Abstract

We consider the planning problem of a flexible optical network. Given the traffic matrix and the transponders’ feasible configurations that account for the physical layer, we formulate the planning problem considering both the use or not of regenerators. Demands are served for their requested rates by choosing the route, selecting the transmission configuration, breaking the transmissions in more than one connection and placing regenerators, if needed, and allocating the spectrum to them. The objective is to serve the traffic and find a solution that is Pareto optimal with respect to the maximum spectrum used and the cost (number and type) of transponders used. The problem definition and the proposed algorithms are general and applicable to flex-grid as well as fixed-grid networks. We start by presenting algorithms based on integer linear programming formulations for transparent and translucent networks (without or with regenerators) and we continue by presenting heuristic algorithms. Using input driven by transmission studies on optical orthogonal frequency-division multiplexing (OFDM)-based networks we evaluate the performance gains that can be obtained by an OFDM over a mixed line rate fixed-grid WDM optical network.

© 2013 Optical Society of America

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2013

K. Christodoulopoulos, I. Tomkos, and E. Varvarigos, “Time-varying spectrum allocation policies and blocking analysis in flexible optical networks,” IEEE J. Sel. Areas Commun., vol.  30, no. 1, pp. 1–13, 2013.

2012

O. Gerstel, M. Jinno, A. Lord, and S. J. Y. Ben, “Elastic optical networking: A new dawn for the optical layer?” IEEE Commun. Mag., vol.  50, no. 2, pp. s12–s20, 2012.
[CrossRef]

B. T. Teipen, M. H. Eiselt, K. Grobe, and J.-P. Elbers, “Adaptive data rates for flexible transceivers in optical networks,” J. Netw., vol.  7, no. 5, pp. 776–782, May 2012.
[CrossRef]

A. N. Patel, P. N. Ji, J. P. Jue, and T. Wang, “Routing, wavelength assignment, and spectrum allocation algorithms in transparent flexible optical WDM networks,” Opt. Switching Networking, vol.  9, no. 3, pp. 191–2042012.

L. Velasco, M. Klinkowski, M. Ruiz, and J. Comellas, “Modeling the routing and spectrum allocation problem for flexgrid optical networks,” Photonic Network Commun., vol.  24, pp. 177–186, 2012.
[CrossRef]

A. Klekamp, U. Gebhard, and F. Ilchmann, “Energy and cost efficiency of adaptive and mixed-line-rate IP over DWDM networks,” J. Lightwave Technol., vol.  30, no. 2, pp. 215–221, 2012.
[CrossRef]

E. Palkopoulou, M. Angelou, D. Klonidis, K. Christodoulopoulos, A. Klekamp, F. Buchali, E. Varvarigos, and I. Tomkos, “Quantifying spectrum, cost, and energy efficiency in fixed-grid and flex-grid networks [Invited],” J. Opt. Commun. Netw., vol.  4, no. 11, pp. B42–B51, Oct. 2012.
[CrossRef]

R. Borkowski, F. Karinou, M. Angelou, V. Arlunno, D. Zibar, D. Klonidis, N. G. Gonzalez, A. Caballero, I. Tomkos, and I. T. Monroy, “Experimental study on OSNR requirements for spectrum-flexible optical networks,” J. Opt. Commun. Netw., vol.  4, no. 11, pp. B85–B93, 2012.
[CrossRef]

2011

2010

A. Nag, M. Tornatore, and B. Mukherjee, “Optical network design with mixed line rates and multiple modulation formats,” J. Lightwave Technol., vol.  28, no. 4, pp. 466–475, 2010.
[CrossRef]

B. Kozicki, H. Takara, Y. Tsukishima, T. Yoshimatsu, K. Yonenaga, and M. Jinno, “Experimental demonstration of spectrum-sliced elastic optical path network (SLICE),” Opt. Express, vol.  18, no. 21, pp. 22105–22118, Oct. 2010.
[CrossRef]

S. Gringeri, B. Basch, V. Shukla, R. Egorov, and T. J. Xia, “Flexible architectures for optical transport nodes and networks,” IEEE Commun. Mag., vol.  48, no. 7, pp. 40–50, 2010.
[CrossRef]

M. Jinno, B. Kozick, H. Takara, A. Watanabe, and Y. Sone, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, 2010.
[CrossRef]

K. Christodoulopoulos, K. Manousakis, and E. Varvarigos, “Offline routing and wavelength assignment in transparent WDM networks,” IEEE/ACM Trans. Netw., vol.  18, pp. 1557–1570, 2010.
[CrossRef]

2009

M. Gagnaire and S. Zahr, “Impairment-aware routing and wavelength assignment in translucent networks: State of the art,” IEEE Commun. Mag., vol.  47, no. 5, pp. 55–61, 2009.
[CrossRef]

2007

G. Shen and R. Tucker, “Translucent optical networks: The way forward,” IEEE Commun. Mag., vol.  45, no. 2, pp. 48–54, 2007.

