Distributed and decentralized approaches for rail freight car management

CHAPTER 7

Distributed and decentralized approaches for rail freight car management Contents 7.1 Distributed model predictive rail freight car management 7.1.1 Problem description 7.1.2 Cooperative MPC for freight car flow planning 7.2 Decentralized model predictive rail freight car management 7.3 Numerical example

232 233 235 239 240

Previous chapters treat the problem of rail freight car fleet management in a centralized way. However, the rail freight car fleet system represents a very large-scale system which involves a number of interacting subsystems. The railway network is usually composed of regions and each region can be further divided into districts. The flows of freight cars are controlled by local agents (controllers) responsible for efficient freight car management. Centralized control of these very complex subsystems can be very difficult despite the development of high-speed computers and fast algorithms. Also, there are issues related to the reliability of information sharing between subsystems as well as communication limitations. On the other hand, in case of a completely decentralized freight car fleet system, there is a problem related to control and a lack of interaction, which generates suboptimal freight car control actions. In this case, a cooperative distributed model predictive control (DMPC) may represent the solution. Local agents or controllers, having some knowledge on the flows of freight cars from other subnetworks, will be charged with freight car flow optimization on their subnetworks. Fig. 7.1 illustrates centralized (a), decentralized (b), and distributed (c) MPC architecture for a rail freight car fleet management system comprised from two subsystems. The aim of this chapter is to design a DMPC scheme and to compare it with decentralized and centralized schemes on a common benchmark process related to freight car fleet size and allocation. Optimization Models for Rail Car Fleet Management https://doi.org/10.1016/B978-0-12-815154-9.00007-1

© 2020 Elsevier Inc. All rights reserved.

231

232

Optimization models for rail car fleet management

MPC

MPC U1(n)

X1(n)

Freight car fleet subsystem 1

Freight car fleet subsystem 2

Rail freight car fleet system

U2(n)

U1(n)

U2(n)

Freight car fleet subsystem 1

MPC

X1(n)

X2(n)

(A)

Rail freight car fleet system

Freight car fleet subsystem 2 X2(n)

(B) MPC

MPC

U1(n)

U2(n)

Freight car fleet subsystem 1 X1(n)

Rail freight car fleet system

Freight car fleet subsystem 2 X2(n)

(C) Fig. 7.1 Centralized (A), decentralized (B), and distributed MPC (C) architecture for rail freight car fleet management.

7.1 Distributed model predictive rail freight car management Cooperative DMPC (CDMPC) approach is also aligned with recently proposed cooperative strategies for improving the rail freight car fleet utilization as well as the position of single freight car management on the EU railway market (Smart-Rail, 2016). DMPC is a control approach that copes with control problems of large- scale systems in which there are interorganizational connections between different subsystems involved in a joint activity, limited measurement capability and control access of different subsystems and also different potentially conflicting objective criteria of individual subsystems (Li et al., 2016). The general concept, industrial application, and future research directions of DMPC are presented in the books of Camponogara et al. (2002), Maestre and Negenborn (2014), Li and Zheng (2015), and Olaru et al. (2015). In this chapter, CDMPC approach is proposed for solving the rail freight car fleet management problem. The overall optimization problem is decomposed into subproblems, which consider the freight car allocation decisions in related subsystems. Individual decisionmaking units or controllers solve their own, local problems using the model

Distributed and decentralized approaches for rail freight car management

233

covering only their part of the railway network. Model predictive control (MPC) is applied as a solution approach for every local problem. All regional controllers are mutually connected by interconnecting links, in this case, railway lines on which the empty and loaded freight car flows between regions are interchanged. More precisely, these local problems are not considered as isolated but dependent on the solution of the MPC problem of the surrounding controllers. In order to solve the CDMPC problem, a parallel augmented Lagrangian relaxation approach (Negenborn et al., 2008) is applied. In this case, input parameters are taken as exact values and only one type of freight cars was considered.

