# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2019019

## Deteriorating inventory with preservation technology under price- and stock-sensitive demand

 1 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India 2 Faculty of Engineering Management, Chair of Marketing and Economic Engineering, Poznan University of Technology, ul. Strzelecka 11, 60-965 Poznan, Poland

* Corresponding Author: sankroy2006@gmail.com

Received  July 2018 Revised  September 2018 Published  March 2019

Fund Project: The author, Magfura Pervin is very much thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under [MANF(UGC)] scheme: Sanctioned letter number [F1-17.1/2012-13/MANF-2012-13-MUS-WES-19170 /(SA-Ⅲ/Website)] dated 28/02/2013.
The research of Sankar Kumar Roy is partially supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), through the CIDMA - Center for Research and Development in Mathematics and Applications, University of Aveiro, Portugal, within project UID/MAT/04106/2019.
The research of Gerhard-Wilhelm Weber (Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey) is partially supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), through the CIDMA - Center for Research and Development in Mathematics and Applications.

In this paper, we formulate and solve an Economic Production Quantity inventory model with deteriorating items. To reduce the rate of deterioration, we apply a preservation technology and calculate the amount for preservation technology investment. The demand function is dependent on stock-level and price. We assume that the production rate is linearly dependent on time, based on customer demand. Shortages are allowed in our consideration, and the shortages amount is partially backlogged for the interested customers for the next slot. The effect of inflation is incorporated, which indicates a critical factor in modern days. Our main objective is to find the optimal cycle length and the optimal amount of preservation technology investment by adjusting the inflation rate with maximizing the profit. A numerical example is provided to illustrate the features and advances of the model. A sensitivity analysis with respect to major parameters is performed in order to assess the stability of our model. The paper ends with a conclusion and an outlook at possible future directions.

Citation: Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Deteriorating inventory with preservation technology under price- and stock-sensitive demand. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019019
##### References:
 [1] M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001, European Journal of Operational Research, 221 (2012), 275-284. doi: 10.1016/j.ejor.2012.03.004. Google Scholar [2] S. Bardhan, H. Pal and B. C. Giri, Optimal replenishment policy and preservation technology investment for a non-instantaneous deteriorating item with stock-dependent demand, Operational Research: An International Journal, (2017), 1–22. doi: 10.1007/s12351-017-0302-0. Google Scholar [3] A. K. Bhunia, S. Kundu, T. Sannigrahi and S. K. Goyal, An application of tournament genetic algorithm in a marketing oriented economic production lot-size model for deteriorating items, International Journal of Production Economics, 119 (2009), 112-121. doi: 10.1016/j.ijpe.2009.01.010. Google Scholar [4] J. A. Buzacott, Economic order quantities with inflation, Operational Research Quaterly, 26 (1975), 553-558. Google Scholar [5] A. Cambini and L. Martein, Generalized Convexity and Optimization: Theory and Application, Lecture Notes in Economics and Mathematical Systems, 616. Springer-Verlag, Berlin, 2009. Google Scholar [6] U. Dave and L. K. Patel, $(T, S_i)$ policy inventory model for deteriorating items with time proportional demand, The Journal of the Operational Research Society, 32 (1981), 137-142. Google Scholar [7] C. Y. Dye, The effect of preservation technology investment on a non-instantaneous deteriorating inventory model, OMEGA, 41 (2013), 872-880. doi: 10.1016/j.omega.2012.11.002. Google Scholar [8] P. M. Ghare and G. P. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243. Google Scholar [9] S. K. Goyal and B. C. Giri, Recent trend in modeling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4. Google Scholar [10] R. Gupta and P. Vrat, Inventory model with multi-items under constraint systems for stock dependent consumption rate, Opsearch, 23 (1986), 19-24. Google Scholar [11] P. H. Hsu, H. M. Wee and H. M. Teng, Preservation technology investment for deteriorating inventory, International Journal of Production Economics, 124 (2010), 388-394. doi: 10.1016/j.ijpe.2009.11.034. Google Scholar [12] U. K. Khedlekar, D. Shukla and A. Namdeo, Pricing policy for declining demand using item preservation technology, Springer Plus, 5 (2016), 1957. doi: 10.1186/s40064-016-3627-x. Google Scholar [13] R. I. Levin, C. P. Mclaughlin, R. P. Lamone and J. F. Kottas, Productions Operations Management: Contemporary Policy for Managing Operating System, McGraw-Hill Series in Management, New York, 1972.Google Scholar [14] H. C. Liao, C. H. Tsai and C. T. Su, An inventory model with deteriorating items under inflation when a delay in payment is permissible, International Journal of Production Economics, 63 (2000), 207-214. doi: 10.1016/S0925-5273(99)00015-8. Google Scholar [15] R. Maihami and N. K. Abadi, Joint control of inventory and its pricing for non-instantaneouly deteriorating items under permissible delay in payments and partial backlogging, Mathematical and Computer Modelling, 55 (2012), 1722-1733. doi: 10.1016/j.mcm.2011.11.017. Google Scholar [16] D. P. Murr and L. L. Morris, Effect of storage temperature on post change in mushrooms, Journal of the American Society for Horticultural Science, 100 (1975), 16-19. Google Scholar [17] L. Y. Ouyang, K. S. Wu and C. T. Yang, A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments, Computers & Industrial Engineering, 51 (2006), 637-651. doi: 10.1016/j.cie.2006.07.012. Google Scholar [18] S. Pal, G. S. Mahapatra and G. P. Samanta, An inventory model of price and stock dependent demand rate with deterioration under inflation and delay in payment, International Journal of System Assurance Engineering and Management, 5 (2014), 591-601. doi: 10.1007/s13198-013-0209-y. Google Scholar [19] P. Papachristos and K. Skouri, A discrete-in-time probabilistic inventory model for deteriorating items under conditions of permissible delay in paymentsAn inventory model with deteriorating items, quantity discount, pricing and time-dependent partial backlogging, International Journal of Production Economics, 83 (2003), 247-256. Google Scholar [20] M. Pervin, G. C. Mahata and S. K. Roy, An inventory model with demand declining market for deteriorating items under trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251. doi: 10.1080/17509653.2015.1081082. Google Scholar [21] M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460. doi: 10.1007/s10479-016-2355-5. Google Scholar [22] M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50. doi: 10.3934/naco.2017002. Google Scholar [23] M. Pervin, S. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control and Optimization, 8 (2018), 169-191. doi: 10.3934/naco.2018010. Google Scholar [24] M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy, Journal of Industrial and Management Optimization, 7 (2017), 21-50. doi: 10.3934/naco.2017002. Google Scholar [25] J. Ray and K. S. Chaudhuri, An EOQ model with stock-dependent demand, shortage, inflation and time discounting, International Journal of Production Economics, 53 (1997), 171-180. doi: 10.1016/S0925-5273(97)00112-6. Google Scholar [26] S. K. Roy, M. Pervin and G. W. Weber, A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy, Journal of Industrial and Management Optimization, 8 (2018), 169-191. doi: 10.3934/naco.2018010. Google Scholar [27] S. Schaible, Fractional programming, Mathematical Methods of Operations Research, 27 (1983), 39-54. doi: 10.1007/bf01916898. Google Scholar [28] B. Sarkar and I. Moon, An EPQ model with inflation in an imperfect manufacturing system, Applied Mathematics and Computation, 217 (2011), 6159-6167. doi: 10.1016/j.amc.2010.12.098. Google Scholar [29] N. H. Shah, H. N. Soni and K. A. Patel, Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates, OMEGA, 41 (2013), 421-430. doi: 10.1016/j.omega.2012.03.002. Google Scholar [30] N. H. Shah, M. Y. Jani and U. Chaudhari, Study of imperfect manufacturing system with preservation technology investment under inflationary environment for quadratic demand: a reverse logistic approach, Journal of Advanced Manufacturing Systems, 16 (2017), 17-34. doi: 10.1142/S0219686717500020. Google Scholar [31] J. T. Teng and C. T. Chang, Economic production quantity models for deteriorating items with price- and stock-dependent demand, Computers and Operations Research, 32 (2005), 297-308. doi: 10.1016/S0305-0548(03)00237-5. Google Scholar [32] Y. C. Tsao, Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits, Journal of Industrial and Management Optimization, 13 (2017), 327-345. doi: 10.3934/jimo.2016020. Google Scholar [33] H. M. Wee, A replenishment policy for items with a price-dependent demand and a varying rate of deterioration, Production Planning & Control, 8 (1997), 494-499. doi: 10.1080/095372897235073. Google Scholar [34] G. A. Widyadana and H. M. Wee, Production inventory models for deteriorating items with stochastic machine unavailability time, lost sales and price-dependent demand, Journal Teknik Industri, 12 (2010), 61-68. Google Scholar [35] G. Zauberman, R. Ronen, M. Akerman and Y. Fuchs, Low PH treatment protects litchi fruit colour, Acta Hortic, 269 (1990), 309-314. Google Scholar

show all references

##### References:
 [1] M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001, European Journal of Operational Research, 221 (2012), 275-284. doi: 10.1016/j.ejor.2012.03.004. Google Scholar [2] S. Bardhan, H. Pal and B. C. Giri, Optimal replenishment policy and preservation technology investment for a non-instantaneous deteriorating item with stock-dependent demand, Operational Research: An International Journal, (2017), 1–22. doi: 10.1007/s12351-017-0302-0. Google Scholar [3] A. K. Bhunia, S. Kundu, T. Sannigrahi and S. K. Goyal, An application of tournament genetic algorithm in a marketing oriented economic production lot-size model for deteriorating items, International Journal of Production Economics, 119 (2009), 112-121. doi: 10.1016/j.ijpe.2009.01.010. Google Scholar [4] J. A. Buzacott, Economic order quantities with inflation, Operational Research Quaterly, 26 (1975), 553-558. Google Scholar [5] A. Cambini and L. Martein, Generalized Convexity and Optimization: Theory and Application, Lecture Notes in Economics and Mathematical Systems, 616. Springer-Verlag, Berlin, 2009. Google Scholar [6] U. Dave and L. K. Patel, $(T, S_i)$ policy inventory model for deteriorating items with time proportional demand, The Journal of the Operational Research Society, 32 (1981), 137-142. Google Scholar [7] C. Y. Dye, The effect of preservation technology investment on a non-instantaneous deteriorating inventory model, OMEGA, 41 (2013), 872-880. doi: 10.1016/j.omega.2012.11.002. Google Scholar [8] P. M. Ghare and G. P. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243. Google Scholar [9] S. K. Goyal and B. C. Giri, Recent trend in modeling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4. Google Scholar [10] R. Gupta and P. Vrat, Inventory model with multi-items under constraint systems for stock dependent consumption rate, Opsearch, 23 (1986), 19-24. Google Scholar [11] P. H. Hsu, H. M. Wee and H. M. Teng, Preservation technology investment for deteriorating inventory, International Journal of Production Economics, 124 (2010), 388-394. doi: 10.1016/j.ijpe.2009.11.034. Google Scholar [12] U. K. Khedlekar, D. Shukla and A. Namdeo, Pricing policy for declining demand using item preservation technology, Springer Plus, 5 (2016), 1957. doi: 10.1186/s40064-016-3627-x. Google Scholar [13] R. I. Levin, C. P. Mclaughlin, R. P. Lamone and J. F. Kottas, Productions Operations Management: Contemporary Policy for Managing Operating System, McGraw-Hill Series in Management, New York, 1972.Google Scholar [14] H. C. Liao, C. H. Tsai and C. T. Su, An inventory model with deteriorating items under inflation when a delay in payment is permissible, International Journal of Production Economics, 63 (2000), 207-214. doi: 10.1016/S0925-5273(99)00015-8. Google Scholar [15] R. Maihami and N. K. Abadi, Joint control of inventory and its pricing for non-instantaneouly deteriorating items under permissible delay in payments and partial backlogging, Mathematical and Computer Modelling, 55 (2012), 1722-1733. doi: 10.1016/j.mcm.2011.11.017. Google Scholar [16] D. P. Murr and L. L. Morris, Effect of storage temperature on post change in mushrooms, Journal of the American Society for Horticultural Science, 100 (1975), 16-19. Google Scholar [17] L. Y. Ouyang, K. S. Wu and C. T. Yang, A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments, Computers & Industrial Engineering, 51 (2006), 637-651. doi: 10.1016/j.cie.2006.07.012. Google Scholar [18] S. Pal, G. S. Mahapatra and G. P. Samanta, An inventory model of price and stock dependent demand rate with deterioration under inflation and delay in payment, International Journal of System Assurance Engineering and Management, 5 (2014), 591-601. doi: 10.1007/s13198-013-0209-y. Google Scholar [19] P. Papachristos and K. Skouri, A discrete-in-time probabilistic inventory model for deteriorating items under conditions of permissible delay in paymentsAn inventory model with deteriorating items, quantity discount, pricing and time-dependent partial backlogging, International Journal of Production Economics, 83 (2003), 247-256. Google Scholar [20] M. Pervin, G. C. Mahata and S. K. Roy, An inventory model with demand declining market for deteriorating items under trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251. doi: 10.1080/17509653.2015.1081082. Google Scholar [21] M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460. doi: 10.1007/s10479-016-2355-5. Google Scholar [22] M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50. doi: 10.3934/naco.2017002. Google Scholar [23] M. Pervin, S. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control and Optimization, 8 (2018), 169-191. doi: 10.3934/naco.2018010. Google Scholar [24] M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy, Journal of Industrial and Management Optimization, 7 (2017), 21-50. doi: 10.3934/naco.2017002. Google Scholar [25] J. Ray and K. S. Chaudhuri, An EOQ model with stock-dependent demand, shortage, inflation and time discounting, International Journal of Production Economics, 53 (1997), 171-180. doi: 10.1016/S0925-5273(97)00112-6. Google Scholar [26] S. K. Roy, M. Pervin and G. W. Weber, A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy, Journal of Industrial and Management Optimization, 8 (2018), 169-191. doi: 10.3934/naco.2018010. Google Scholar [27] S. Schaible, Fractional programming, Mathematical Methods of Operations Research, 27 (1983), 39-54. doi: 10.1007/bf01916898. Google Scholar [28] B. Sarkar and I. Moon, An EPQ model with inflation in an imperfect manufacturing system, Applied Mathematics and Computation, 217 (2011), 6159-6167. doi: 10.1016/j.amc.2010.12.098. Google Scholar [29] N. H. Shah, H. N. Soni and K. A. Patel, Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates, OMEGA, 41 (2013), 421-430. doi: 10.1016/j.omega.2012.03.002. Google Scholar [30] N. H. Shah, M. Y. Jani and U. Chaudhari, Study of imperfect manufacturing system with preservation technology investment under inflationary environment for quadratic demand: a reverse logistic approach, Journal of Advanced Manufacturing Systems, 16 (2017), 17-34. doi: 10.1142/S0219686717500020. Google Scholar [31] J. T. Teng and C. T. Chang, Economic production quantity models for deteriorating items with price- and stock-dependent demand, Computers and Operations Research, 32 (2005), 297-308. doi: 10.1016/S0305-0548(03)00237-5. Google Scholar [32] Y. C. Tsao, Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits, Journal of Industrial and Management Optimization, 13 (2017), 327-345. doi: 10.3934/jimo.2016020. Google Scholar [33] H. M. Wee, A replenishment policy for items with a price-dependent demand and a varying rate of deterioration, Production Planning & Control, 8 (1997), 494-499. doi: 10.1080/095372897235073. Google Scholar [34] G. A. Widyadana and H. M. Wee, Production inventory models for deteriorating items with stochastic machine unavailability time, lost sales and price-dependent demand, Journal Teknik Industri, 12 (2010), 61-68. Google Scholar [35] G. Zauberman, R. Ronen, M. Akerman and Y. Fuchs, Low PH treatment protects litchi fruit colour, Acta Hortic, 269 (1990), 309-314. Google Scholar
The inventory path-line
Concavity of the profit function; including are $x^*$, $r_1^*$ and $TP$, as x-axis, y-axis and z-axis, respectively
Concavity of the profit function; including are $r_1^*$, $r_2^*$ and $TP$, as x-axis, y-axis and z-axis, respectively
Concavity of the profit function; including are $x^*$, $r_2^*$ and $TP$, as x-axis, y-axis and z-axis, respectively
Effect of $\theta$ on total profit $TP$
Effect of $\theta$ on selling price $r_1$
Effect of demand parameter $b_1$ on preservation amount $x$
Contributions of some authors related to inventory model
 Authors Deterio-rations Preservation Technology Stock-dependent Price-dependent Inflation Partial backorder Bhunia et al. (2009) $\surd$ Dave & Patel (1981) $\surd$ Papachristos & Skouri (2003) $\surd$ $\surd$ Wee (1997) $\surd$ $\surd$ Hsu et al. (2010) $\surd$ $\surd$ Maihami and Abadi (2012) $\surd$ $\surd$ Khedlekar et al. (2016) $\surd$ $\surd$ Murr and Morris (1975) $\surd$ $\surd$ Buzacott (1975) $\surd$ Gupta and Vrat (1986) $\surd$ Liao et al. (2000) $\surd$ $\surd$ $\surd$ Ray and Chaudhuri (1997) $\surd$ $\surd$ Pal et al. (2014) $\surd$ $\surd$ $\surd$ $\surd$ Sarkar and Moon (2011) $\surd$ Shah et al. (2017) $\surd$ $\surd$ Teng and Chang (2005) $\surd$ $\surd$ $\surd$ Widyadana and Wee (2010) $\surd$ $\surd$ This work $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