Angelou, M.

Arlunno, V.

Basch, B.

S. Gringeri, B. Basch, V. Shukla, R. Egorov, and T. J. Xia, “Flexible architectures for optical transport nodes and networks,” IEEE Commun. Mag., vol.  48, no. 7, pp. 40–50, 2010.
[CrossRef]

Ben, S. J. Y.

O. Gerstel, M. Jinno, A. Lord, and S. J. Y. Ben, “Elastic optical networking: A new dawn for the optical layer?” IEEE Commun. Mag., vol.  50, no. 2, pp. s12–s20, 2012.
[CrossRef]

Borkowski, R.

Buchali, F.

E. Palkopoulou, M. Angelou, D. Klonidis, K. Christodoulopoulos, A. Klekamp, F. Buchali, E. Varvarigos, and I. Tomkos, “Quantifying spectrum, cost, and energy efficiency in fixed-grid and flex-grid networks [Invited],” J. Opt. Commun. Netw., vol.  4, no. 11, pp. B42–B51, Oct. 2012.
[CrossRef]

A. Klekamp, R. Dischler, and F. Buchali, “Limits of spectral efficiency and transmission reach of optical-OFDM superchannels for adaptive networks,” IEEE Photon. Technol. Lett., vol.  23, no. 20, pp. 1526–1528, Oct. 2011.
[CrossRef]

Caballero, A.

Cai, X.

D. J. Geisler, R. Proietti, Y. Yawei, R. P. Scott, X. Cai, N. K. Fontaine, L. Paraschis, O. Gerstel, and S. J. B. Yoo, “The first testbed demonstration of a flexible bandwidth network with a real-time adaptive control plane,” in European Conf. on Optical Communication (ECOC), 2011.

Cao, X.

Y. Wang, X. Cao, and Y. Pan, “A study of the routing and spectrum allocation in spectrum-sliced elastic optical path networks,” in IEEE INFOCOM, 2011, pp. 1503–1511.

Christodoulopoulos, K.

K. Christodoulopoulos, I. Tomkos, and E. Varvarigos, “Time-varying spectrum allocation policies and blocking analysis in flexible optical networks,” IEEE J. Sel. Areas Commun., vol.  30, no. 1, pp. 1–13, 2013.

E. Palkopoulou, M. Angelou, D. Klonidis, K. Christodoulopoulos, A. Klekamp, F. Buchali, E. Varvarigos, and I. Tomkos, “Quantifying spectrum, cost, and energy efficiency in fixed-grid and flex-grid networks [Invited],” J. Opt. Commun. Netw., vol.  4, no. 11, pp. B42–B51, Oct. 2012.
[CrossRef]

K. Christodoulopoulos, I. Tomkos, and E. A. Varvarigos, “Elastic nandwidth allocation in flexible OFDM-based optical networks,” J. Lightwave Technol., vol.  29, no. 9, pp. 1354–1366, May 2011.
[CrossRef]

K. Christodoulopoulos, K. Manousakis, and E. Varvarigos, “Offline routing and wavelength assignment in transparent WDM networks,” IEEE/ACM Trans. Netw., vol.  18, pp. 1557–1570, 2010.
[CrossRef]

K. Christodoulopoulos, P. Soumplis, and E. Varvarigos, “Trading off transponders for spectrum in flexgrid networks,” in Optical Fiber Communication Conference (OFC), 2013, paper OTu2A.3.

Comellas, J.

L. Velasco, M. Klinkowski, M. Ruiz, and J. Comellas, “Modeling the routing and spectrum allocation problem for flexgrid optical networks,” Photonic Network Commun., vol.  24, pp. 177–186, 2012.
[CrossRef]

Dischler, R.

A. Klekamp, R. Dischler, and F. Buchali, “Limits of spectral efficiency and transmission reach of optical-OFDM superchannels for adaptive networks,” IEEE Photon. Technol. Lett., vol.  23, no. 20, pp. 1526–1528, Oct. 2011.
[CrossRef]

Egorov, R.

