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Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns
May
2020, 16(3): 1415-1433.
doi: 10.3934/jimo.2019009
A nonhomogeneous quasi-birth-death process approach for an $ (s, S) $ policy for a perishable inventory system with retrial demands
Department of Industrial Engineering, Konkuk University, Seoul, Korea |
In this paper, an $ (s, S) $ continuous inventory model with perishable items and retrial demands is proposed. In addition, replenishment lead times that are independent and identically distributed according to phase-type distribution are implemented. The proposed system is modeled as a three-dimensional Markov process using a level-dependent quasi-birth-death (QBD) process. The ergodicity of the modeled Markov system is demonstrated and the best method for efficiently approximating the steady-state distribution at the inventory level is determined. This paper also provides performance measure formulas based on the steady-state distribution of the proposed approximation method. Furthermore, in order to minimize the system cost, the optimum values of $ s $ and $ S $ are determined numerically and sensitivity analysis is performed on the main parameters.
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C35.
Citation:
Sung-Seok Ko. A nonhomogeneous quasi-birth-death process approach for an $ (s, S) $ policy for a perishable inventory system with retrial demands. Journal of Industrial and Management Optimization,
2020, 16
(3)
: 1415-1433.
doi: 10.3934/jimo.2019009
![]()
References:
| [1] |
M. Alizadeh, H. Eskandari and S. M. Sajadifar,
A modified $(S-1, S)$ inventory system for deteriorating items with Poisson demand and non-zero lead time, Applied Mathematical Modelling, 38 (2014), 699-711.
doi: 10.1016/j.apm.2013.07.014. |
| [2] |
W. J. Anderson, Continuous-time Markov Chains: An Applications-oriented Approach, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-3038-0. |
| [3] |
J. R. Artalejo,
Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 51 (2010), 1071-1081.
doi: 10.1016/j.mcm.2009.12.011. |
| [4] |
J. Artalejo and G. Falin,
Standard and retrial queueing systems: A comparative analysis, Revista Matematica Complutense, 15 (2002), 101-129.
doi: 10.5209/rev_REMA.2002.v15.n1.16950. |
| [5] |
J. R. Artalejo, A. Krishnamoorthy and M. J. Lopez-Herrero,
Numerical analysis of $(s, S)$ inventory systems with repeated attempts, Annals of Operations Research, 141 (2006), 67-83.
doi: 10.1007/s10479-006-5294-8. |
| [6] |
J. R. Artalejo and M. J. Lopez-Herrero,
A simulation study of a discrete-time multiserver retrial queue with finite population, Journal of Statistical Planning and Inference, 137 (2007), 2536-2542.
doi: 10.1016/j.jspi.2006.04.018. |
| [7] |
O. Baron, O. Berman and D. Perry,
Continuous review inventory models for perishable items ordered in batches, Mathematical Methods of Operations Research, 72 (2010), 217-247.
doi: 10.1007/s00186-010-0318-1. |
| [8] |
L. Bright and P. G. Taylor,
Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Stochastic Models, 11 (1995), 497-525.
doi: 10.1080/15326349508807357. |
| [9] |
B. D. Choi and B. Kim,
Non-ergodicity criteria for denumerable continuous time Markov processes, Operations Research Letters, 32 (2004), 574-580.
doi: 10.1016/j.orl.2004.03.001. |
| [10] |
G. Falin and J. G. Templeton, Retrial Queues (Vol. 75). CRC Press, 1997. |
| [11] |
A. Gómez-Corral,
A bibliographical guide to the analysis of retrial queues through matrix analytic techniques, Annals of Operations Research, 141 (2006), 163-191.
doi: 10.1007/s10479-006-5298-4. |
| [12] |
Ü. Gürler and B. Y. Özkaya,
Analysis of the $(s, S)$ policy for perishables with a random shelf life, IIE Transactions, 40 (2008), 759-781.
|
| [13] |
S. Kalpakam and G. Arivarignan,
A continuous review perishable inventory model, Statistics, 19 (1988), 389-398.
doi: 10.1080/02331888808802112. |
| [14] |
S. Kalpakam and G. Arivarignan,
Inventory system with random supply quantity, Operations Research Spektrum, 12 (1990), 139-145.
