0123456789
Bulletin of the JSME
Vol.10, No.3, 2016
Journal of Advanced Mechanical Design, Systems, and Manufacturing
Multistage stochastic programming model for optimizing allocation of running time supplements Takayuki SHIINA*, Susumu MORITO* and Jun IMAIZUMI** *School of Creative Science and Engineering, Waseda University 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan E-mail:
[email protected] **Faculty of Business Administration, Toyo University 5-28-20 Hakusan, Bunkyo-ku, Tokyo 112-8606, Japan
Received 29 October 2015
Abstract
We consider the allocation of a running time supplement to a railway timetable. Previously, Vekas et al. examined the optimal way to allocate the running time supplement. The uncertain disturbances in a railway were modeled using random variables. In their model, it was assumed that there was an upper limit to the total supplement, but its allocation was not restricted. In this paper, we suggest an improvement to the previous model and present a new stochastic programming model in which there is a constraint on the running time supplement allocated to each trip to minimize the expected delay. Then a solution algorithm to solve the problem is developed. In the previous model, allocation of the running time supplement was biased because it was not allocated to all trips. We balance the amounts of supplements for trips by adding upper and lower bounds. The uctuations of the supplements for trips become small, and the probability of a delay decreases using our new model. Then the calculation times using the L-shaped algorithm and the former method solving a deterministic equivalent of large-scale problems are compared. It is shown that the large-scale problems can be solved effectively by using the L-shaped method.
Key words : Railway timetable, Running time supplement, Optimization, Multistage stochastic programming, Lshaped method
1.
Introduction
In Japan, there is high demand for travel by railway, and this is especially true during weekday rush hours, when the trains become highly congested due to commuters who work in the metropolitan areas. There is a positive correlation between the congestion rate and the times at which travelers embark and disembark at train stations. On the other hand, it is important to determine an appropriate supplement to the running times between stations and arrival times at each station, in order to create a schedule that is robust against unanticipated delays. In the present study, we develop a way to distribute the running time supplement in such a way that the train can operate according to the schedule, in spite of delays. A previous study by Vekas et al. (2012) examined the optimal way to allocate the running time supplement for a train. The uncertain disturbances were modeled as a random variable. In their model, it was assumed that there was an upper limit to the total supplement, but its allocation was not restricted. In this paper, we suggest an improvement to the previous model and present a new mathematical programming model in which there is a constraint on the running time supplement allocated to each trip; this is done in order to minimize the expected delay. In addition, we compare the expected total delay for each model and examine the differences in the delay ratios. 2.
Allocation of supplement in railway line
We assume the existence of a railway line with a sequence of T + 1 stations, and the stations are assigned numbers 0, 1, . . . , T as shown in Fig. 1. Thus, we will consider T trips between stations. The expected total delay time of a train traveling from station 0 to station T is set as the objective function that is to be minimized. Paper No.15-00601 [DOI: 10.1299/jamdsm.2016jamdsm0043]
© 2016 The Japan Society of Mechanical Engineers
1
Shiina, Morito and Imaizumi, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.10, No.3 (2016)
Starting Station 0 Trip 1
Station Station 1 T −1 Trip 2 Trip T Fig. 1 Railway line and trip
Terminal Station T
-
To formulate the problem, we make the following assumptions. ( 1 ) A stop event in station t − 1 and the running event between station t − 1 and t are brought together in one event, and this is dened as trip t(1 ≤ t ≤ T ). Each trip corresponds to a stage in the multistage stochastic programming problem. ( 2 ) A disturbance that occurs in trip t is dened by the random variable ξt . It is assumed that ξ1 , . . . , ξT are independent. ( 3 ) The total running time supplement is limited, and the amount that can be distributed to each trip has both a lower bound and an upper bound. ( 4 ) The actual delay in a trip can be calculated by subtracting the running time supplement from the sum of the delay of the previous trip and the disturbance of the present trip. The planned time for the trip is composed of the running time supplement and the ideal time. The relationship between actual and planned travel time is shown in Fig. 2. ( 5 ) The minimum unit is assumed to be one second for both the running time supplement and the delay. 6
Actual travel time
Disturbance + Delay from previous trip
?
Ideal time
Actual delay Supplement Ideal time
6
Planned time
?
