doi: 10.3934/naco.2019032

Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model

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

The author, Magfura Pervin is very 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

Received  August 2018 Revised  April 2019 Published  May 2019

Fund Project: The research of Gerhard-Wilhelm Weber (Institute of Applied Mathematics, METU, 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, within project UID/MAT/ 04106/2013

A deteriorating inventory model with imperfect product and variable demand is formulated in this paper. A time-dependent deterioration factor is considered because the rate of deterioration is highly hinging on time. We introduce imperfect quality of production which leads to imperfect items in our proposed model. The retailer adopts inspection policy to pick over the perfect items from imperfect. Type Ⅰ and Type Ⅱ, both type of errors are included and the retailer invest some capital to improve the production process quality of the supplier. There is also a penalty cost for the retailer if they deliver some defective items by mistake. Sometime, there is a high amount of demand and, consequently, we assume shortages and partial backorder in our formulated model. The retailer adopts the trade-credit policy for his customers in order to promote market competition. The main objective of the paper is to show that the total cost is globally minimized and we have aimed at reducing the total cycle length, defectiveness of the system and the optimal order size by maximizing the total profit of the system. Then, we present three theorems and prove them to find an easy solution procedure to reduce the total cost of a system. The results are discussed with the help of numerical examples to approve the proposed model. A sensitivity analysis of the optimal solutions for the parameters is also provided. The paper ends with the conclusions and an outlook to possible future studies.

Citation: Sankar Kumar Roy, Magfura Pervin, Gerhard Wilhelm Weber. Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2019032
References:
[1]

S. P. Aggarwal and C. K. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, The Journal of the Operational Research Society, 46 (1995), 658-662. Google Scholar

[2]

M. BakkerJ. 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

[3]

Z. T. Balkhi, An optimal solution of a general lot size inventory model with deteriorated and imperfect products, taking into account inflation and time value of money, International Journal of Systems Science, 35 (2004), 87-96. doi: 10.1016/S0377-2217(00)00133-8. Google Scholar

[4]

L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders, Computers and Industrial Engineering, 57 (2009), 1105-1113. Google Scholar

[5]

H. J. Chang and C. Y. Dye, An inventory model for deteriorating items with partial back logging and permissible delay in payments, International Journal of Systems Science, 32 (2001), 345-352. doi: 10.1080/002077201300029700. Google Scholar

[6]

K. Chung and P. Ting, An heuristic for replenishment of deteriorating items with a linear trend in demand, Journal of the Operational Research Society, 44 (1993), 1235-1241. Google Scholar

[7]

U. Dave and L. K. Patel, $(T, S_j)$ policy inventory model for deteriorating items with time proportional demand, Journal of the Operational Research Society, 32 (1981), 137-142. doi: 10.1016/0377-2217(80)90190-3. Google Scholar

[8]

W. A. Donaldson, Inventory replenishment policy for a linear trend in demand-an analytical solution, Operational Research Quaterly, 28 (1977), 663-670. Google Scholar

[9]

C. Y. Dye, The effect of preservation technology investment on a non-instantaneous deteriorating inventory model, Omega, 41 (2013), 872-880. Google Scholar

[10]

P. M. Ghare and G. P. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243. Google Scholar

[11]

S. K. Ghosh and K. S. Chaudhuri, An order-level inventory model for a deteriorating item with Weibull distribution deterioration, time-quadratic demand and shortages, Advanced Modelling and Optimization, 6 (2004), 21-35. Google Scholar

[12]

A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand, The Journal of the Operational Research Society, 42 (1991), 1105-1110. Google Scholar

[13]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, The Journal of Operational Research Society, 36 (1985), 335-338. Google Scholar

[14]

S. K. Goyal and B. C. Giri, Recents trends in modelling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4. Google Scholar

[15]

R. W. Hall, Zero Inventories, Illinois: Dow Jones-Irwin, Homewood.Google Scholar

[16]

M. A. Hargia and L. Benkherouf, Optimal and heuristic inventory replenishment models for deteriorating items with exponential time-varying demand, European Journal of Operational Research, 79 (1994), 123-137. Google Scholar

[17]

A. K. JalanR. R. Giri and K. S. Chaudhuri, EOQ model for items with Weibull distribution deterioration, shortages and trended demand, International Journal of Systems Science, 27 (1996), 851-855. Google Scholar

[18]

M. KhanM. Y. Jaber and A. R. Ahmad, An integrated supply chain model with errors in quality inspection and learning in production, Omega, 42 (2014), 16-24. Google Scholar

[19]

H. L. Lee and M. J. Rosenblatt, Simultaneous determination of production cycles and inspection schedules in a production system, Management Science, 33 (1987), 1125-1136. Google Scholar

[20]

J. J. Liao, An EOQ model with non instantaneous receipt and exponentially deteriorating items under two-level trade credit, International Journal of Production Economics, 113 (2008), 852-861. Google Scholar

[21]

T. Y. Lin and K. L. Hou, An imperfect quality economic order quantity with advanced receiving, TOP, 23 (2015), 535-551. doi: 10.1007/s11750-014-0352-x. Google Scholar

[22]