doi: 10.1007/BF01719709. |
| [15] |
S. Kalpakam and K. P. Sapna,
Continuous review $(s, S)$ inventory system with random lifetimes and positive leadtimes, Operations Research Letters, 16 (1994), 115-119.
doi: 10.1016/0167-6377(94)90066-3. |
| [16] |
S. Kalpakam and K. P. Sapna,
$(S-1, S)$ Perishable systems with stochastic leadtimes, Mathematical and Computer Modelling, 21 (1995), 95-104.
doi: 10.1016/0895-7177(95)00026-X. |
| [17] |
T. Karthick, B. Sivakumar and G. Arivarignan,
An inventory system with two types of customers and retrial demands, International Journal of Systems Science: Operations & Logistics, 2 (2015), 90-112.
|
| [18] |
C. Kouki, E. Sahin, Z. Jemai and Y. Dallery, Periodic Review Inventory Policy for Perishables with Random Lifetime, In Eighth International Conference of Modeling and Simulation, 2010. |
| [19] |
A. Krishnamoorthy and P. V. Ushakumari,
Reliability of a k-out-of-n system with repair and retrial of failed units, Top, 7 (1999), 293-304.
doi: 10.1007/BF02564728. |
| [20] |
S. Kumaraswamy and E. Sankarasubramanian,
A continuous review of $(s, S)$ inventory systems in which depletion is due to demand and failure of units, Journal of the Operational Research Society, 32 (1981), 997-1001.
|
| [21] |
G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, Society for Industrial and Applied Mathematics, 1999.
doi: 10.1137/1.9780898719734. |
| [22] |
A. S. Lawrence, B. Sivakumar and G. Arivarignan,
A perishable inventory system with service facility and finite source, Applied Mathematical Modelling, 37 (2013), 4771-4786.
doi: 10.1016/j.apm.2012.09.018. |
| [23] |
P. Vijaya Laxmi and M. L. Soujanya,
Perishable inventory system with service interruptions, retrial demands and negative customers, Applied Mathematics and Computation, 262 (2015), 102-110.
doi: 10.1016/j.amc.2015.04.013. |
| [24] |
Z. Lian and L. Liu,
Continuous review perishable inventory systems: Models and heuristics, IIE Transactions, 33 (2001), 809-822.
|
| [25] |
L. Liu,
(s, S) Continuous Review Models for Inventory with Random Lifetimes, Operations Research Letters, 9 (1990), 161-167.
doi: 10.1016/0167-6377(90)90014-V. |
| [26] |
L. Liu and D. H. Shi,
An $(s, S)$ model for inventory with exponential lifetimes and renewal demands, Naval Research Logistics, 46 (1999), 39-56.
doi: 10.1002/(SICI)1520-6750(199902)46:1<39::AID-NAV3>3.0.CO;2-G. |
| [27] |
L. Liu and T. Yang,
An $(s, S)$ random lifetime inventory model with a positive lead time, European Journal of Operational Research, 113 (1999), 52-63.
doi: 10.1016/0167-6377(90)90014-V. |
| [28] |
E. Mohebbi and M. J. Posner,
A continuous review inventory system with lost sales and variable lead time, Naval Research Logistics, 45 (1998), 259-278.
doi: 10.1002/(SICI)1520-6750(199804)45:3<259::AID-NAV2>3.0.CO;2-6. |
| [29] |
S. Nahmias,
Perishable inventory theory: A review, Operations Research, 30 (1982), 680-708.
|
| [30] |
S. Nahmias, Perishable Inventory Systems, Springer Science & Business Media, 2011. |
| [31] |
M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, Courier Corporation, 1981. |
| [32] |
F. Olsson and P. Tydesjö,
Inventory problems with perishable items: Fixed lifetimes and backlogging, European Journal of Operational Research, 202 (2010), 131-137.
doi: 10.1016/j.ejor.2009.05.010. |
| [33] |
C. Periyasamy,
A finite source perishable inventory system with retrial demands and multiple server vacation, International Journal of Engineering Research and Technology, 2 (2013), 3802-3815.
|
| [34] |
G. P. Prastacos,
Blood inventory management: An overview of theory and practice, Management Science, 30 (1984), 777-800.