S. Gringeri, B. Basch, V. Shukla, R. Egorov, and T. J. Xia, “Flexible architectures for optical transport nodes and networks,” IEEE Commun. Mag., vol.  48, no. 7, pp. 40–50, 2010.
[CrossRef]

Eiselt, M. H.

B. T. Teipen, M. H. Eiselt, K. Grobe, and J.-P. Elbers, “Adaptive data rates for flexible transceivers in optical networks,” J. Netw., vol.  7, no. 5, pp. 776–782, May 2012.
[CrossRef]

Elbers, J.-P.

B. T. Teipen, M. H. Eiselt, K. Grobe, and J.-P. Elbers, “Adaptive data rates for flexible transceivers in optical networks,” J. Netw., vol.  7, no. 5, pp. 776–782, May 2012.
[CrossRef]

Fontaine, N. K.

D. J. Geisler, R. Proietti, Y. Yawei, R. P. Scott, X. Cai, N. K. Fontaine, L. Paraschis, O. Gerstel, and S. J. B. Yoo, “The first testbed demonstration of a flexible bandwidth network with a real-time adaptive control plane,” in European Conf. on Optical Communication (ECOC), 2011.

Gagnaire, M.

M. Gagnaire and S. Zahr, “Impairment-aware routing and wavelength assignment in translucent networks: State of the art,” IEEE Commun. Mag., vol.  47, no. 5, pp. 55–61, 2009.
[CrossRef]

Gebhard, U.

Geisler, D. J.

D. J. Geisler, R. Proietti, Y. Yawei, R. P. Scott, X. Cai, N. K. Fontaine, L. Paraschis, O. Gerstel, and S. J. B. Yoo, “The first testbed demonstration of a flexible bandwidth network with a real-time adaptive control plane,” in European Conf. on Optical Communication (ECOC), 2011.

Gerstel, O.

O. Gerstel, M. Jinno, A. Lord, and S. J. Y. Ben, “Elastic optical networking: A new dawn for the optical layer?” IEEE Commun. Mag., vol.  50, no. 2, pp. s12–s20, 2012.
[CrossRef]

D. J. Geisler, R. Proietti, Y. Yawei, R. P. Scott, X. Cai, N. K. Fontaine, L. Paraschis, O. Gerstel, and S. J. B. Yoo, “The first testbed demonstration of a flexible bandwidth network with a real-time adaptive control plane,” in European Conf. on Optical Communication (ECOC), 2011.

Gonzalez, N. G.

Gringeri, S.

S. Gringeri, B. Basch, V. Shukla, R. Egorov, and T. J. Xia, “Flexible architectures for optical transport nodes and networks,” IEEE Commun. Mag., vol.  48, no. 7, pp. 40–50, 2010.
[CrossRef]

Grobe, K.

B. T. Teipen, M. H. Eiselt, K. Grobe, and J.-P. Elbers, “Adaptive data rates for flexible transceivers in optical networks,” J. Netw., vol.  7, no. 5, pp. 776–782, May 2012.
[CrossRef]

Hasegawa, H.

T. Takagi, H. Hasegawa, K. Sato, Y. Sone, B. Kozicki, A. Hirano, and M. Jinno, “Dynamic routing and frequency slot assignment for elastic optical path networks that adopt distance adaptive modulation,” in Optical Fiber Communication Conf., 2011.

Hirano, A.

T. Takagi, H. Hasegawa, K. Sato, Y. Sone, B. Kozicki, A. Hirano, and M. Jinno, “Dynamic routing and frequency slot assignment for elastic optical path networks that adopt distance adaptive modulation,” in Optical Fiber Communication Conf., 2011.

Ilchmann, F.

Ji, P. N.

A. N. Patel, P. N. Ji, J. P. Jue, and T. Wang, “Routing, wavelength assignment, and spectrum allocation algorithms in transparent flexible optical WDM networks,” Opt. Switching Networking, vol.  9, no. 3, pp. 191–2042012.

A. N. Patel, P. N. Ji, J. P. Jue, and W. Ting, “Defragmentation of transparent flexible optical WDM (FWDM) networks,” in Optical Fiber Communication Conf. (OFC), 2011, paper OTuI.

Jinno, M.