7.1.1 Problem description Let’s consider a railway transport network M(N, L) composed of a set of nodes N ¼ [s¼1, …, Nsub Ns and railway lines L ¼ [s¼1, …, Nsub Ls . Cooperative freight car management can be established between a set of railway operators Nsub in the M(N, L) network where every railway operator manages freight flows on their own subnetwork which does not include common nodes and links with any other subnetwork Ns \ Nk ¼ ∅, Ls \ Lk ¼ ∅, 8 s, k ¼ 1, …, Nsub. Subnetworks mutually interact through interconnecting lines. Freight car manager s plans the flows of freight cars on subnetwork Ns by solving the optimal freight car allocation problem over P periods of the planning horizon

ð7:1Þ subject to system dynamics and planning constraints Xs ðn + 1Þ ¼ Fs ðnÞXs ðnÞ + Gs Us ðnÞ + Φs ds ðnÞ, n ¼ 0,1,…, P  1

(7.2)

Ds Xs ðnÞ  Ks

(7.3)

Xs ðnÞ, Us ðnÞ  O

(7.4)

The stated objective criteria and constraints are based on the linear discretetime rail freight car fleet sizing and allocation model presented in the previous chapter. Therefore, the objective function (7.1) minimizes the quadratic costs of owning and distributing empty and loaded freight cars but only on a considered subnetwork. The first term is the cost of the rail freight car

234

Optimization models for rail car fleet management

subsystem Ns in the last period of the planning horizon, which is the function of the state vector Xs(n) resulting from control actions Us(n) implemented during the previous periods. The second term (7.2) represents the sum of costs over the P  1 periods of the planning horizon and it is the function of the state vector Xs(n) and control vector Us(n) in a given subnetwork Ns. Matrices in objective functional are given as follows: As(n): Composed of rail freight car supplying costs π si for each station i on a subnetwork s, and quadratic cost coefficients of freight cars in transit, Q. Supplying costs are determined based on the following formula: π si ¼

N   1 X χ  eji + ð1  χ Þlji , t ¼ 1,…, T N  1 j¼1

(7.5)

eji and lji are the unit costs of empty and loaded trips whereas χ represents the empty freight car-running coefficient (ratio of the number of empty freight cars to the total number of freight cars). Traveling costs are given as a linear function of the number of cars on the considered route: xjim ðnÞQ ¼ xjim ðnÞðq0 + qxjim ðnÞ

P X

sajis ðn + s  mÞ

(7.6)

sbjis ðn + s  mÞ

(7.7)

s¼m

yjim ðnÞQ ¼ yjim ðnÞðq0 + qyjitm ðnÞ

P X s¼m

Bs: matrix with costs of unmet demand π ui . Ls: matrix with freight car holding costs π hi . Γ s(P): matrix that contains rail freight car supplying costs π si and the costs of cars in transit which are considered as a linear function of the number of cars on a considered section s. The dynamic model of the subsystem s is represented by the constraint (7.2). The state in the next period Xs(n + 1) is a function of the state in the previous period, Xs(n), the control actions Us(n) and the vector of demand rates, d(n) which are considered as disturbances in the subsystem. Matrices in the dynamic model are defined as follows: Fs(n): state transition matrix; Gs: system control matrix; Φs: system disturbance matrix.

Distributed and decentralized approaches for rail freight car management

235

The dynamic model (5.2) incorporates the following state and unmet demand relations: Sis ðn + 1Þ ¼ Sis ðnÞ +

N X  X  Fjis ðmÞ  θjis ðm, n + 1Þ + Ejis ðmÞ  αjis ðm, n + 1Þ j¼1 m


N  X

Fijs ðn + 1Þ + Eijs ðn + 1Þ



ð7:8Þ

j¼1

Vijs ðn + 1Þ ¼ Vijs ðnÞ + Dijs ðn + 1Þ  Fijs ðn + 1Þ

(7.9)