U. MishraL. E. Cárdenas-BarrónS. TiwariA. A. Shaikh and G. Treviño-Garza, An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment, Annals of Operations Research, 9 (2015), 351-365. doi: 10.1007/s10479-017-2419-1. Google Scholar

[23]

L. Y. OuyangJ. T. Teng and L. H. Chen, Optimum ordering policy for deteriorating items with partial backlogging under permissible delay in payments, Journal of Global Optimization, 34 (2005), 245-271. doi: 10.1007/s10898-005-2604-7. Google Scholar

[24]

L. Y. OuyangL. Y. Chen and C. T. Yang, Impacts of collaborative investment and inspection policies on the integrated inventory model with defective items, International Journal of Production Research, 51 (2013), 5789-5802. Google Scholar

[25]

G. Padmanabhan and P. Vrat, EOQ models for perishable items under stock dependent selling rate, European Journal of Operational Research, 86 (1995), 281-292. Google Scholar

[26]

M. PervinG. 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. Google Scholar

[27]

M. PervinS. 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

[28]

M. PervinS. 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

[29]

M. PervinS. 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

[30]

M. PervinS. 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, 15 (2019), 1345-1373. doi: 10.3934/jimo.2018098. Google Scholar

[31]

M. Pervin, S. K. Roy and G. W. Weber, Deteriorating inventory with preservation technology under price- and stock-sensitive demand, Journal of Industrial and Management Optimization, DOI: 10.3934/jimo.2019019. doi: 10.3934/jimo.2019019. Google Scholar

[32]

E. L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Operations Research, 34 (1986), 137-144. Google Scholar

[33]

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, DOI: 10.3934/jimo.2018167. doi: 10.3934/jimo.2018167. Google Scholar

[34]

M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, International Journal of Production Economics, 64 (2000), 59-64. Google Scholar

[35]

S. S. Sana, An economic production lot size model in an imperfect production system, European Journal of Operational Research, 201 (2010), 158-170. doi: 10.1504/IJMOR.2010.033441. Google Scholar

[36]

S. S. SanaS. K. Goyal and K. S. Chaudhuri, A production inventory model for a deteriorating item with trended demand and shortages, European Journal of Operational Research, 157 (2004), 357-371. doi: 10.1016/S0377-2217(03)00222-4. Google Scholar

[37]

B. SarkarS. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Annals of Operations Research, 229 (2015), 677-702. doi: 10.1007/s10479-014-1745-9. Google Scholar

[38]

J. T. TengH. J. ChangC. Y. Dye and C. H. Hung, An optimal replenishment policy for deterioratng items with time-varying demand and partial backlogging, Operations Research Letters, 30 (2002), 387-393. doi: 10.1016/S0167-6377(02)00150-5. Google Scholar

[39]

R. P. Tripathi, Inventory model with stock-level dependent demand rate and shortages under trade credits, International Journal of Modern Mathematical Sciences, 13 (2015), 122-136. Google Scholar

[40]

H. M. Wee, A deterministic lot-size inventory model for deteriorating items with shortages and a declining market, Computers and Operations Research, 22 (1995), 345-356. Google Scholar

show all references

References:
[1]

S. P. Aggarwal and C. K. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, The Journal of the Operational Research Society, 46 (1995), 658-662. Google Scholar

[2]

M. BakkerJ. 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

[3]

Z. T. Balkhi, An optimal solution of a general lot size inventory model with deteriorated and imperfect products, taking into account inflation and time value of money, International Journal of Systems Science, 35 (2004), 87-96. doi: 10.1016/S0377-2217(00)00133-8. Google Scholar

[4]

L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders, Computers and Industrial Engineering, 57 (2009), 1105-1113. Google Scholar

[5]

H. J. Chang and C. Y. Dye, An inventory model for deteriorating items with partial back logging and permissible delay in payments, International Journal of Systems Science, 32 (2001), 345-352. doi: 10.1080/002077201300029700. Google Scholar

[6]

K. Chung and P. Ting, An heuristic for replenishment of deteriorating items with a linear trend in demand, Journal of the Operational Research Society, 44 (1993), 1235-1241. Google Scholar

[7]

U. Dave and L. K. Patel, $(T, S_j)$ policy inventory model for deteriorating items with time proportional demand, Journal of the Operational Research Society, 32 (1981), 137-142. doi: 10.1016/0377-2217(80)90190-3. Google Scholar

[8]

W. A. Donaldson, Inventory replenishment policy for a linear trend in demand-an analytical solution, Operational Research Quaterly, 28 (1977), 663-670. Google Scholar

[9]

C. Y. Dye, The effect of preservation technology investment on a non-instantaneous deteriorating inventory model, Omega, 41 (2013), 872-880. Google Scholar

[10]

P. M. Ghare and G. P. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243. Google Scholar

[11]

S. K. Ghosh and K. S. Chaudhuri, An order-level inventory model for a deteriorating item with Weibull distribution deterioration, time-quadratic demand and shortages, Advanced Modelling and Optimization, 6 (2004), 21-35. Google Scholar

[12]

A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand, The Journal of the Operational Research Society, 42 (1991), 1105-1110. Google Scholar

[13]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, The Journal of Operational Research Society, 36 (1985), 335-338. Google Scholar

[14]