O. Gerstel, M. Jinno, A. Lord, and S. J. Y. Ben, “Elastic optical networking: A new dawn for the optical layer?” IEEE Commun. Mag., vol.  50, no. 2, pp. s12–s20, 2012.
[CrossRef]

M. Jinno, B. Kozick, H. Takara, A. Watanabe, and Y. Sone, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, 2010.
[CrossRef]

B. Kozicki, H. Takara, Y. Tsukishima, T. Yoshimatsu, K. Yonenaga, and M. Jinno, “Experimental demonstration of spectrum-sliced elastic optical path network (SLICE),” Opt. Express, vol.  18, no. 21, pp. 22105–22118, Oct. 2010.
[CrossRef]

T. Takagi, H. Hasegawa, K. Sato, Y. Sone, B. Kozicki, A. Hirano, and M. Jinno, “Dynamic routing and frequency slot assignment for elastic optical path networks that adopt distance adaptive modulation,” in Optical Fiber Communication Conf., 2011.

Jue, J. P.

A. N. Patel, P. N. Ji, J. P. Jue, and T. Wang, “Routing, wavelength assignment, and spectrum allocation algorithms in transparent flexible optical WDM networks,” Opt. Switching Networking, vol.  9, no. 3, pp. 191–2042012.

A. N. Patel, P. N. Ji, J. P. Jue, and W. Ting, “Defragmentation of transparent flexible optical WDM (FWDM) networks,” in Optical Fiber Communication Conf. (OFC), 2011, paper OTuI.

Karinou, F.

Klekamp, A.

Klinkowski, M.

L. Velasco, M. Klinkowski, M. Ruiz, and J. Comellas, “Modeling the routing and spectrum allocation problem for flexgrid optical networks,” Photonic Network Commun., vol.  24, pp. 177–186, 2012.
[CrossRef]

Klonidis, D.

Kozick, B.

M. Jinno, B. Kozick, H. Takara, A. Watanabe, and Y. Sone, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, 2010.
[CrossRef]

Kozicki, B.

B. Kozicki, H. Takara, Y. Tsukishima, T. Yoshimatsu, K. Yonenaga, and M. Jinno, “Experimental demonstration of spectrum-sliced elastic optical path network (SLICE),” Opt. Express, vol.  18, no. 21, pp. 22105–22118, Oct. 2010.
[CrossRef]

T. Takagi, H. Hasegawa, K. Sato, Y. Sone, B. Kozicki, A. Hirano, and M. Jinno, “Dynamic routing and frequency slot assignment for elastic optical path networks that adopt distance adaptive modulation,” in Optical Fiber Communication Conf., 2011.

Lord, A.

O. Gerstel, M. Jinno, A. Lord, and S. J. Y. Ben, “Elastic optical networking: A new dawn for the optical layer?” IEEE Commun. Mag., vol.  50, no. 2, pp. s12–s20, 2012.
[CrossRef]

Manousakis, K.

K. Christodoulopoulos, K. Manousakis, and E. Varvarigos, “Offline routing and wavelength assignment in transparent WDM networks,” IEEE/ACM Trans. Netw., vol.  18, pp. 1557–1570, 2010.
[CrossRef]

Monroy, I. T.

Morea, A.

Mukherjee, B.

Nag, A.

Palkopoulou, E.

Pan, Y.

Y. Wang, X. Cao, and Y. Pan, “A study of the routing and spectrum allocation in spectrum-sliced elastic optical path networks,” in IEEE INFOCOM, 2011, pp. 1503–1511.

Papadimitriou, C.

C. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Dover, 1998.

Paraschis, L.

D. J. Geisler, R. Proietti, Y. Yawei, R. P. Scott, X. Cai, N. K. Fontaine, L. Paraschis, O. Gerstel, and S. J. B. Yoo, “The first testbed demonstration of a flexible bandwidth network with a real-time adaptive control plane,” in European Conf. on Optical Communication (ECOC), 2011.

Patel, A. N.

A. N. Patel, P. N. Ji, J. P. Jue, and T. Wang, “Routing, wavelength assignment, and spectrum allocation algorithms in transparent flexible optical WDM networks,” Opt. Switching Networking, vol.  9, no. 3, pp. 191–2042012.

A. N. Patel, P. N. Ji, J. P. Jue, and W. Ting, “Defragmentation of transparent flexible optical WDM (FWDM) networks,” in Optical Fiber Communication Conf. (OFC), 2011, paper OTuI.