doi: 10.1287/mnsc.30.7.777. |
| [35] |
F. Raafat,
Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research society, 42 (1991), 27-37.
|
| [36] |
N. Ravichandran,
Stochastic analysis of a continuous review perishable inventory system with positive lead time and Poisson demand, European Journal of Operational Research, 84 (1995), 444-457.
|
| [37] |
G. E. H. Reuter, Competition processes, In Proc. 4th Berkeley Symp. Math. Statist. Prob, 2
(1961), 421–430. |
| [38] |
C. P. Schmidt and S. Nahmias,
$(S-1, S)$ Policies for perishable inventory, Management Science, 31 (1985), 719-728.
doi: 10.1287/mnsc.31.6.719. |
| [39] |
L. I. Sennott, P. A. Humblet and R. L. Tweedie,
Mean drifts and the non-ergodicity of Markov chains, Operations Research, 31 (1983), 783-789.
doi: 10.1287/opre.31.4.783. |
| [40] |
B. Sivakumar,
Two-commodity inventory system with retrial demand, European Journal of Operational Research, 187 (2008), 70-83.
doi: 10.1016/j.ejor.2007.02.036. |
| [41] |
B. Sivakumar,
A perishable inventory system with retrial demands and a finite population, Journal of Computational and Applied Mathematics, 224 (2009), 29-38.
doi: 10.1016/j.cam.2008.03.041. |
| [42] |
R. L. Tweedie,
Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes, Journal of Applied Probability, 18 (1981), 122-130.
doi: 10.2307/3213172. |
| [43] |
P. V. Ushakumari, On $(s, S)$ inventory system with random lead time and repeated demands, International Journal of Stochastic Analysis, 2006 (2006), Art. ID 81508, 22 pp.
doi: 10.1155/JAMSA/2006/81508. |
| [44] |
H. J. Weiss,
Optimal ordering policies for continuous review perishable inventory models, Operations Research, 28 (1980), 365-374.
doi: 10.1287/opre.28.2.365. |
show all references
References:
| [1] |
M. Alizadeh, H. Eskandari and S. M. Sajadifar,
A modified $(S-1, S)$ inventory system for deteriorating items with Poisson demand and non-zero lead time, Applied Mathematical Modelling, 38 (2014), 699-711.
doi: 10.1016/j.apm.2013.07.014. |
| [2] |
W. J. Anderson, Continuous-time Markov Chains: An Applications-oriented Approach, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-3038-0. |
| [3] |
J. R. Artalejo,
Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 51 (2010), 1071-1081.
doi: 10.1016/j.mcm.2009.12.011. |
| [4] |
J. Artalejo and G. Falin,
Standard and retrial queueing systems: A comparative analysis, Revista Matematica Complutense, 15 (2002), 101-129.
doi: 10.5209/rev_REMA.2002.v15.n1.16950. |
| [5] |
J. R. Artalejo, A. Krishnamoorthy and M. J. Lopez-Herrero,
Numerical analysis of $(s, S)$ inventory systems with repeated attempts, Annals of Operations Research, 141 (2006), 67-83.
doi: 10.1007/s10479-006-5294-8. |
| [6] |
J. R. Artalejo and M. J. Lopez-Herrero,
A simulation study of a discrete-time multiserver retrial queue with finite population, Journal of Statistical Planning and Inference, 137 (2007), 2536-2542.
doi: 10.1016/j.jspi.2006.04.018. |
| [7] |
O. Baron, O. Berman and D. Perry,
Continuous review inventory models for perishable items ordered in batches, Mathematical Methods of Operations Research, 72 (2010), 217-247.
doi: 10.1007/s00186-010-0318-1. |
| [8] |
L. Bright and P. G. Taylor,
Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Stochastic Models, 11 (1995), 497-525.
doi: 10.1080/15326349508807357. |
| [9] |
B. D. Choi and B. Kim,
Non-ergodicity criteria for denumerable continuous time Markov processes, Operations Research Letters, 32 (2004), 574-580.