S. K. Goyal and B. C. Giri, Recents trends in modelling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4. Google Scholar

[15]

R. W. Hall, Zero Inventories, Illinois: Dow Jones-Irwin, Homewood.Google Scholar

[16]

M. A. Hargia and L. Benkherouf, Optimal and heuristic inventory replenishment models for deteriorating items with exponential time-varying demand, European Journal of Operational Research, 79 (1994), 123-137. Google Scholar

[17]

A. K. JalanR. R. Giri and K. S. Chaudhuri, EOQ model for items with Weibull distribution deterioration, shortages and trended demand, International Journal of Systems Science, 27 (1996), 851-855. Google Scholar

[18]

M. KhanM. Y. Jaber and A. R. Ahmad, An integrated supply chain model with errors in quality inspection and learning in production, Omega, 42 (2014), 16-24. Google Scholar

[19]

H. L. Lee and M. J. Rosenblatt, Simultaneous determination of production cycles and inspection schedules in a production system, Management Science, 33 (1987), 1125-1136. Google Scholar

[20]

J. J. Liao, An EOQ model with non instantaneous receipt and exponentially deteriorating items under two-level trade credit, International Journal of Production Economics, 113 (2008), 852-861. Google Scholar

[21]

T. Y. Lin and K. L. Hou, An imperfect quality economic order quantity with advanced receiving, TOP, 23 (2015), 535-551. doi: 10.1007/s11750-014-0352-x. Google Scholar

[22]

U. MishraL. E. Cárdenas-BarrónS. TiwariA. A. Shaikh and G. Treviño-Garza, An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment, Annals of Operations Research, 9 (2015), 351-365. doi: 10.1007/s10479-017-2419-1. Google Scholar

[23]

L. Y. OuyangJ. T. Teng and L. H. Chen, Optimum ordering policy for deteriorating items with partial backlogging under permissible delay in payments, Journal of Global Optimization, 34 (2005), 245-271. doi: 10.1007/s10898-005-2604-7. Google Scholar

[24]

L. Y. OuyangL. Y. Chen and C. T. Yang, Impacts of collaborative investment and inspection policies on the integrated inventory model with defective items, International Journal of Production Research, 51 (2013), 5789-5802. Google Scholar

[25]

G. Padmanabhan and P. Vrat, EOQ models for perishable items under stock dependent selling rate, European Journal of Operational Research, 86 (1995), 281-292. Google Scholar

[26]

M. PervinG. 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. Google Scholar

[27]

M. PervinS. 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

[28]

M. PervinS. 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

[29]

M. PervinS. 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

[30]

M. PervinS. 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, 15 (2019), 1345-1373. doi: 10.3934/jimo.2018098. Google Scholar

[31]

M. Pervin, S. K. Roy and G. W. Weber, Deteriorating inventory with preservation technology under price- and stock-sensitive demand, Journal of Industrial and Management Optimization, DOI: 10.3934/jimo.2019019. doi: 10.3934/jimo.2019019. Google Scholar

[32]

E. L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Operations Research, 34 (1986), 137-144. Google Scholar

[33]

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, DOI: 10.3934/jimo.2018167. doi: 10.3934/jimo.2018167. Google Scholar

[34]

M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, International Journal of Production Economics, 64 (2000), 59-64. Google Scholar

[35]

S. S. Sana, An economic production lot size model in an imperfect production system, European Journal of Operational Research, 201 (2010), 158-170. doi: 10.1504/IJMOR.2010.033441. Google Scholar

[36]

S. S. SanaS. K. Goyal and K. S. Chaudhuri, A production inventory model for a deteriorating item with trended demand and shortages, European Journal of Operational Research, 157 (2004), 357-371. doi: 10.1016/S0377-2217(03)00222-4. Google Scholar

[37]

B. SarkarS. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Annals of Operations Research, 229 (2015), 677-702. doi: 10.1007/s10479-014-1745-9. Google Scholar

[38]

J. T. TengH. J. ChangC. Y. Dye and C. H. Hung, An optimal replenishment policy for deterioratng items with time-varying demand and partial backlogging, Operations Research Letters, 30 (2002), 387-393. doi: 10.1016/S0167-6377(02)00150-5. Google Scholar

[39]

R. P. Tripathi, Inventory model with stock-level dependent demand rate and shortages under trade credits, International Journal of Modern Mathematical Sciences, 13 (2015), 122-136. Google Scholar

[40]

H. M. Wee, A deterministic lot-size inventory model for deteriorating items with shortages and a declining market, Computers and Operations Research, 22 (1995), 345-356. Google Scholar