Proietti, R.

D. J. Geisler, R. Proietti, Y. Yawei, R. P. Scott, X. Cai, N. K. Fontaine, L. Paraschis, O. Gerstel, and S. J. B. Yoo, “The first testbed demonstration of a flexible bandwidth network with a real-time adaptive control plane,” in European Conf. on Optical Communication (ECOC), 2011.

Rival, O.

Ruiz, M.

L. Velasco, M. Klinkowski, M. Ruiz, and J. Comellas, “Modeling the routing and spectrum allocation problem for flexgrid optical networks,” Photonic Network Commun., vol.  24, pp. 177–186, 2012.
[CrossRef]

Sato, K.

T. Takagi, H. Hasegawa, K. Sato, Y. Sone, B. Kozicki, A. Hirano, and M. Jinno, “Dynamic routing and frequency slot assignment for elastic optical path networks that adopt distance adaptive modulation,” in Optical Fiber Communication Conf., 2011.

Scott, R. P.

D. J. Geisler, R. Proietti, Y. Yawei, R. P. Scott, X. Cai, N. K. Fontaine, L. Paraschis, O. Gerstel, and S. J. B. Yoo, “The first testbed demonstration of a flexible bandwidth network with a real-time adaptive control plane,” in European Conf. on Optical Communication (ECOC), 2011.

Shen, G.

G. Shen and R. Tucker, “Translucent optical networks: The way forward,” IEEE Commun. Mag., vol.  45, no. 2, pp. 48–54, 2007.

Shukla, V.

S. Gringeri, B. Basch, V. Shukla, R. Egorov, and T. J. Xia, “Flexible architectures for optical transport nodes and networks,” IEEE Commun. Mag., vol.  48, no. 7, pp. 40–50, 2010.
[CrossRef]

Sone, Y.

M. Jinno, B. Kozick, H. Takara, A. Watanabe, and Y. Sone, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, 2010.
[CrossRef]

T. Takagi, H. Hasegawa, K. Sato, Y. Sone, B. Kozicki, A. Hirano, and M. Jinno, “Dynamic routing and frequency slot assignment for elastic optical path networks that adopt distance adaptive modulation,” in Optical Fiber Communication Conf., 2011.

Soumplis, P.

K. Christodoulopoulos, P. Soumplis, and E. Varvarigos, “Trading off transponders for spectrum in flexgrid networks,” in Optical Fiber Communication Conference (OFC), 2013, paper OTu2A.3.

Steiglitz, K.

C. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Dover, 1998.

Takagi, T.

T. Takagi, H. Hasegawa, K. Sato, Y. Sone, B. Kozicki, A. Hirano, and M. Jinno, “Dynamic routing and frequency slot assignment for elastic optical path networks that adopt distance adaptive modulation,” in Optical Fiber Communication Conf., 2011.

Takara, H.

M. Jinno, B. Kozick, H. Takara, A. Watanabe, and Y. Sone, “Distance-adaptive spectrum resource allocation in spectrum-sliced elastic optical path network,” IEEE Commun. Mag., vol.  48, no. 8, pp. 138–145, 2010.
[CrossRef]

B. Kozicki, H. Takara, Y. Tsukishima, T. Yoshimatsu, K. Yonenaga, and M. Jinno, “Experimental demonstration of spectrum-sliced elastic optical path network (SLICE),” Opt. Express, vol.  18, no. 21, pp. 22105–22118, Oct. 2010.
[CrossRef]

Teipen, B. T.

B. T. Teipen, M. H. Eiselt, K. Grobe, and J.-P. Elbers, “Adaptive data rates for flexible transceivers in optical networks,” J. Netw., vol.  7, no. 5, pp. 776–782, May 2012.
[CrossRef]

Ting, W.

A. N. Patel, P. N. Ji, J. P. Jue, and W. Ting, “Defragmentation of transparent flexible optical WDM (FWDM) networks,” in Optical Fiber Communication Conf. (OFC), 2011, paper OTuI.

Tomkos, I.

Tornatore, M.

Tsukishima, Y.

Tucker, R.

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Figures (6)

Fig. 1.
Fig. 1.

Transmission reach as a function of the rate and spectrum used for a specific flexgrid transponder.

Fig. 2.
Fig. 2.

Transmission reach as a function of the rate and modulation format used by a specific flexgrid transponder.

Fig. 3.
Fig. 3.