doi: 10.1016/j.orl.2004.03.001. |
| [10] |
G. Falin and J. G. Templeton, Retrial Queues (Vol. 75). CRC Press, 1997. |
| [11] |
A. Gómez-Corral,
A bibliographical guide to the analysis of retrial queues through matrix analytic techniques, Annals of Operations Research, 141 (2006), 163-191.
doi: 10.1007/s10479-006-5298-4. |
| [12] |
Ü. Gürler and B. Y. Özkaya,
Analysis of the $(s, S)$ policy for perishables with a random shelf life, IIE Transactions, 40 (2008), 759-781.
|
| [13] |
S. Kalpakam and G. Arivarignan,
A continuous review perishable inventory model, Statistics, 19 (1988), 389-398.
doi: 10.1080/02331888808802112. |
| [14] |
S. Kalpakam and G. Arivarignan,
Inventory system with random supply quantity, Operations Research Spektrum, 12 (1990), 139-145.
doi: 10.1007/BF01719709. |
| [15] |
S. Kalpakam and K. P. Sapna,
Continuous review $(s, S)$ inventory system with random lifetimes and positive leadtimes, Operations Research Letters, 16 (1994), 115-119.
doi: 10.1016/0167-6377(94)90066-3. |
| [16] |
S. Kalpakam and K. P. Sapna,
$(S-1, S)$ Perishable systems with stochastic leadtimes, Mathematical and Computer Modelling, 21 (1995), 95-104.
doi: 10.1016/0895-7177(95)00026-X. |
| [17] |
T. Karthick, B. Sivakumar and G. Arivarignan,
An inventory system with two types of customers and retrial demands, International Journal of Systems Science: Operations & Logistics, 2 (2015), 90-112.
|
| [18] |
C. Kouki, E. Sahin, Z. Jemai and Y. Dallery, Periodic Review Inventory Policy for Perishables with Random Lifetime, In Eighth International Conference of Modeling and Simulation, 2010. |
| [19] |
A. Krishnamoorthy and P. V. Ushakumari,
Reliability of a k-out-of-n system with repair and retrial of failed units, Top, 7 (1999), 293-304.
doi: 10.1007/BF02564728. |
| [20] |
S. Kumaraswamy and E. Sankarasubramanian,
A continuous review of $(s, S)$ inventory systems in which depletion is due to demand and failure of units, Journal of the Operational Research Society, 32 (1981), 997-1001.
|
| [21] |
G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, Society for Industrial and Applied Mathematics, 1999.
doi: 10.1137/1.9780898719734. |
| [22] |
A. S. Lawrence, B. Sivakumar and G. Arivarignan,
A perishable inventory system with service facility and finite source, Applied Mathematical Modelling, 37 (2013), 4771-4786.
doi: 10.1016/j.apm.2012.09.018. |
| [23] |
P. Vijaya Laxmi and M. L. Soujanya,
Perishable inventory system with service interruptions, retrial demands and negative customers, Applied Mathematics and Computation, 262 (2015), 102-110.
doi: 10.1016/j.amc.2015.04.013. |
| [24] |
Z. Lian and L. Liu,
Continuous review perishable inventory systems: Models and heuristics, IIE Transactions, 33 (2001), 809-822.
|
| [25] |
L. Liu,
(s, S) Continuous Review Models for Inventory with Random Lifetimes, Operations Research Letters, 9 (1990), 161-167.
doi: 10.1016/0167-6377(90)90014-V. |
| [26] |
L. Liu and D. H. Shi,
An $(s, S)$ model for inventory with exponential lifetimes and renewal demands, Naval Research Logistics, 46 (1999), 39-56.
doi: 10.1002/(SICI)1520-6750(199902)46:1<39::AID-NAV3>3.0.CO;2-G. |
| [27] |
L. Liu and T. Yang,
An $(s, S)$ random lifetime inventory model with a positive lead time, European Journal of Operational Research, 113 (1999), 52-63.
doi: 10.1016/0167-6377(90)90014-V. |
| [28] |
E. Mohebbi and M. J. Posner,
A continuous review inventory system with lost sales and variable lead time, Naval Research Logistics, 45 (1998), 259-278.
doi: 10.1002/(SICI)1520-6750(199804)45:3<259::AID-NAV2>3.0.CO;2-6. |
| [29] |
S. Nahmias,
Perishable inventory theory: A review, Operations Research, 30 (1982), 680-708.