Fig. 2 Running time supplement The actual delay in a trip must be greater than or equal zero and is calculated as equation (1). Actual delay {
}
= max Delay of the previous trip + Disturbance of the present trip − Running time supplement, 0
3.
(1)
Basic approach of stochastic programming
Stochastic programming (Birge 1997, Birge and Louveaux 1997, Kall and Wallace 1994) deals with optimization under uncertainty. A stochastic programming problem with recourse (SPR) is referred to as a two-stage stochastic problem, and it is formulated as follows. (SPR): min c⊤ x + Q( x) subject to Ax = b x≥0 where Q( x) = Eξ [Q( x, ξ)] Q( x, ξ) = min{q⊤ y(ξ) | W y(ξ) = ξ − T x, y(ξ) ≥ 0}, ξ ∈ Ξ In the formulation of the SPR problem, c is a known n1 -vector, b is a known m1 -vector, q(> 0) is a known n2 -vector, and A and W are known matrices of size m1 × n1 and m2 × n2 , respectively. In the rst stage, the decisions are represented by the n1 -vector x. We assume the m2 -random vector ξ is dened on a known probability space. Let Ξ be the support of ξ, i.e., the smallest closed set such that P(Ξ) = 1. Given a rst-stage decision x, the realization of the random vector ξ of ξ is observed. The second-stage data ξ then become known. The second-stage decision y(ξ) must be taken so as to satisfy the constraints W y(ξ) = ξ − T x and y(ξ) ≥ 0. The second-stage decision y(ξ) is assumed to cause a penalty of q. The objective function contains a deterministic term c⊤ x and the expectation of the second-stage objective. The symbol Eξ [DOI: 10.1299/jamdsm.2016jamdsm0043]
© 2016 The Japan Society of Mechanical Engineers
2
Shiina, Morito and Imaizumi, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.10, No.3 (2016)
represents the mathematical expectation with respect to ξ, and the function Q( x, ξ) is called the recourse function in state ξ. The value of the recourse function is given by solving a second-stage linear programming problem. It is assumed that the random vector ξ has a discrete distribution with nite support Ξ = {ξ1 , . . . , ξS } with Prob(ξ = ξ s ) = p s , s = 1, . . . , S . A particular realization ξ of the random vector ξ is called a scenario. Given the nite discrete distribution, the SPR problem is restated as the deterministic equivalent problem (DEP). (DEP): min c⊤ x +
S ∑ s=1
p s Q( x, ξ s )
subject to Ax = b x≥0 where Q( x, ξ s ) = min{q⊤ y(ξ s ) | W y(ξ s ) = ξ s − T x, y(ξ s ) ≥ 0}, s = 1, . . . , S To solve the DEP, an L-shaped method by Van Slyke and Wets (Van Slyke and Wets 1969) has been used. This approach is based on Benders' decomposition (Benders 1962). The expected recourse function is piecewise linear and convex, but it is not given explicitly in advance. In the algorithm of the L-shaped method, we solve the MASTER problem, ∑ stated below. The new variable θ denotes the upper bound on the expected recourse function, and θ ≥ Ss=1 p s Q( x, ξ s ). The given parameter θ0 indicates the lower bound. (MASTER): min c⊤ x + θ subject to Ax = b x≥0 θ ≥ θ0
Q(x) Optimality Cut Feasibility Cut
θ
x
Fig. 3 L-shaped method Let x∗ , θ∗ be the optimal solution of the MASTER problem; then the following second-stage problem is solved for s = 1, . . . , S .