Relations (7.8) represent the conservation of flow constraint for freight cars at each station and in each time period, which also include the effects of traveling times for freight car movements through θ and α terms. The system state represents the number of freight cars available at each station during every period of the planning horizon. The state in the next period Sis(n + 1) is a function of the number of available cars in the previous period Sis(n) increased for the difference between the sum of the number of loaded Fjis(m) and empty car flows Ejis(m) arrived during the period n + 1 but dispatched in previous time periods (θjis(m, n + 1), αjis(m, n + 1)) and the number of cars dispatched during the period n + 1, Fijs(n + 1), and Eijs(n + 1), respectively. In case of impossibility to satisfy the requests for freight cars in period, n + 1 some of these requests will be backordered. According to the relation (7.9) unmet demand in period n + 1, Vijs(n + 1) is equal to unmet demand in the previous period Vijs(n) increased for the unsatisfied transport requests in period n + 1, Dijs(n + 1)  Fijs(n + 1). The MPC model of the subsystem s also considers station capacity constraints. Ds is a matrix that includes the length of freight cars. Ks is a column vector with station capacities or in other words, the maximum number of freight cars on stock in every station of the considered subnetwork s. Non-negativity (7.4) is assumed for control actions and system state for each period n ¼ 0, 1, …, P  1.

7.1.2 Cooperative MPC for freight car flow planning The proposed approach considers a cooperative railway network in which every railway undertaking or operator employs a MPC to determine the appropriate actions regarding the efficient freight car flow utilization.

236

Optimization models for rail car fleet management

Fig. 7.2 Cooperative railway network for three operators.

The partition of the railway network is considered as fixed during the cooperative freight car management. The cooperative rail network may include neighboring national railway networks or regions within one national railway network. Fig. 7.2 represents an example of a cooperative railway network composed of three subnets managed by separate railway operators. Subnetworks are connected by railway lines or interconnecting links. For each subnetwork, s, the set of neighboring subnetworks is denoted as Nsnei. For each subnetwork, optimal actions have been found based on the model (7.1)–(7.9) for each subsystem s, taking into account the behavior of the surrounding railway systems and additional constraints. Therefore, cooperative freight car planning can be formulated as the cooperative MPC as follows: min

Nsub X

Js ðXs ðn + 1Þ, Us ðnÞ, ds ðnÞÞ

(7.10)

s¼1

for s ¼ 1, …, Nsub subject to Xs ðn + i + 1Þ ¼ Fs ðnÞXs ðn + iÞ + Gs Us ðn + iÞ + Φs ds ðn + iÞ, i ¼ 0,1,…, P  1 Ds X s ðnÞ  K s

(7.11) (7.12)

Distributed and decentralized approaches for rail freight car management

237

X s ðnÞ, U s ðnÞ  O

(7.13)

vin, s ðnÞ ¼ K in ðsÞU s ðnÞ

(7.14)

vout, s ðnÞ ¼ K out ðsÞU s ðnÞ

(7.15)

vin, k, s ðnÞ ¼ vout, s, k ðnÞ, 8k 2 Nsnei

(7.16)

Total costs of owning and distributing of empty and loaded flows over the cooperative freight car network are integrated by the objective functional (7.10). Dynamics of the subnetwork s is given by Eq. (7.11). Eq. (7.12) represents a station capacity constraint within each subnetwork s. Network and state control vectors over the prediction horizon are denoted as X s ðnÞ ¼ T T ½X s ðnÞ, …, X s ðn + P Þ and U s ðnÞ ¼ ½U s ðnÞ, …, U s ðn + P  1Þ , respectively. Constraints (5.13) enforce non-negativity of state and control actions. Input and output of loaded and empty freight car flows for operator s on interconnecting railway links are represented by constraints (7.14) and (7.15). The input of loaded freight car flows on interconnecting lines of subnetwork s with respect to the neighboring operator k (k 2 N nei s ) in the prediction horizon P are denoted as νin, s ðnÞ ¼ ½νin, s ðnÞ, …, νin, s ðn + P  1ÞT whereas the empty freight car flows are given by νout, s ðnÞ ¼ ½νout, s ðnÞ, …, νout, s ðn + P  1ÞT . K in ðsÞ and K out ðsÞ are zeroone matrices that serve to select input and output of empty and loaded flows of freight cars on interconnecting links. Interconnecting constraints between freight car flows that enter or leave subnetwork s from or to its neighboring subnetwork k are presented by the set of coupling constraints (7.16). Coupling constraints (7.16) include the flows of two neighboring subnetworks and thus prevent independent problem solving by the operator. This means that the solution of one subnetwork’s MPC problem depends on the solution of the surrounding MPC controllers. More precisely, MPC problems must be solved in a cooperative way by enabling communication between controllers. As a solution approach for this problem, a parallel augmented Lagrangian relaxation algorithm which is based on a combination of the augmented Lagrangian formulation and auxiliary problem principle, has been suggested (Li et al., 2016). The first step involves the relaxation of interconnecting constraints and adding them to the objective functionals of different controllers min Js ðxs ðn + 1Þ, us ðnÞ, ds ðnÞÞ + us ðnÞ, s¼1, …, Nsub

