design and LH design to enhance the performance of SAA is promising. A direct etension of this wor is to eplore other sampling schemes to control scenario generation and thus improe the quality of solutions. Another possible direction for future research is to study the conergence rate of these samplings for stochastic programming. REFERECES [] A. Olio, P. Zuddas,. D. Francesco, and A. anca, An Operational odel for Empty Container anagement, aritime Economics and Logistics, ol. 7, pp , [2] C.. Feng and C. H. Chang, Empty Container Reposition Planning for intra-asia Liner Shipping, aritime Policy and anagement, ol. 35, pp , [3] R. K. Cheung and C. Y. Chen, A Two-Stage Stochastic etwor odel and Solution ethod for the Dynamic Empty Container Allocation Problem, Transportation Science, ol. 32, pp , 998. [4] A. L. Erera, J. C. orales, and. Saelsbergh, Robust optimization for empty repositioning problems, Operations Research, ol. 57, pp , [5]. D. Francesco, T. G. Crainic, and P. Zuddas, The effect of multi-scenario policies on empty container repositioning, Transportation Research Part E: Logistics and Transportation Reiew, ol. 45, pp , [6] A. J. Kleywegt, A. Shapiro, and T. Homem-De-ello, The Sample Aerage Approimation ethod for Stochastic Discrete Optimization, SIA Journal on Optimization, ol. 2, pp , [7] J. Wei and. J. Realff, Sample aerage approimation methods for stochastic ILPs, Computers and Chemical Engineering, ol. 28, pp , [8] T. Homem-De-ello, On Rates of Conergence for Stochastic Optimization Programs on-independent and Identically distributed Sampling, SIA J. OPTI, ol. 9, pp , [9] H. Xu, Uniform Eponential Conergence of Sample aerage random functions under general sampling with allocation in stochastic programming, Journal of athematical Analysis and Applications, ol. 368, pp , 200. [0]. B. Freimer, J. T. Linderoth, and D. Thomas, The Impact of sampling methods on bias and ariance in stochastic linear programs. Computational Optimization and Applications. 200, DOI: 0.007/s

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