|
| [30] |
S. Nahmias, Perishable Inventory Systems, Springer Science & Business Media, 2011. |
| [31] |
M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, Courier Corporation, 1981. |
| [32] |
F. Olsson and P. Tydesjö,
Inventory problems with perishable items: Fixed lifetimes and backlogging, European Journal of Operational Research, 202 (2010), 131-137.
doi: 10.1016/j.ejor.2009.05.010. |
| [33] |
C. Periyasamy,
A finite source perishable inventory system with retrial demands and multiple server vacation, International Journal of Engineering Research and Technology, 2 (2013), 3802-3815.
|
| [34] |
G. P. Prastacos,
Blood inventory management: An overview of theory and practice, Management Science, 30 (1984), 777-800.
doi: 10.1287/mnsc.30.7.777. |
| [35] |
F. Raafat,
Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research society, 42 (1991), 27-37.
|
| [36] |
N. Ravichandran,
Stochastic analysis of a continuous review perishable inventory system with positive lead time and Poisson demand, European Journal of Operational Research, 84 (1995), 444-457.
|
| [37] |
G. E. H. Reuter, Competition processes, In Proc. 4th Berkeley Symp. Math. Statist. Prob, 2
(1961), 421–430. |
| [38] |
C. P. Schmidt and S. Nahmias,
$(S-1, S)$ Policies for perishable inventory, Management Science, 31 (1985), 719-728.
doi: 10.1287/mnsc.31.6.719. |
| [39] |
L. I. Sennott, P. A. Humblet and R. L. Tweedie,
Mean drifts and the non-ergodicity of Markov chains, Operations Research, 31 (1983), 783-789.
doi: 10.1287/opre.31.4.783. |
| [40] |
B. Sivakumar,
Two-commodity inventory system with retrial demand, European Journal of Operational Research, 187 (2008), 70-83.
doi: 10.1016/j.ejor.2007.02.036. |
| [41] |
B. Sivakumar,
A perishable inventory system with retrial demands and a finite population, Journal of Computational and Applied Mathematics, 224 (2009), 29-38.
doi: 10.1016/j.cam.2008.03.041. |
| [42] |
R. L. Tweedie,
Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes, Journal of Applied Probability, 18 (1981), 122-130.
doi: 10.2307/3213172. |
| [43] |
P. V. Ushakumari, On $(s, S)$ inventory system with random lead time and repeated demands, International Journal of Stochastic Analysis, 2006 (2006), Art. ID 81508, 22 pp.
doi: 10.1155/JAMSA/2006/81508. |
| [44] |
H. J. Weiss,
Optimal ordering policies for continuous review perishable inventory models, Operations Research, 28 (1980), 365-374.
doi: 10.1287/opre.28.2.365. |
Figure 2.
Contour Plot of TCR
Figure 3.
The effect of $ \lambda $
Figure 4.
The effect of $ \mu $
Table 1.