Ns X

  J inter , k vin, k, s ðnÞ, vout, s, k ðnÞ, λin, k, s ðnÞ

k¼1

(7.17)

238

Optimization models for rail car fleet management

The cost associated with the coordination between controllers s and k, Jinter, k is defined as follows: 2 θ   Jinter, k ¼ λTin, k, s ðnÞðvin, k, s ðnÞ  vout, s, k Þ + vin, k, s ðnÞ  vout, s, k ðnÞ (7.18) 2 2 λin, k, s(n) and θ are Lagrangian multipliers related to interconnecting constraints and the penalty parameter, respectively. However, this formulation contains a non-separable quadratic term that involves the difference between the input vin, k, s ðnÞ and output vout, s, k ðnÞ empty and loaded rail freight car flows between subnetworks. In order to solve the issue of decoupling the quadratic terms in the augmented Lagrangian, the parallel DMPC scheme based on the auxiliary problem principle is applied (Cohen, 1980). The auxiliary problem principle will enable simultaneous solving of a sequence of auxiliary problems by all operators (Negenborn et al., 2006). Problem (5.10) will be solved by the operators using the following additional objective functional term for interconnecting constraints. " #  v‘ ðnÞ  vout, k, s ðnÞ 2 θ   in , s , k Jinter, k ¼ λTin, k, s ðnÞðvin, k, s ðnÞ  vout, s, k Þ +  ‘  2  vout, s, k ðnÞ  vin, k, s ðnÞ  2

 2 υ  θ  vin, k, s  v‘in, s, k ðnÞ  + (7.19) ðnÞ  2  vout, k, s  v‘ out, k, s

2

where ν represents positive scalar. Now, the distributed MPC can be summarized as follows: 1. For the time interval n, every operator s evaluates the current state of the rail freight car subsystem xs(n) and estimates the expected disturbance ds(n) for n ¼ 0, …, P  1. 2. In the initialization phase, the iteration counter ‘ is set to 1 and Lagrange multipliers, control actions, and interconnecting constraints are initialized. 3. The iteration process includes the simultaneous optimization of the local MPC problem (7.19) subject to subnetwork dynamics and planning constraints (7.11)–(7.16). Adjacent controllers send v‘in+, k1, s ðnÞ and v‘out+,1k, s ðnÞ receive v‘in+, s,1k ðnÞ and v‘out+,1s, k ðnÞ. 4. Operators update the Lagrange multipliers using the following expressions   λ‘in+, k1, s ¼ λ‘in, k, s + δ v‘in+, k1, s ðnÞ  v‘out+,1s, k ðnÞ (7.20)

Distributed and decentralized approaches for rail freight car management

5. 6. 7.

239

‘ Distributed MPC iterations terminate when kλ‘+1 in, k, s  λin, k, sk∞  ε, where ε represents the stated tolerance and kk∞ denotes the infinity norm. The output is u‘s ,v‘in, k, s ðnÞ,v‘out, k, s ðnÞ, λ‘in+, k1, s ,s ¼ 1,…, Nsub , k 2 Nsnei . Optimal control actions have been implemented by the controllers until the beginning of the next planning period. The next time period starts.