Q( x∗ , ξ s ) = min{q⊤ y(ξ s ) | W y(ξ s ) ≥ ξ s − T x∗ , y(ξ s ) ≥ 0} = max{(ξ s − T x∗ )⊤ π(ξ s ) | π(ξ s )⊤ W ≤ q⊤ }
(2) (3)
If the minimization problem (2) is infeasible for some scenario ξ s , then either the optimal objective value of the maximization problem (3) is unbounded above or problem (3) is infeasible. Ignoring the latter case, we obtain a dual solution π¯ (ξ s ) ≥ 0, which satises the following inequalities. (ξ s − T x∗ )⊤ π¯ (ξ s ) > 0 and π¯ (ξ s )⊤ W ≤ 0
(4)
To cut off solution x∗ , the feasibility cut (5) is added to the formulation of the MASTER problem. (ξ s − T x)⊤ π¯ (ξ s ) ≤ 0
[DOI: 10.1299/jamdsm.2016jamdsm0043]
(5)
© 2016 The Japan Society of Mechanical Engineers
23
Shiina, Morito and Imaizumi, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.10, No.3 (2016)
If the minimization problem (2) is feasible for ∀ξ ∈ Ξ, and θ∗ < Ss=1 p s Q( x∗ , ξ s ), let π∗ (ξ s ) be the solution of problem (3). ∑ In this case, the optimality cut (6) is added as an outer approximation of Ss=1 p s Q( x, ξ s ). ∑
θ≥
S ∑
s=1
ps (ξ s − T x)⊤ π∗ (ξ s )
(6)
The recourse function is given by an outer linearization using a set of optimality cuts, as shown in Fig. 3. For the continuous stochastic programming problem with recourse, an L-shaped method is well known. The L-shaped method has been used to solve stochastic programs that have discrete decisions in the rst stage by Laporte and Louveaux (1993). This method was also used to solve a stochastic concentrator location problem (Shiina 2000a, 2000b) and a problem with integer recourse (Shiina et al. 2007). We assume the stochastic elements are dened over a nite discrete probability space (Ξ, σ(Ξ), P), where Ξ = Ξ1 × · · · × ΞT is the support of the random data in each stage, with Ξt = {ξts , s = 1, . . . , kt }. The possible sequences of the realization of random variables (ξ1 , . . . , ξT ) are called scenarios. The scenarios are often described using a scenario tree, as shown in Fig. 4. In stages t ≤ T , we have a limited number of possible realizations, which we call the stage t scenarios. * 1lScenario * 1l XXX X z 2l l 1 HH : 3l HH l j 2 H HH H j 4l ξ1 ξ2 r r r -
0
1
2 Stage Fig. 4 Scenario tree
In a scenario tree, the stage t scenario connected to the stage t − 1 scenario s is referred to as a successor of the stage t − 1 scenario s. The set of all successors of the stage t − 1 scenario s is denoted by Dt ( s). Similarly, the predecessor of the stage t scenario s is denoted by α( s, t). These relationships are illustrated in Fig. 5. 1 '$ - s s α ( s , t ) D ( s ) PP PP &% P q P Stage t − 1 Stage t Stage t − 1 t
Stage t
Fig. 5 Successor and predecessor We can formulate a deterministic equivalent problem for the multistage stochastic linear programming problem by replicating the constraints for each possible event in Ξt , t = 1, . . . , T , since the probability space is nite and discrete. To solve the deterministic equivalent problem, the nested decomposition method (Birge 1985, Birge et al. 1996) can be used. For multistage stochastic programming with recourse, Louveaux (1986) introduced the concept of a block-separable recourse. By utilizing this property, the problem can be transformed into a two-stage stochastic program with recourse. Such a typical problem is the multistage electric power capacity expansion problem. Shiina and Birge (2003) proposed an L-shaped algorithm to solve the problem by reformulating the problem into one that has integer variables only in the rst-stage decisions. 4.