Figure 1.  Graphical representation of our proposed inventory control model
Figure 2.  Graphical description to show the convexity of the total cost. Included are $ T $, $ \alpha $ and the total cost $ R_2(T) $, along the x-axis, the y-axis and the z-axis, respectively
Figure 3.  Graphical presentation to show the convexity of the total cost. The Portrayed are $ T $, $ q $ and the total cost $ R_1(T) $, along the x-axis, the y-axis and the z-axis, respectively
Figure 4.  Graphical representation to show the convexity of the total cost. Represented are $ \alpha $, $ q $ and the total cost $ R_3(T) $, along the x-axis, the y-axis and the z-axis, respectively
Table 1.  Contributions of many authors related to inventory model
Author(s) Imperfec tness Variable demand Deterio rations Trade-credit policy Short ages
Sana et al. (2004) $ \surd $ $ \surd $ $ \surd $
Khan et al. (2014) $ \surd $
Lin and Hou (2015) $ \surd $
Teng et al. (2002) $ \surd $ $ \surd $ $ \surd $
Ghosh and Chaudhuri (2004) $ \surd $ $ \surd $ $ \surd $
Kumar et al. (2012) $ \surd $ $ \surd $
Pervin et al. (2016, a) $ \surd $ $ \surd $ $ \surd $
Mahata (2012) $ \surd $ $ \surd $
Goswami and Chaudhuri (1991) $ \surd $ $ \surd $ $ \surd $
Aggarwal and Jaggi (1995) $ \surd $ $ \surd $
Ting (2015) $ \surd $ $ \surd $
Ouyang et al. (2005) $ \surd $ $ \surd $ $ \surd $
Tripathi (2015) $ \surd $ $ \surd $ $ \surd $
Annadurai and Uthayakumar (2015) $ \surd $ $ \surd $ $ \surd $
Balkhi (2004) $ \surd $ $ \surd $
Pervin et al. (2016, b) $ \surd $ $ \surd $ $ \surd $
Our paper $ \surd $ $ \surd $ $ \surd $ $ \surd $ $ \surd $
Author(s) Imperfec tness Variable demand Deterio rations Trade-credit policy Short ages
Sana et al. (2004) $ \surd $ $ \surd $ $ \surd $
Khan et al. (2014) $ \surd $
Lin and Hou (2015) $ \surd $
Teng et al. (2002) $ \surd $ $ \surd $ $ \surd $
Ghosh and Chaudhuri (2004) $ \surd $ $ \surd $ $ \surd $
Kumar et al. (2012) $ \surd $ $ \surd $
Pervin et al. (2016, a) $ \surd $ $ \surd $ $ \surd $
Mahata (2012) $ \surd $ $ \surd $
Goswami and Chaudhuri (1991) $ \surd $ $ \surd $ $ \surd $
Aggarwal and Jaggi (1995) $ \surd $ $ \surd $
Ting (2015) $ \surd $ $ \surd $
Ouyang et al. (2005) $ \surd $ $ \surd $ $ \surd $
Tripathi (2015) $ \surd $ $ \surd $ $ \surd $
Annadurai and Uthayakumar (2015) $ \surd $ $ \surd $ $ \surd $
Balkhi (2004) $ \surd $ $ \surd $
Pervin et al. (2016, b) $ \surd $ $ \surd $ $ \surd $
Our paper $ \surd $ $ \surd $ $ \surd $ $ \surd $ $ \surd $
Table 2.  Computational result. Bold style represents the optimal solution
$ q $ $ \alpha^* $ $ T_1^* $ $ T_2^* $ $ T_3^* $ $ R_1(T,q,\alpha) $ $ R_2(T,q,\alpha) $ $ R_3(T,q,\alpha) $
$ 1 $ 0.1077 1.235 1.302 1.187 3478.12 3599.45 3517.30
$ 2 $ 0.1125 1.301 1.213 1.171 3560.82 3011.67 3069.57
$ 3 $ 0.1092 1.420 1.387 1.432 3673.19 3095.07 3112.70
$ {\bf{4}} $ 0.1089 1.324 1.112 1.108 3488.22 2973.16 2844.93
$ 5 $ 0.1107 1.403 1.274 1.392 3579.08 3077.45 3205.71
$ 6 $ 0.1073 1.449 1.171 1.321 3678.99 3360.21 3119.54
$ 7 $ 0.128 1.610 1.534 1.558 3879.04 3232.70 3374.56
$ 8 $ 0.116 1.791 1.645 1.325 3927.34 3306.18 3317.84
$ 9 $ 0.125 1.570 1.349 1.102 4012.57 3812.05 3518.38
$ 10 $ 0.144 1.397 1.747 1.560 3939.55 3670.10 3575.68
$ q $ $ \alpha^* $ $ T_1^* $ $ T_2^* $ $ T_3^* $ $ R_1(T,q,\alpha) $ $ R_2(T,q,\alpha) $ $ R_3(T,q,\alpha) $
$ 1 $ 0.1077 1.235 1.302 1.187 3478.12 3599.45 3517.30
$ 2 $ 0.1125 1.301 1.213 1.171 3560.82 3011.67 3069.57
$ 3 $ 0.1092 1.420 1.387 1.432 3673.19 3095.07 3112.70
$ {\bf{4}} $ 0.1089 1.324 1.112 1.108 3488.22 2973.16 2844.93
$ 5 $ 0.1107 1.403 1.274 1.392 3579.08 3077.45 3205.71
$ 6 $ 0.1073 1.449 1.171 1.321 3678.99 3360.21 3119.54
$ 7 $ 0.128 1.610 1.534 1.558 3879.04 3232.70 3374.56