7.2 Decentralized model predictive rail freight car management Decentralized model predictive control (DecMPC) for freight car flow planning represents the simplest generalization of the MPC approach for a large-scale rail freight car fleet system. In other words, decentralized MPC relies on a set of regional or national railway network models, which neglect inter-network freight car flows, leading to multiple detached optimization problems. The freight car flows on each individual subnetwork are controlled by an independent MPC controller. Incoming and outgoing flows among the subnetworks are considered as external unknown parameters or disturbances. Comparing to the distributed approach there are no negotiations between individual agents or controllers during the decision process. The time needed for control action determining is not affected by communication issues, and information exchange is only allowed before and after the decision-making process. Usually, this non-centralized MPC strategy leads to low control performances due to the fact that coupling between adjacent networks are not considered by the control law. Therefore, the decentralized control approach is appropriate only in cases where there are no couplings between subsystems or when the influence of exogenous inputs is weak (Christofides et al., 2013). In the decentralized modeling, framework behavior of the surrounding railway systems on the local subnetwork is assumed to be negligible. Therefore, for each subnetwork s local optimal control actions Us(n) are based on the decentralized objective functional (7.1) subject to dynamic model (7.2) and planning constraints (7.3)– (7.4) for subnetwork s. The proposed MPC optimization problem is defined as min J ¼ X,U

Nsub X s¼1

Js

(7.21)

240

Optimization models for rail car fleet management

for s ¼ 1, …, Nsub subject to Xs ðn + i + 1Þ ¼ Fs ðnÞXs ðn + iÞ + Gs Us ðn + iÞ + Φs ds ðn + iÞ, i ¼ 0,1, …,P  1

where

(7.22)

Ds X s ðnÞ  K s

(7.23)

X s ðnÞ, U s ðnÞ  O

(7.24)

ð7:25Þ With J being the overall objective function of the DecMPC network. In every time period, i each MPC controller estimates the state Xs(n) and solves the problem (7.21)–(7.24) and obtains the optimizer U∗s (n), which contains control actions (empty and loaded flows freight car flows) within the subnetwork s. After all Nsub controllers have calculated their local optimal control sequence U∗s (n), the collection of all inputs U(n) ¼ [U1(n), …, Us(n)], s ¼ 1, …, Nsub is applied to the global railway freight car fleet system (7.26): X ðn + 1Þ ¼ F ðnÞX ðnÞ + GU ðnÞ + ΦdðnÞ and the whole procedure is repeated at the next time instant.

(7.26)

7.3 Numerical example The application of the developed cooperative freight car planning model is described in this section. The railway network illustrated in Fig. 7.2 includes three subnetworks managed cooperatively by different railway coordinators. The entire network includes ten railway nodes belonging to subnetworks in which one or more railway undertakings or operators manage the railway freight transport service. One type of freight cars is considered for rail freight operations. Planning horizon includes four periods (P ¼ 4) where each day represents one decision period. Unit costs of empty (eji) and loaded trips (lji), as well as the unit car shortage costs (pij) for all origin-destination pairs, are given in Table 7.1. The coefficient of empty trips χ is 0.35. Table 7.2 contains proportions of arrivals of empty [αji(m, n + 1)] and loaded car [θji(m, n + 1)] flows. Daily unit car ownership and station holding costs are assumed to be 20 monetary units.

Distributed and decentralized approaches for rail freight car management

241

Table 7.1 Cost parameters for the rail freight car fleet sizing and allocation problem.

