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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

Sung-Seok Ko

Department of Industrial Engineering, Konkuk University, Seoul, Korea

Received  March 2018 Revised  October 2018 Published  May 2020 Early access  March 2019

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.

Keywords: Inventory, perishable item, retrial demand, level-dependent QBD process.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. Falin and J. G. Templeton, Retrial Queues (Vol. 75). CRC Press, 1997. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

S. Kalpakam and G. Arivarignan, A continuous review perishable inventory model, Statistics, 19 (1988), 389-398.  doi: 10.1080/02331888808802112.  Google Scholar

[14]

S. Kalpakam and G. Arivarignan, Inventory system with random supply quantity, Operations Research Spektrum, 12 (1990), 139-145.  doi: 10.1007/BF01719709.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Z. Lian and L. Liu, Continuous review perishable inventory systems: Models and heuristics, IIE Transactions, 33 (2001), 809-822.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

S. Nahmias, Perishable inventory theory: A review, Operations Research, 30 (1982), 680-708.   Google Scholar

[30]

S. Nahmias, Perishable Inventory Systems, Springer Science & Business Media, 2011. Google Scholar

[31]

M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, Courier Corporation, 1981.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[35]

F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research society, 42 (1991), 27-37.   Google Scholar

[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.   Google Scholar

[37]

G. E. H. Reuter, Competition processes, In Proc. 4th Berkeley Symp. Math. Statist. Prob, 2 (1961), 421–430.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. Falin and J. G. Templeton, Retrial Queues (Vol. 75). CRC Press, 1997. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

S. Kalpakam and G. Arivarignan, A continuous review perishable inventory model, Statistics, 19 (1988), 389-398.  doi: 10.1080/02331888808802112.  Google Scholar

[14]

S. Kalpakam and G. Arivarignan, Inventory system with random supply quantity, Operations Research Spektrum, 12 (1990), 139-145.  doi: 10.1007/BF01719709.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Z. Lian and L. Liu, Continuous review perishable inventory systems: Models and heuristics, IIE Transactions, 33 (2001), 809-822.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

S. Nahmias, Perishable inventory theory: A review, Operations Research, 30 (1982), 680-708.   Google Scholar

[30]

S. Nahmias, Perishable Inventory Systems, Springer Science & Business Media, 2011. Google Scholar

[31]

M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, Courier Corporation, 1981.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[35]

F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research society, 42 (1991), 27-37.   Google Scholar

[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.   Google Scholar

[37]

G. E. H. Reuter, Competition processes, In Proc. 4th Berkeley Symp. Math. Statist. Prob, 2 (1961), 421–430.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Inventory Model
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Figure 2.  Contour Plot of TCR
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Figure 3.  The effect of $ \lambda $
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Figure 4.  The effect of $ \mu $
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Table 1.  Total Cost Rate(TCR)
$S \diagdown s$ 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
Table Options

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$S \diagdown s$ 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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