Formulation of the problem
In the formulation of the problem, we will use the following notation for parameters and random variables. [DOI: 10.1299/jamdsm.2016jamdsm0043]
© 2016 The Japan Society of Mechanical Engineers
24
Shiina, Morito and Imaizumi, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.10, No.3 (2016)
Parameters
T Number of trips kt Number of realizations of disturbances in trip t Kt Number of stage t scenarios Kt = ∏ti= ki pts Probability of stage t scenario s Ut Upper bound on the running time supplement allocated to trip t Lt Lower bound on the running time supplement allocated to trip t M Upper bound on total running time supplement 1
Random variables
ξt Time of a disturbance in trip t ξts Realization of time of disturbance in trip t for stage t scenario s α(t, s) Predecessor of stage t scenario s; α(1, s) = 1 Ξt Support of ξt Decison variables
xt Amount of running time supplement exceeding Lt that is allocated to trip t yts Actual delay in stage t scenario s The stochastic programming problem is formulated as follows. K T ∑ ∑
pts Qts (yαt−(t1,s) , xt , ξts ) t=1 s=1 subject to Qts (yαt−(t1, s) , xt , ξts ) = min{yts |yts ≥ yαt−(t1, s) + ξts − ( xt + Lt ), yts ≥ 0}, t = 1, . . . , T , s = 1, . . . , Kt T ∑ ( xt + Lt ) ≤ M t=1 ( xt + Lt ) ≤ Ut , t = 1, . . . , T min
t
y =0 1 0
x t ≥ 0, t = 1, . . . , T yts ≥ 0, t = 1, . . . , T , s = 1, . . . , Kt ξts ∈ Ξt , t = 1, . . . , T , s = 1, . . . , Kt
(7) (8) (9) (10) (11) (12) (13)
The objective function (7) minimizes the expectation of the total delay. Constraint (8) represents the denition of the recourse function that denotes the recursion of the actual delay. The actual delay yts (≥ 0) can be calculated by subtracting the running time supplement xt + Lt from the sum of the delay of the previous trip yαt−(t1, s) and the disturbance of the present trip ξts . Constraint (9) shows that the total running time supplement is bounded by M . Constraint (10) expresses the upper and lower bounds on the running time supplement for trip t. Constraint (11) indicates the initial delay. Constraints (12) and (13) ensure that the variables are nonnegative. The previously described L-shaped algorithm is applied to the above problem. The MASTER problem uses y as the upper bound of the expected value for the recourse function, as shown below. (MASTER): min subject to
K T ∑ ∑ t
pts yts
t=1 s=1 T ∑ ( xt + Lt ) ≤ M t=1 ( xt + Lt ) ≤ Ut , t = 1, . . . , T
x t ≥ 0, t = 1, . . . , T yts ≥ 0, t = 1, . . . , T , s = 1, . . . , Kt [DOI: 10.1299/jamdsm.2016jamdsm0043]
(14) (15) (16) (17) (18) © 2016 The Japan Society of Mechanical Engineers
25
Shiina, Morito and Imaizumi, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.10, No.3 (2016)
The relation between the next y and the recourse function is not included in this problem, and so the optimal cuts are added to a later iteration. yts ≥
Qts (yαt−t,s , xt , ξts ) (
(19)
)
1
The MASTER problem is a linear programing (LP) problem, and various types of algorithms can be used to seek the optimal solution. Here, the MASTER problem is solved. Next, the solution to the MASTER problem ( x, y ) is employed to solve a second-stage problem that denes the recourse function. The dual problem in the second-stage problem is dened as follows.
Qts (yαt−t,s , xt , ξts ) = min{yts | yts ≥ yαt−t,s + ξts − ( xt + Lt ), yts ≥ 0} = max{πts {yαt−t, s + ξts − ( xt + Lt )} | 0 ≤ πts ≤ 1} (
)
(
1
(20) (21)
)
1
(
)
1
The second-stage problem has complete recourse, and so this is a feasible problem. In other words, there exist feasible yts ≥ 0 in an arbitrary ξts . Also, as long as y is not at the upper bound of the recourse function, that is, when y < Qts (yαt−(t1, s) , xt , ξts ), the dual optimal solution π ts of the second-stage problem can be used to generate the following optimality cut, and this is added to the MASTER problem. yts ≥ π ts {yαt−(t1, s) + ξts − ( xt + Lt )}
(22)
The MASTER problem is solved using the L-shaped algorithm, and the cut is added; this is repeated until y ts ≥ + ξts − ( xt + Lt ) is satised. When the solution x, y that is obtained in the solution to the MASTER problem satises y ts ≥ y αt−(t1, s) + ξts − ( xt + Lt ), we can assume that we have obtained an acceptable approximation of the recourse function Qts (yαt−(t1,s) , xt , ξts ) in the neighborhood of ( x, y ). Then, we nd a rst-stage feasible solution, solve a second-stage problem using the rst-stage solution, and nd the expected value for the recourse function. The L-shaped algorithm for obtaining the optimal solution can be summarized as follows. y αt−(t1, s)
L-shaped algorithm
Step 1:Solve the MASTER problem. Let xt , t = 1, . . . , T , y ts , s = 1, . . . , Kt , t = 1, . . . , T be the optimal solution of the MASTER problem. Step 2:Solve the recourse problem for stage t scenario s, s = 1, . . . , Kt , t = 1, . . . , T . ′
′
′
′
′
Step 3:Calculate Qts (yts−1 , xt , ξts ), ∀ s ∈ Dt ( s), s = 1, . . . , Kt , t = 0, . . . , T − 1. If y ts < (1 − ε) Qts (yts−1 , xt , ξts ), s ∈ Dt ( s), the optimality cut (22) is added to the formulation of the MASTER problem (ε > 0: tolerance). Go to Step 1. ′
′
Step 4:If no optimality cuts are added, then stop. 5.