Origin

Destination

Unit cost of empty trip (m.u./car)

1 1 2 2 3 3 4 4 4 5 5 5 6 6 7 7 7 8 8 8 9 9 10 10

2 3 1 4 1 4 2 3 5 4 6 7 5 7 5 6 8 7 9 10 8 10 8 9

200 200 200 300 200 250 300 250 300 300 200 150 200 250 150 250 300 200 180 150 180 200 150 200

Unit cost of loaded trip (m.u./car)

Unit car shortage cost (m.u./car)

150 150 150 200 150 180 200 180 200 200 150 100 150 180 100 180 200 200 180 150 180 200 150 200

300 300 300 400 300 300 400 300 400 400 300 200 300 300 200 300 400 400 300 300 300 400 300 400

The demand Dij(n) is considered as deterministic (Table 7.3). Table 7.3 also contains unmet demand from the preceding period for all origindestination pairs. Loaded (xjim(n)) and empty (yjim(n)) freight car flows that were initially in transit at the beginning of the horizon are given in Table 7.4. Table 7.5 contains the initial state of cars (Si(0)) in stations. For the sake of comparison, the obtained results from CDMPC were compared with the centralized (MPC) and decentralized (DecMPC) approach. Centralized planning performed by one central authority involves a QP optimization problem given by the objective functional (7.10) and linear dynamic constraints (7.11)–(7.13). The decentralized approach involves

242

Table 7.2 Proportions of empty and loaded car flow arrivals. Route 1-2

1-3

2-1

2-4

3-1

3-4

4-2

4-3

4-5

5-4

5-6

5-7

0.55 0.30 0.15

0.70 0.25 0.05

0.60 0.30 0.10

0.80 0.15 0.05

0.50 0.30 0.20

0.75 0.15 0.05

0.60 0.30 0.10

0.65 0.25 0.20

0.75 0.20 0.05

0.70 0.25 0.05

0.70 0.25 0.05

0.60 0.30 0.10

0.70 0.25 0.05

0.80 0.15 0.05

0.70 0.25 0.05

0.75 0.15 0.10

0.65 0.30 0.10

0.70 0.20 0.10

0.70 0.25 0.05

0.80 0.15 0.05

0.70 0.25 0.05

0.60 0.30 0.10

0.60 0.30 0.10

0.50 0.35 0.15

Loaded

1 2 3 Empty

1 2 3

Route N

6-5

6-7

7-5

7-6

7-8

8-7

8-9

8-10

9-8

9-10

10-8

10-9

0.70 0.10 0.20

0.50 0.35 0.15

0.60 0.25 0.15

0.60 0.30 0.10

0.50 0.30 0.20

0.65 0.30 0.05

0.60 0.30 0.10

0.50 0.30 0.20

0.60 0.30 0.10

0.65 0.30 0.05

0.45 0.40 0.15

0.40 0.50 0.10

0.60 0.30 0.10

0.60 0.25 0.15

0.50 0.35 0.15

0.60 0.30 0.10

0.60 0.30 0.10

0.70 0.20 0.10

0.60 0.30 0.10

0.60 0.30 0.10

0.55 0.35 0.10

0.50 0.40 0.10

0.70 0.25 0.05

0.65 0.30 0.05

Loaded

1 2 3 Empty

1 2 3

Optimization models for rail car fleet management

N

Daily demand Route

1

2

3

4

Unmet demand

1-2 1-3 2-1 2-4 3-1 3-4 4-2 4-3 4-5 5-4 5-6 5-7

20 22 17 15 15 16 15 16 17 18 19 10

15 17 13 9 5 14 18 11 17 18 19 10

8 6 13 9 15 9 10 7 14 11 9 13

9 11 15 4 10 11 10 11 11 16 13 15

2 4 1 3 5 6 4 4 2 3 4 3

Daily demand Route

1

2

3

4

Unmet demand

6-5 6-7 7-5 7-6 7-8 8-7 8-9 8-10 9-8 9-10 10-8 10-9

11 12 10 11 5 16 11 15 19 10 11 14

13 15 10 11 15 10 12 18 11 20 10 12

10 14 14 10 12 11 13 10 13 15 11 10

17 5 19 8 10 15 10 9 12 12 17 8

9 10 7 8 9 3 4 2 4 4 5 4

Distributed and decentralized approaches for rail freight car management

Table 7.3 Daily demand and unmet demand from the preceding period.