$ 8 $ 0.116 1.791 1.645 1.325 3927.34 3306.18 3317.84
$ 9 $ 0.125 1.570 1.349 1.102 4012.57 3812.05 3518.38
$ 10 $ 0.144 1.397 1.747 1.560 3939.55 3670.10 3575.68
Table 3.  Sensitivity analysis for various parameters involved in Example 1
Para meter $ \% $ change value $ T_1^* $ $ T_2^* $ $ T_3^* $ $ R_1(T) $ $ R_2(T) $ $ R_3(T) $ Total Cost Profit
$ A $ +50 1350 1.832 1.572 1.624 3122.05 2705.91 2812.83 $ \downarrow $ $ \uparrow $
+30 1170 1.510 1.480 1.613 2988.71 2633.61 2694.74 $ \downarrow $ $ \uparrow $
+10 990 1.257 1.437 1.589 2734.00 2482.09 2503.15 $ \downarrow $ $ \uparrow $
-10 810 0.941 1.428 1.541 2613.47 2290.41 2387.40 $ \downarrow $ $ \uparrow $
-30 630 0.703 1.410 1.516 2497.38 2100.16 2206.85 $ \downarrow $ $ \uparrow $
-50 450 0.627 1.377 1.487 2206.16 2000.00 2183.57 $ \downarrow $ $ \uparrow $
$ s $ +50 600 0.904 0.867 0.880 3385.10 2834.61 2791.38 $ \downarrow $ $ \uparrow $
+30 520 0.760 0.843 0.861 3174.51 2690.42 2614.50 $ \downarrow $ $ \uparrow $
+10 440 0.601 0.829 0.847 2893.67 2517.83 2576.31 $ \downarrow $ $ \uparrow $
-10 360 0.472 0.820 0.831 2635.85 2385.76 2344.10 $ \downarrow $ $ \uparrow $
-30 280 0.275 0.816 0.822 2483.77 2189.02 2208.47 $ \downarrow $ $ \uparrow $
-50 200 0.064 0.803 0.815 2305.38 1985.47 2083.17 $ \downarrow $ $ \uparrow $
$ h_1 $ +50 150 1.966 1.973 1.989 3192.70 2911.56 2810.55 $ \downarrow $ $ \uparrow $
+30 130 1.728 1.854 1.910 2948.16 2738.64 2655.18 $ \downarrow $ $ \uparrow $
+10 110 1.601 1.779 1.878 2749.03 2500.53 2451.38 $ \downarrow $ $ \uparrow $
-10 90 1.486 1.611 1.763 2605.74 2374.95 2180.76 $ \downarrow $ $ \uparrow $
-30 70 1.310 1.572 1.683 2477.19 2285.31 2090.59 $ \downarrow $ $ \uparrow $
-50 50 1.173 1.489 1.557 2204.60 2069.73 1877.92 $ \downarrow $ $ \uparrow $
$ h_2 $ +50 300 2.765 2.581 2.593 3592.00 2746.31 2522.20 $ \downarrow $ $ \uparrow $
+30 260 2.303 2.505 2.512 3386.17 2506.57 2257.46 $ \downarrow $ $ \uparrow $
+10 220 2.007 2.488 2.502 3152.04 2344.10 2037.18 $ \downarrow $ $ \uparrow $
-10 180 1.841 2.394 2.581 2819.11 2183.51 1734.62 $ \downarrow $ $ \uparrow $
-30 140 1.573 2.353 2.473 2540.83 1982.86 1504.27 $ \downarrow $ $ \uparrow $
-50 100 1.250 2.279 2.371 2274.62 1710.27 1374.02 $ \downarrow $ $ \uparrow $
$ a $ +50 22.5 2.791 2.880 2.898 2916.47 3721.40 2883.45 $ \downarrow $ $ \uparrow $
+30 19.5 2.675 2.741 2.805 2803.15 3569.00 2761.74 $ \downarrow $ $ \uparrow $
+10 16.5 2.533 2.689 2.775 2785.32 3477.54 2653.14 $ \downarrow $ $ \uparrow $
-10 13.5 2.412 2.537 2.643 2511.63 3307.11 2578.39 $ \downarrow $ $ \uparrow $
-30 10.5 2.370 2.481 2.550 2479.59 3258.46 2386.55 $ \downarrow $ $ \uparrow $
-50 7.5 2.264 2.372 2.445 2333.16 3117.35 2290.28 $ \downarrow $ $ \uparrow $
$ b $ +50 30 2.672 2.503 2.475 3599.89 2800.00 2795.64 $ \downarrow $ $ \uparrow $
+30 26 2.538 2.484 2.449 3512.04 2715.35 2683.05 $ \downarrow $ $ \uparrow $
+10 22 2.460 2.336 2.429 3475.68 2610.97 2500.34 $ \downarrow $ $ \uparrow $
-10 18 2.397 2.260 2.405 3381.62 2573.61 2435.65 $ \downarrow $ $ \uparrow $
-30 14 2.200 2.175 2.391 3260.91 2483.29 2362.41 $ \downarrow $ $ \uparrow $
-50 10 2.822 2.124 2.364 3129.56 2321.79 2155.20 $ \downarrow $ $ \uparrow $
$ \gamma $ +50 0.60 1.754 2.134 2.431 3529.87 3138.56 2490.18 $ \downarrow $ $ \uparrow $
+30 0.52 1.694 2.089 2.379 3348.72 2763.79 2317.93 $ \downarrow $ $ \uparrow $
+10 0.44 1.504 1.905 2.248 3095.23 2480.58 2188.37 $ \downarrow $ $ \uparrow $
-10 0.36 1.438 1.881 2.179 2860.93 2201.49 2020.99 $ \downarrow $ $ \uparrow $
-30 0.28 1.215 1.753 2.098 2547.65 1973.10 1845.73 $ \downarrow $ $ \uparrow $
-50 0.20 1.093 1.668 1.979 2196.37 1748.53 1549.27 $ \downarrow $ $ \uparrow $