 Authors Deterio-rations Preservation Technology Stock-dependent Price-dependent Inflation Partial backorder Bhunia et al. (2009) $\surd$ Dave & Patel (1981) $\surd$ Papachristos & Skouri (2003) $\surd$ $\surd$ Wee (1997) $\surd$ $\surd$ Hsu et al. (2010) $\surd$ $\surd$ Maihami and Abadi (2012) $\surd$ $\surd$ Khedlekar et al. (2016) $\surd$ $\surd$ Murr and Morris (1975) $\surd$ $\surd$ Buzacott (1975) $\surd$ Gupta and Vrat (1986) $\surd$ Liao et al. (2000) $\surd$ $\surd$ $\surd$ Ray and Chaudhuri (1997) $\surd$ $\surd$ Pal et al. (2014) $\surd$ $\surd$ $\surd$ $\surd$ Sarkar and Moon (2011) $\surd$ Shah et al. (2017) $\surd$ $\surd$ Teng and Chang (2005) $\surd$ $\surd$ $\surd$ Widyadana and Wee (2010) $\surd$ $\surd$ This work $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Sensitivity analysis on decision variables with respect to major parameters
 Input Parameters Output Variables and Parameters $-50\%$ $-20\%$ $0\%$ $+20\%$ $+50\%$ $T$ 1.213 1.120 0.932 0.730 0.691 $x$ 4.5 4.0 3.5 2.7 2.1 $a_1$ $r_1$ 12.7212 15.1139 16.2681 17.2527 18.1462 $r_2$ 11.58 12.86 13.42 14.23 15.70 $TP$ 1980.21 2378.00 2549.00 2645.72 2868.43 $T$ 0.647 0.720 0.839 0.906 0.987 $x$ 1.8 2.2 2.6 2.9 3.3 $c_1$ $r_1$ 19.3510 18.7220 17.5701 16.0148 14.7370 $r_2$ 15.65 13.76 12.59 11.38 10.44 $TP$ 3376.30 3048.00 2849.21 2432.43 2135.67 $T$ 1.211 1.053 0.916 0.848 0.720 $x$ 3.4 3.0 2.9 2.6 2.1 $h_1$ $r_1$ 15.5108 16.7233 17.3604 17.9451 18.4817 $r_2$ 11.55 11.89 12.10 12.93 13.41 $TP$ 3142.31 2991.00 2635.21 2307.10 2069.38 $T$ 1.120 0.984 0.764 0.793 0.807 $x$ 3.6 3.4 3.1 2.8 2.5 $b_1$ $r_1$ 16.4707 16.8574 17.3612 18.7185 20.7354 $r_2$ 10.73 11.81 12.46 12.79 13.99 $TP$ 2376.00 2578.82 2854.06 3075.56 3243.61 $T$ 0.867 0.867 0.867 0.867 0.835 $x$ 2.7 2.8 2.9 3.0 3.2 $\delta$ $r_1$ 19.4603 18.8746 18.2163 17.7563 17.3356 $r_2$ 11.77 12.13 12.85 13.23 13.79 $TP$ 2649.01 2546.24 2480.23 2465.35 2435.31 $T$ 0.979 0.964 0.932 0.881 0.831 $x$ 3.1 3.0 2.8 2.7 2.5 $c$ $r_1$ 16.3745 16.8100 17.1344 17.6007 17.9103 $r_2$ 11.45 11.79 11.97 12.32 12.85 $TP$ 3036.54 2850.23 2662.04 2567.54 2387.37 $T$ 1.158 1.074 0.964 0.793 0.549 $x$ 3.45 3.16 2.90 2.75 2.53 $\theta$ $r_1$ 14.0826 16.7284 18.2352 20.3577 23.1437 $r_2$ 9.76 10.45 13.12 15.67 17.88 $TP$ 2719.03 2680.25 2559.47 2478.53 2376.82
 Input Parameters Output Variables and Parameters $-50\%$ $-20\%$ $0\%$ $+20\%$ $+50\%$ $T$ 1.213 1.120 0.932 0.730 0.691 $x$ 4.5 4.0 3.5 2.7 2.1 $a_1$ $r_1$ 12.7212 15.1139 16.2681 17.2527 18.1462 $r_2$ 11.58 12.86 13.42 14.23 15.70 $TP$ 1980.21 2378.00 2549.00 2645.72 2868.43 $T$ 0.647 0.720 0.839 0.906 0.987 $x$ 1.8 2.2 2.6 2.9 3.3 $c_1$ $r_1$ 19.3510 18.7220 17.5701 16.0148 14.7370 $r_2$ 15.65 13.76 12.59 11.38 10.44 $TP$ 3376.30 3048.00 2849.21 2432.43 2135.67 $T$ 1.211 1.053 0.916 0.848 0.720 $x$ 3.4 3.0 2.9 2.6 2.1 $h_1$ $r_1$ 15.5108 16.7233 17.3604 17.9451 18.4817 $r_2$ 11.55 11.89 12.10 12.93 13.41 $TP$ 3142.31 2991.00 2635.21 2307.10 2069.38 $T$ 1.120 0.984 0.764 0.793 0.807 $x$ 3.6 3.4 3.1 2.8 2.5 $b_1$ $r_1$ 16.4707 16.8574 17.3612 18.7185 20.7354 $r_2$ 10.73 11.81 12.46 12.79 13.99 $TP$ 2376.00 2578.82 2854.06 3075.56 3243.61 $T$ 0.867 0.867 0.867 0.867 0.835 $x$ 2.7 2.8 2.9 3.0 3.2 $\delta$ $r_1$ 19.4603 18.8746 18.2163 17.7563 17.3356 $r_2$ 11.77 12.13 12.85 13.23 13.79 $TP$ 2649.01 2546.24 2480.23 2465.35 2435.31 $T$ 0.979 0.964 0.932 0.881 0.831 $x$ 3.1 3.0 2.8 2.7 2.5 $c$ $r_1$ 16.3745 16.8100 17.1344 17.6007 17.9103 $r_2$ 11.45 11.79 11.97 12.32 12.85 $TP$ 3036.54 2850.23 2662.04 2567.54 2387.37 $T$ 1.158 1.074 0.964 0.793 0.549 $x$ 3.45 3.16 2.90 2.75 2.53 $\theta$ $r_1$ 14.0826 16.7284 18.2352 20.3577 23.1437 $r_2$ 9.76 10.45 13.12 15.67 17.88 $TP$ 2719.03 2680.25 2559.47 2478.53 2376.82
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