Numerical experiments
The running time supplement was allocated by using the model proposed in the present study. The number of trips in the rail line was set to 6. The maximum number of disturbances per trip was set to 8. It was assumed that there was a possibility of either a big or a small disturbance in any trip. The expected values of these big and small disturbance were assumed to be 60 and 20 seconds, respectively. We assumed three cases: big disturbances occur in each of two of the early trips, big disturbances occur in each of two of the central trips, and big disturbances occur in each of two of the nal trips as shown in Fig. 6. The upper bound M on the total running time supplement for the entire rail line was assumed to be 450 seconds. The bounds for the supplement were set to Lt = 55 and Ut = 100. The probability of a disturbance in each trip was set to 1/kt for all trips. The disturbances followed a discretized exponential distribution with expected value 1/λt . The values of the disturbances were generated from the cumulative distribution function in which the cumulative probability equals 1/kt (i − 1/2), i = 1, . . . , kt . Experiments were carried out using AMPL (Fourer et al. 1993) CPLEX on a Xeon E5507 2.00 GHz (two processors; memory: 12.0 GB). Table 1 shows the results of the experiment with three disturbance cases. In this table, two types of optimization models which include bound constraints for total supplements or not are compared. The models without bound constraints coincide with the previous model (Vekas et al. 2012). The value of xt + Lt denotes the planned time for trip t. The total supplement time was equal to the upper bound M , except for one case that had a big disturbance in [DOI: 10.1299/jamdsm.2016jamdsm0043]
© 2016 The Japan Society of Mechanical Engineers
26
Shiina, Morito and Imaizumi, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.10, No.3 (2016)
Big disturbance in forward trips Big Big Small Small Small Small λ1 = 1/60 λ2 = 1/60 λ3 = 1/20 λ4 = 1/20 λ5 = 1/20 λ6 = 1/20 Big disturbance in central trips Big Small Small Small Small Big λ1 = 1/20 λ2 = 1/20 λ3 = 1/60 λ4 = 1/60 λ5 = 1/20 λ6 = 1/20 Big disturbance in backward trips Big Small Small Small Small Big λ1 = 1/20 λ2 = 1/20 λ3 = 1/20 λ4 = 1/20 λ5 = 1/60 λ6 = 1/60 Trip 1 Trip 2 Trip 3 Trip 4 Trip 5 Fig. 6 Three cases of disturbances
Trip 6
the nal trips and with bound constraints. The probability of delay is the probability of the scenario in which the delay is caused in either of trip in the rail line. It can be seen that the probability that a delay occurs decreases when upper and lower bounds are added, even though the value of the objective function increases; with upper and lower bounds, the objective function increased by a factor of 1.5. Because the expectation of the total delay is small compared to the upper bound on the total supplement M = 450, it has little inuence on the schedule. In a previous model, allocation of the running time supplement was biased because it was not allocated to all trips. We balanced the amounts of supplements for trips by adding upper and lower bounds. The uctuations of the supplements for trips thus became small, and the probability of a delay thus decreased. A certain amount of supplement is necessary for safe operation and as a buffer against delays. An overcrowded schedule can be avoided by setting a lower bound on the supplement for each trip.
Table 1 Result of experiments Position of Big Disturbance forward forward central Bound Constraints Yes No Yes Trip 1 x1 + L1 100 108 55 Trip 2 x2 + L2 100 166 55 Trip 3 x3 + L3 85 55 100 Trip 4 x4 + L4 55 33 100 Trip 5 x5 + L5 55 55 85 Trip 6 x6 + L6 55 33 55 Objective 22.10 14.71 21.89 Probability of Delay Occurrence 23.44% 33.01% 23.44% Computing Time (s) 42 55 29
central No 33 51 122 156 55 33 14.60 48.71% 57
backward Yes 55 55 55 55 100 100 19.39 23.44% 6
backward No 33 55 55 41 166 100 13.42 33.01% 46
Then, we compare the calculation times using the L-shaped algorithm and branch-and-bound for a deterministic equivalent LP on large-scale problems. The number of scenarios are increased to 12. So the total number of scenarios becomes 126 = 2985984 which is approximately three million. Both methods showed calculation times increasing with the problem scale, but the L-shaped algorithm had shorter times as shown in Table 2.