Total Cost Rate(TCR)
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 15 | 367.40 | 363.25 | 361.55 | 362.46 | 366.06 | 372.45 | 381.92 | 394.31 | 409.75 | 429.18 |
| 16 | 366.82 | 362.49 | 360.49 | 360.94 | 363.82 | 369.16 | 377.05 | 387.83 | 400.29 | 415.71 |
| 17 | 366.83 | 362.40 | 360.21 | 360.34 | 362.72 | 367.30 | 374.09 | 383.24 | 394.32 | 407.17 |
| 18 | 367.32 | 362.86 | 360.56 | 360.46 | 362.49 | 366.53 | 372.52 | 380.51 | 390.68 | 401.73 |
| 19 | 368.23 | 363.77 | 361.42 | 361.18 | 362.95 | 366.59 | 372.00 | 379.14 | 388.10 | 398.40 |
| 20 | 369.49 | 365.07 | 362.70 | 362.38 | 363.97 | 367.32 | 372.29 | 378.80 | 386.87 | 396.60 |
| 21 | 371.06 | 366.69 | 364.35 | 363.98 | 365.45 | 368.58 | 373.22 | 379.25 | 386.65 | 395.44 |
| 22 | 372.88 | 368.60 | 366.29 | 365.91 | 367.30 | 370.27 | 374.65 | 380.32 | 387.20 | 395.29 |
| 23 | 374.93 | 370.74 | 368.49 | 368.12 | 369.45 | 372.31 | 376.50 | 381.89 | 388.37 | 395.91 |
| 24 | 377.18 | 373.09 | 370.91 | 370.56 | 371.87 | 374.65 | 378.69 | 383.85 | 390.02 | 397.14 |
| 25 | 379.60 | 375.62 | 373.52 | 373.21 | 374.51 | 377.23 | 381.16 | 386.14 | 392.06 | 398.84 |
| 26 | 382.17 | 378.31 | 376.29 | 376.03 | 377.34 | 380.02 | 383.86 | 388.71 | 394.42 | 400.93 |
| 27 | 384.88 | 381.13 | 379.20 | 379.00 | 380.33 | 382.99 | 386.76 | 391.50 | 397.05 | 403.34 |
| 28 | 387.71 | 384.07 | 382.24 | 382.10 | 383.46 | 386.10 | 389.83 | 394.48 | 399.91 | 406.01 |
| 29 | 390.64 | 387.12 | 385.38 | 385.31 | 386.70 | 389.35 | 393.05 | 397.63 | 402.94 | 408.90 |
| 30 | 393.66 | 390.27 | 388.62 | 388.62 | 390.05 | 392.71 | 396.39 | 400.91 | 406.14 | 411.98 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 15 | 367.40 | 363.25 | 361.55 | 362.46 | 366.06 | 372.45 | 381.92 | 394.31 | 409.75 | 429.18 |
| 16 | 366.82 | 362.49 | 360.49 | 360.94 | 363.82 | 369.16 | 377.05 | 387.83 | 400.29 | 415.71 |
| 17 | 366.83 | 362.40 | 360.21 | 360.34 | 362.72 | 367.30 | 374.09 | 383.24 | 394.32 | 407.17 |
| 18 | 367.32 | 362.86 | 360.56 | 360.46 | 362.49 | 366.53 | 372.52 | 380.51 | 390.68 | 401.73 |
| 19 | 368.23 | 363.77 | 361.42 | 361.18 | 362.95 | 366.59 | 372.00 | 379.14 | 388.10 | 398.40 |
| 20 | 369.49 | 365.07 | 362.70 | 362.38 | 363.97 | 367.32 | 372.29 | 378.80 | 386.87 | 396.60 |
| 21 | 371.06 | 366.69 | 364.35 | 363.98 | 365.45 | 368.58 | 373.22 | 379.25 | 386.65 | 395.44 |
| 22 | 372.88 | 368.60 | 366.29 | 365.91 | 367.30 | 370.27 | 374.65 | 380.32 | 387.20 | 395.29 |
| 23 | 374.93 | 370.74 | 368.49 | 368.12 | 369.45 | 372.31 | 376.50 | 381.89 | 388.37 | 395.91 |
| 24 | 377.18 | 373.09 | 370.91 | 370.56 | 371.87 | 374.65 | 378.69 | 383.85 | 390.02 | 397.14 |
| 25 | 379.60 | 375.62 | 373.52 | 373.21 | 374.51 | 377.23 | 381.16 | 386.14 | 392.06 | 398.84 |
| 26 | 382.17 | 378.31 | 376.29 | 376.03 | 377.34 | 380.02 | 383.86 | 388.71 | 394.42 | 400.93 |
| 27 | 384.88 | 381.13 | 379.20 | 379.00 | 380.33 | 382.99 | 386.76 | 391.50 | 397.05 | 403.34 |
| 28 | 387.71 | 384.07 | 382.24 | 382.10 | 383.46 | 386.10 | 389.83 | 394.48 | 399.91 | 406.01 |
| 29 | 390.64 | 387.12 | 385.38 | 385.31 | 386.70 | 389.35 | 393.05 | 397.63 | 402.94 | 408.90 |
| 30 | 393.66 | 390.27 | 388.62 | 388.62 | 390.05 | 392.71 | 396.39 | 400.91 | 406.14 | 411.98 |
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