$ r_1 $ +50 75 2.402 1.402 1.657 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
+30 65 2.763 1.435 1.661 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
+10 55 2.911 1.492 1.704 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
-10 45 3.250 1.555 1.783 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
-30 35 3.619 1.784 2.044 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
-50 25 3.873 1.976 2.423 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
Para meter $ \% $ change value $ T_1^* $ $ T_2^* $ $ T_3^* $ $ R_1(T) $ $ R_2(T) $ $ R_3(T) $ Total Cost Profit
$ A $ +50 1350 1.832 1.572 1.624 3122.05 2705.91 2812.83 $ \downarrow $ $ \uparrow $
+30 1170 1.510 1.480 1.613 2988.71 2633.61 2694.74 $ \downarrow $ $ \uparrow $
+10 990 1.257 1.437 1.589 2734.00 2482.09 2503.15 $ \downarrow $ $ \uparrow $
-10 810 0.941 1.428 1.541 2613.47 2290.41 2387.40 $ \downarrow $ $ \uparrow $
-30 630 0.703 1.410 1.516 2497.38 2100.16 2206.85 $ \downarrow $ $ \uparrow $
-50 450 0.627 1.377 1.487 2206.16 2000.00 2183.57 $ \downarrow $ $ \uparrow $
$ s $ +50 600 0.904 0.867 0.880 3385.10 2834.61 2791.38 $ \downarrow $ $ \uparrow $
+30 520 0.760 0.843 0.861 3174.51 2690.42 2614.50 $ \downarrow $ $ \uparrow $
+10 440 0.601 0.829 0.847 2893.67 2517.83 2576.31 $ \downarrow $ $ \uparrow $
-10 360 0.472 0.820 0.831 2635.85 2385.76 2344.10 $ \downarrow $ $ \uparrow $
-30 280 0.275 0.816 0.822 2483.77 2189.02 2208.47 $ \downarrow $ $ \uparrow $
-50 200 0.064 0.803 0.815 2305.38 1985.47 2083.17 $ \downarrow $ $ \uparrow $
$ h_1 $ +50 150 1.966 1.973 1.989 3192.70 2911.56 2810.55 $ \downarrow $ $ \uparrow $
+30 130 1.728 1.854 1.910 2948.16 2738.64 2655.18 $ \downarrow $ $ \uparrow $
+10 110 1.601 1.779 1.878 2749.03 2500.53 2451.38 $ \downarrow $ $ \uparrow $
-10 90 1.486 1.611 1.763 2605.74 2374.95 2180.76 $ \downarrow $ $ \uparrow $
-30 70 1.310 1.572 1.683 2477.19 2285.31 2090.59 $ \downarrow $ $ \uparrow $
-50 50 1.173 1.489 1.557 2204.60 2069.73 1877.92 $ \downarrow $ $ \uparrow $
$ h_2 $ +50 300 2.765 2.581 2.593 3592.00 2746.31 2522.20 $ \downarrow $ $ \uparrow $
+30 260 2.303 2.505 2.512 3386.17 2506.57 2257.46 $ \downarrow $ $ \uparrow $
+10 220 2.007 2.488 2.502 3152.04 2344.10 2037.18 $ \downarrow $ $ \uparrow $
-10 180 1.841 2.394 2.581 2819.11 2183.51 1734.62 $ \downarrow $ $ \uparrow $
-30 140 1.573 2.353 2.473 2540.83 1982.86 1504.27 $ \downarrow $ $ \uparrow $
-50 100 1.250 2.279 2.371 2274.62 1710.27 1374.02 $ \downarrow $ $ \uparrow $
$ a $ +50 22.5 2.791 2.880 2.898 2916.47 3721.40 2883.45 $ \downarrow $ $ \uparrow $
+30 19.5 2.675 2.741 2.805 2803.15 3569.00 2761.74 $ \downarrow $ $ \uparrow $
+10 16.5 2.533 2.689 2.775 2785.32 3477.54 2653.14 $ \downarrow $ $ \uparrow $
-10 13.5 2.412 2.537 2.643 2511.63 3307.11 2578.39 $ \downarrow $ $ \uparrow $
-30 10.5 2.370 2.481 2.550 2479.59 3258.46 2386.55 $ \downarrow $ $ \uparrow $
-50 7.5 2.264 2.372 2.445 2333.16 3117.35 2290.28 $ \downarrow $ $ \uparrow $
$ b $ +50 30 2.672 2.503 2.475 3599.89 2800.00 2795.64 $ \downarrow $ $ \uparrow $
+30 26 2.538 2.484 2.449 3512.04 2715.35 2683.05 $ \downarrow $ $ \uparrow $
+10 22 2.460 2.336 2.429 3475.68 2610.97 2500.34 $ \downarrow $ $ \uparrow $
-10 18 2.397 2.260 2.405 3381.62 2573.61 2435.65 $ \downarrow $ $ \uparrow $
-30 14 2.200 2.175 2.391 3260.91 2483.29 2362.41 $ \downarrow $ $ \uparrow $
-50 10 2.822 2.124 2.364 3129.56 2321.79 2155.20 $ \downarrow $ $ \uparrow $
$ \gamma $ +50 0.60 1.754 2.134 2.431 3529.87 3138.56 2490.18 $ \downarrow $ $ \uparrow $
+30 0.52 1.694 2.089 2.379 3348.72 2763.79 2317.93 $ \downarrow $ $ \uparrow $
+10 0.44 1.504 1.905 2.248 3095.23 2480.58 2188.37 $ \downarrow $ $ \uparrow $