Table 2 Large Scale Problems
Upper Bound U ∞ ∞ Lower Bound L 0 55 Computing time (s) of deterministic equivalent MIP 43810 8904 Computing time (s) of L-shaped 10846 3539 t
t
100 0 17561 1018
100 55 1086 672
The results we have presented here are encouraging and the application to real problem is necessary in the future. If we consider T = 20 trips with kt = 2, the total number of scenario is 220 = 1048576. But with kt = 4, the number of scenarios becomes 420 = 1099511627776 which is impossible to solve. There are many points which must be solved in scenario generation.
[DOI: 10.1299/jamdsm.2016jamdsm0043]
© 2016 The Japan Society of Mechanical Engineers
27
Shiina, Morito and Imaizumi, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.10, No.3 (2016)
6.
Concluding remarks
In the present study, we showed that our method reduced the probability of delay for a rail line, even though the objective function was slightly larger than that of a previous optimization model. Moreover, it was shown that the problem can be solved effectively by using the L-shaped method. As a next step, considering the application, denition of the objective function is an important problem. Instead of the unweighted sum of expected delays, some other measure of the importance of stations on the railway line can be applied to the model. In addition to the importance of stations, the probability of delay can be dened as the objective function. However, the problem becomes a large-scale mixed integer programming problem. It is evident that more work using different approach is necessary. References
Benders, J. F., Partitioning procedures for solving mixed-variables programming problems, Numerische Mathematik, Vol. 4 (1962), pp. 238252. Birge, J. R., Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research, Vol. 33 (1985), pp. 9891007. Birge, J. R., Stochastic programming computation and applications, INFORMS Journal on Computing, Vol. 9 (1997), pp. 111133. Birge, J. R., Donohue, C. J., Holmes, D. F., and Svintsitski, O. G., A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs, Mathematical Programming, Vol. 75 (1996), pp. 327352. Birge, J. R. and Louveaux, F. V., Introduction to Stochastic Programming (1997), Springer-Verlag. Fourer, R., Gay, D. M., and Kernighan, B. W., AMPL: A Modeling Language for Mathematical Programming (1993), Scientic Press. Kall, P. and Wallace, S. W., Stochastic Programming (1994), John Wiley & Sons. Laporte, G. and Louveaux, F. V., The integer L-shaped method for stochastic integer programs with complete recourse, Operations Research Letters, Vol. 13 (1993), pp. 133142. Louveaux, F. V., Multistage stochastic programs with block-separable recourse, Mathematical Programming Study, Vol. 28 (1986), pp. 4862. Shiina, T., L-shaped decomposition method for multi-stage stochastic concentrator location problem, Journal of the Operations Research Society of Japan, Vol. 43 (2000a), pp. 317-332. Shiina, T., Stochastic programming model for the design of computer network, Transactions of the Japan Society for Industrial and Applied Mathematics, Vol. 10 (2000b), pp. 3750. (in Japanese) Shiina, T. and Birge, J. R., Multi-stage stochastic programming model for electric power capacity expansion problem, Japan Journal of Industrial and Applied Mathematics, Vol. 20 (2003), pp. 379397. Shiina, T., Tagaya, Y., and Morito, S., Stochastic programming problem with xed charge recourse, Journal of the Operations Research Society of Japan, Vol. 50 (2007), pp. 299314. Van Slyke, R. and Wets, R. J. -B., L-shaped linear programs with applications to optimal control and stochastic linear programs, SIAM Journal on Applied Mathematics, Vol. 17 (1969), pp. 638663. Vekas, P., van der Vlerk, M. H., and Klein Haneveld, W. K., Optimizing existing railway timetables by means of stochastic programming, Stochastic Programming E-print Series (online), available from [http://edoc.huberlin.de/browsing/speps/], (2012).
[DOI: 10.1299/jamdsm.2016jamdsm0043]
© 2016 The Japan Society of Mechanical Engineers
28