243

244

Loaded cars

Empty cars

Route

One period

Two periods

One period

Two periods

Route One period

1-2 1-3 2-1 2-4 3-1 3-4 4-2 4-3 4-5 5-4 5-6 5-7

7 4 5 4 4 5 6 7 6 6 10 4

4 6 2 8 3 7 5 5 7 3 7 4

2 2 1 1 2 2 3 7 2 3 3 2

2 3 1 4 3 5 1 5 1 2 3 4

6-5 6-7 7-5 7-6 7-8 8-7 8-9 8-10 9-8 9-10 10-8 10-9

Loaded cars

Empty cars

Two periods

Two periods

One period

Two periods

4 5 5 8 5 3 8 3 7 4 9 9

3 5 4 4 8 6 7 3 5 4 4 10

3 1 4 2 3 3 3 2 2 1 2 1

2 2 2 4 1 3 1 3 3 2 4 2

Optimization models for rail car fleet management

Table 7.4 Loaded and empty cars dispatched before the beginning of the planning horizon.

Distributed and decentralized approaches for rail freight car management

245

Table 7.5 Initial number of cars in stations. Station 1

2

3

4

5

6

7

8

9

10

45

40

40

30

40

30

45

35

30

45

independent MPC optimization of freight car fleet on three subnetworks. The CDMPC, as well as the decentralized MPC, and the centralized MPC performance benchmarks, were solved by the interior-point-convex algorithm (MathWorks Inc., 2012). All experiments were conducted on a laptop computer with an Intel Core i7-5600 CPU with 2.60 GHz and 8 GB RAM. In order to illustrate the cooperative process of the distributed freight car MPC the evolution of the differences between the interconnecting variables is illustrated in Fig. 7.3. The presented variables represent empty and loaded flows on interconnecting links between subnets 1 and 2 in the first planning period. Fig. 7.3, actually shows the evolution of the differences between the output interconnecting variable of operator 1, vlout, 2, 1(m + n) and the corresponding input interconnecting variable of operator 2, vlin, 1, 2(m + n)

The evolution of the differences related to loaded flows u |(out,2,1)(m + n) – u |(in,1,2)(m + n)

4 2 0 –2 After 20 iterations After 50 iterations After 100 iterations After convergence

–4 –6

1

(A)

3

2 n

The evolution of the differences related to empty flows u |(out,2,1)(m + n) – u |(in,1,2)(m + n)

4 2 0 –2 –4 After 20 iterations After 50 iterations After 100 iterations After convergence

–6 –8

(B)

1

2 n

3

Fig. 7.3 Evolution of the differences in interconnecting flows between railway operator 1 and operator 2.

246

Optimization models for rail car fleet management

Fig. 7.4 Evolution of Lagrange multipliers (freight car flows) for operator 1 for the time interval m ¼ 1 over the whole prediction period n ¼ 1, …, P  1

for the period m ¼ 1 over the prediction horizon n ¼ 1, …, P  1. Loaded freight car flows are given in Fig. 7.3A whereas Fig. 7.3B contains the evolution of the differences for empty freight car flows. Evolution of Lagrangian multipliers associated with interconnecting variables vlin, 1, 2(m + n) and vlout, 2, 1(m + n) for the time interval m ¼ 1 is given in Fig. 7.4. The maximum computation time for operators is limited to 60 min. Cooperation parameters are θ ¼ 0.9 , ν ¼ 2θ, and δ ¼ 5θ. The iteration stopping threshold is ε ¼ 103. The obtained solution, control actions as well as the unmet demand are equal to the outputs from the centralized approach (Table 7.6). The total freight car planning cost obtained on the centralized level is 7.03 106. The proposed CDMPC approach obtains the same total delivery cost. The required fleet of freight cars is 412. In the case of the decentralized approach, the total freight car planning cost is 7.64 106 and the total fleet size required is 454 freight cars. However, the difference between the centralized/decentralized and distributed cooperative approach lies in the communication cost and computation time. Table 7.7 contains the total number of iterations and total communication cost per period in the planning horizon. A set of alternative case studies with a different number of stations was generated in order to additionally validate performances of the proposed cooperative distributed approach (CDMPC) against centralized (MPC) and decentralized decision making (DecMPC). Comparison results are given in Table 7.8.