-10 0.36 1.438 1.881 2.179 2860.93 2201.49 2020.99 $ \downarrow $ $ \uparrow $
-30 0.28 1.215 1.753 2.098 2547.65 1973.10 1845.73 $ \downarrow $ $ \uparrow $
-50 0.20 1.093 1.668 1.979 2196.37 1748.53 1549.27 $ \downarrow $ $ \uparrow $
$ r_1 $ +50 75 2.402 1.402 1.657 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
+30 65 2.763 1.435 1.661 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
+10 55 2.911 1.492 1.704 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
-10 45 3.250 1.555 1.783 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
-30 35 3.619 1.784 2.044 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
-50 25 3.873 1.976 2.423 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
Table 4.  Sensitivity analysis for various parameters involved in Example 1
Para meter $ \% $ change value $ T_1^* $ $ T_2^* $ $ T_3^* $ $ R_1(T) $ $ R_2(T) $ $ R_3(T) $ Total Cost Profit
$ r_2 $ +50 60 2.402 2.265 1.875 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
+30 52 2.257 2.203 1.713 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
+10 44 2.105 2.165 1.681 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
-10 36 1.943 2.081 1.578 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
-30 28 1.870 1.863 1.475 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
-50 20 1.655 1.756 1.409 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
$ r_3 $ +50 15 2.795 2.531 2.320 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
+30 13 2.661 2.457 2.259 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
+10 11 2.573 2.347 2.188 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
-10 9 2.412 2.201 2.096 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
-30 7 2.345 2.179 2.016 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
-50 5 2.235 2.153 1.875 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
$ r_4 $ +50 60 1.987 1.853 1.760 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
+30 52 1.664 1.571 1.483 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
+10 44 1.370 1.356 1.338 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
-10 36 1.297 1.283 1.275 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
-30 28 1.264 1.257 1.241 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
-50 20 1.239 1.221 1.195 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
$ r_5 $ +50 45 2.354 2.279 2.165 3567.10 3049.31 3098.21 $ \downarrow $ $ \uparrow $
+30 39 2.170 2.086 1.952 3417.05 2994.22 2947.18 $ \downarrow $ $ \uparrow $
+10 33 1.941 1.876 1.740 3307.53 2885.31 2856.37 $ \downarrow $ $ \uparrow $
-10 27 1.769 1.635 1.509 3268.37 2765.42 2664.00 $ \downarrow $ $ \uparrow $
-30 21 1.526 1.470 1.358 3191.30 2670.51 2509.11 $ \downarrow $ $ \uparrow $
-50 15 1.391 1.258 1.174 3034.67 2500.00 2481.03 $ \downarrow $ $ \uparrow $
$ \beta $ +50 0.45 1.310 1.465 1.589 2473.11 1730.22 2017.27 $ \uparrow $ $ \downarrow $
+30 0.39 1.574 1.640 1.736 2705.43 1918.47 2384.53 $ \uparrow $ $ \downarrow $
+10 0.33 1.812 1.922 1.989 3058.24 2164.38 2673.40 $ \uparrow $ $ \downarrow $
-10 0.27 2.123 2.070 2.145 3347.80 2351.06 2945.82 $ \uparrow $ $ \downarrow $
-30 0.21 2.492 2.195 2.287 3692.47 2537.64 3255.70 $ \uparrow $ $ \downarrow $
-50 0.15 2.760 2.358 2.460 3903.12 2799.00 3509.35 $ \uparrow $ $ \downarrow $
$ \delta $ +50 30 2.385 2.072 2.110 3392.46 2918.36 2840.79 $ \downarrow $ $ \uparrow $
+30 26 2.200 1.915 2.048 3175.13 2875.42 2711.54 $ \downarrow $ $ \uparrow $
+10 22 2.071 1.854 1.991 2917.00 2704.60 2657.18 $ \downarrow $ $ \uparrow $
-10 18 1.848 1.774 1.867 2739.26 2538.11 2302.63 $ \downarrow $ $ \uparrow $
-30 14 1.639 1.640 1.790 2543.35 2326.83 2199.29 $ \downarrow $ $ \uparrow $
-50 10 1.325 1.573 1.615 2319.67 2100.00 1978.63 $ \downarrow $ $ \uparrow $