Flexgrid network with four nodes and three source–destination demands and the related spectrum slot allocation on the links used. Connections can be established in several transmission configuration options. Demands can be broken into multiple connections that can be regenerated at intermediate nodes (forming a sequence of transparent flexpaths). The guardband that is left between two adjacent flexpaths has to be at least equal to the maximum of the two guardband values of the spectrum-adjacent flexpaths. The nonoverlapping spectrum assignment constraint pertains only to the data-transferring spectrum slots. The spectrum continuity constraint is applied to the transparent flexpaths (subpaths) of the connections.

Fig. 4.
Fig. 4.

Example of serving demand ( s , d ) using regenerators. The demand of 50 Gbps is served by path-transmission tuple ( p , t ) and is broken into two translucent connections, which are all regenerated at node r . The subpaths m 1 (starting at s and ending at r ) and m 2 (starting at r and ending at d ) form the set R p , t that comprise the subpaths of p . Thus, four transparent flexpaths are used in total, indexed as ( p , m , t , i ) , where m = m 1 , m 2 and i = i 1 , i 2 . The utilization of the spectrum of the links comprising path p is also presented. Another connection (blue) is served over the first two links, but since regeneration is performed at node r the spectrum used by ( p , m 1 , t , i 2 ) and ( p , m 2 , t , i 2 ) can change at r .

Fig. 5.
Fig. 5.

Small network topology and link lengths in kilometers.

Fig. 6.
Fig. 6.

(a) Maximum spectrum used (gigahertz), (b) number of transponders, and (c) transponders’ cost, for the flexgrid OFDM and the MLR network, for optimizing the maximum spectrum used ( W = 1 ) and the transponders’ cost ( W = 0.01 ) on the DT network. (d) Trade-off between maximum spectrum used and transponders’ cost in the flexgrid network for year 2018.

Tables (4)

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TABLE I Performance of the Six-Node Network Assuming a Transparent Network Setting and w = 1 (Spectrum Optimization)

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TABLE II Performance of the Six-Node Network Assuming a Transparent Network Setting and w = 0.01 (Transponder Cost Optimization)

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TABLE III Performance of the Six-Node Network Assuming a Translucent Network Setting and w = 1 (Spectrum Optimization)

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TABLE IV Performance of the Six-Node Network Assuming a Translucent Network Setting and w = 0.01 (Transponder Cost Optimization)

Equations (17)

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W p , t = N p , t = Λ s d r t .
C p , t = Λ s d r t · c t + c t rem , S p , t = Λ s d r t · b t + b t rem ,
W p , t = Λ s d r t N p , t = ( Λ s d r t · | R p , t | ) + | R p , t rem | ,
C p , t = ( Λ s d r t · | R p , t | ) · c t + | R p , t rem | · c t rem , S p , t = Λ s d c t · b t + b t rem
( p , t ) > ( p , t ) iff C p , t C p , t and S p , t S p , t .
F total = s d max ( p , t ) Q s d ( i [ 1 , , W p , t ] ( g p , t , i + b p , t , i ) ) = s d max ( p , t ) Q s d ( S p , t ) .
Minimize w · S + ( 1 w ) · C
S f p , m , t , i + b p , t , i ,
C = s d ( p , t ) Q s d C p , t · x p , t .
( p , t ) Q s d x p , t = 1 .
δ p , m , t , i , p , m , t , i + δ p , m , t , i , p , m , t , i = 1 ,
f p , m , t , i f p , m , t , i F total · δ p , m , t , i , p , m , t , i ,
f p , m , t , i f p , m , t , i F total · δ p , m , t , i , p , m , t , i ,
f p , m , t , i ( b p , t , i + max ( g p , t , i , g p , t , i ) ) f p , m , t , i ( F total + max ( g p , t , i , g p , t , i ) ) · ( 1 δ p , m , t , i , p , m , t , i + 2 x p , t x p , t ) ,
f p , m , t , i ( b p , t , i + max ( g p , t , i , g p , t , i ) ) f p , m , t , i ( F total + max ( g p , t , i , g p , t , i ) ) · ( 1 δ p , m , t , i , p , m , t , i + 2 x p , t x p , t ) .
u f u d = u d , u f u g = u g , u f u f = u f , u g u g = u g , u d u d = u d , u d u g = u d .
arg min ( p , t ) Q s d ( w · S ˜ p , t + ( 1 w ) · C ˜ p , t ) .