Table 7.6 Control actions and unmet demand. Period 1

Period 2

Period 3

Unmet demand Uij(1)

Loaded flows Fij(2)

Empty flows Eij(2)

Unmet demand Uij(2)

Loaded flows Fij(3)

Empty flows Eij(2)

Unmet demand Uij(3)

1-2 1-3 2-1 2-4 3-1 3-4 4-2 4-3 4-5 5-4 5-6 5-7 6-5 6-7 7-5 7-6 7-8 8-7 8-9 8-10 9-8 9-10 10-8 10-9

17.54 19.09 16.74 14.18 16.59 18.83 14.37 13.83 16.97 19.24 20.83 15.52 23.27 24.72 23.90 26.60 16.70 20.17 16.00 18.03 19.03 15.90 15.68 15.52

8.45 8.37 10.83 8.72 10.10 10.12 3.65 4.37 5.33 9.20 5.33 5.61 10.12 4.79 6.90 4.60 8.00 8.38 5.41 5.54 4.66 13.17 5.11 4.98

7.85 10.31 4.19 6.01 6.81 6.56 6.12 8.18 3.53 3.91 5.51 2.48 0.00 0.54 0.00 0.00 0.75 0.74 1.93 1.89 6.50 0.00 3.92 5.18

17.56 19.20 15.68 12.78 13.63 16.19 16.02 16.27 12.09 14.30 20.13 13.76 13.03 15.31 10.67 11.44 16.08 11.30 11.65 14.25 15.64 10.98 11.99 14.21

4.58 4.06 6.98 4.30 7.37 5.86 5.14 5.84 0.78 1.37 4.16 4.19 4.80 0.04 0.67 0.45 1.84 0.90 0.98 0.44 0.81 5.32 1.23 3.41

7.48 10.29 3.12 3.45 0.00 6.19 8.83 3.88 9.17 8.97 5.56 0.52 0.00 0.27 0.00 0.00 0.00 0.33 2.64 6.00 2.67 9.61 3.15 3.88

16.60 17.92 14.65 11.06 11.43 15.07 18.38 12.06 6.70 9.69 15.84 13.21 10.02 14.28 14.78 10.52 12.39 7.22 13.38 16.58 12.73 13.57 11.41 13.39

3.39 3.24 6.91 3.89 7.13 5.22 6.23 7.59 0.56 0.97 5.01 3.07 7.59 1.38 0.78 0.52 0.89 0.36 2.62 0.83 0.82 1.59 1.41 3.65

0.49 0.00 2.30 2.01 4.41 0.97 1.33 0.00 17.36 11.25 0.00 2.22 0.00 0.00 0.00 0.00 0.00 4.47 2.83 0.00 3.76 11.66 3.50 1.06

247

Empty flows Eij(1)

Distributed and decentralized approaches for rail freight car management

Route

Loaded flows Fij(1)

248

Total communication cost per period DMPC

Cost of loaded and empty freight car allocation

Operator 1 Operator 2 Operator 3 Total

2.63 2.39 2.01 7.03

   

106 106 106 106

Period

Niteration

Iiteration

Jcost (floats)

Computation time (min) per period

1

13,322

256

3.41  106

36.50

2 3

4524 9974

256 256

1.15  106 2.55  106

9.03 20.59

Optimization models for rail car fleet management

Table 7.7 Distribution costs, communication costs and computational time of the DMPC approach.

Problem

Number of stations

Number of subnetworks

Number of interconnecting links

1 2 3

15 20 30

3 4 5

3 6 7

Total cost of freight car allocation (106)

Total communication cost (106 floats)

CDMPC

DecMPC

MPC

10.68 23.32 40.01

10.97 15.44 27.97

92.32 115.70 149.26

0.07 0.20 0.30

0.08 0.25 0.38

Total computation time (min)

Distributed and decentralized approaches for rail freight car management

Table 7.8 Alternative case studies.

249