$ M $ +50 1.8 1.367 1.470 1.585 3141.47 2758.10 2890.05 $ \uparrow $ $ \downarrow $
+30 1.56 1.532 1.609 1.779 3250.26 2893.55 2918.62 $ \uparrow $ $ \downarrow $
+10 1.32 1.727 1.826 1.963 3319.51 2964.28 3022.37 $ \uparrow $ $ \downarrow $
-10 1.08 1.904 1.995 2.057 3471.43 3027.83 3126.29 $ \uparrow $ $ \downarrow $
-30 0.84 2.157 2.257 2.376 3520.79 3119.44 3257.81 $ \uparrow $ $ \downarrow $
-50 0.6 2.370 2.450 2.538 3752.00 3348.63 3486.90 $ \uparrow $ $ \downarrow $
$ \theta $ +50 0.6 2.752 2.568 2.330 4180.57 3042.78 3506.79 $ \downarrow $ $ \uparrow $
+30 0.52 2.695 2.418 2.257 3912.36 2965.03 3481.20 $ \downarrow $ $ \uparrow $
+10 0.44 2.516 2.343 2.194 3764.25 2860.24 3358.18 $ \downarrow $ $ \uparrow $
-10 0.36 2.404 2.267 2.210 3580.19 2725.57 3293.49 $ \downarrow $ $ \uparrow $
-30 0.28 2.279 2.001 1.935 3318.38 2530.29 3167.16 $ \downarrow $ $ \uparrow $
-50 0.20 2.011 1.883 1.520 3127.93 2350.03 2948.57 $ \downarrow $ $ \uparrow $
Para meter $ \% $ change value $ T_1^* $ $ T_2^* $ $ T_3^* $ $ R_1(T) $ $ R_2(T) $ $ R_3(T) $ Total Cost Profit
$ r_2 $ +50 60 2.402 2.265 1.875 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
+30 52 2.257 2.203 1.713 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
+10 44 2.105 2.165 1.681 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
-10 36 1.943 2.081 1.578 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
-30 28 1.870 1.863 1.475 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
-50 20 1.655 1.756 1.409 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
$ r_3 $ +50 15 2.795 2.531 2.320 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
+30 13 2.661 2.457 2.259 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
+10 11 2.573 2.347 2.188 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
-10 9 2.412 2.201 2.096 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
-30 7 2.345 2.179 2.016 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
-50 5 2.235 2.153 1.875 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
$ r_4 $ +50 60 1.987 1.853 1.760 3891.57 3124.51 3054.36 $ \downarrow $ $ \uparrow $
+30 52 1.664 1.571 1.483 3522.56 2910.28 2839.17 $ \downarrow $ $ \uparrow $
+10 44 1.370 1.356 1.338 3205.84 2764.33 2780.10 $ \downarrow $ $ \uparrow $
-10 36 1.297 1.283 1.275 2935.27 2459.37 2638.61 $ \downarrow $ $ \uparrow $
-30 28 1.264 1.257 1.241 2789.42 2218.06 2529.50 $ \downarrow $ $ \uparrow $
-50 20 1.239 1.221 1.195 2642.17 2049.62 2485.73 $ \downarrow $ $ \uparrow $
$ r_5 $ +50 45 2.354 2.279 2.165 3567.10 3049.31 3098.21 $ \downarrow $ $ \uparrow $
+30 39 2.170 2.086 1.952 3417.05 2994.22 2947.18 $ \downarrow $ $ \uparrow $
+10 33 1.941 1.876 1.740 3307.53 2885.31 2856.37 $ \downarrow $ $ \uparrow $
-10 27 1.769 1.635 1.509 3268.37 2765.42 2664.00 $ \downarrow $ $ \uparrow $
-30 21 1.526 1.470 1.358 3191.30 2670.51 2509.11 $ \downarrow $ $ \uparrow $
-50 15 1.391 1.258 1.174 3034.67 2500.00 2481.03 $ \downarrow $ $ \uparrow $
$ \beta $ +50 0.45 1.310 1.465 1.589 2473.11 1730.22 2017.27 $ \uparrow $ $ \downarrow $
+30 0.39 1.574 1.640 1.736 2705.43 1918.47 2384.53 $ \uparrow $ $ \downarrow $
+10 0.33 1.812 1.922 1.989 3058.24 2164.38 2673.40 $ \uparrow $ $ \downarrow $
-10 0.27 2.123 2.070 2.145 3347.80 2351.06 2945.82 $ \uparrow $ $ \downarrow $
-30 0.21 2.492 2.195 2.287 3692.47 2537.64 3255.70 $ \uparrow $ $ \downarrow $
-50 0.15 2.760 2.358 2.460 3903.12 2799.00 3509.35 $ \uparrow $ $ \downarrow $
$ \delta $ +50 30 2.385 2.072 2.110 3392.46 2918.36 2840.79 $ \downarrow $ $ \uparrow $
+30 26 2.200 1.915 2.048 3175.13 2875.42 2711.54 $ \downarrow $ $ \uparrow $
+10 22 2.071 1.854 1.991 2917.00 2704.60 2657.18 $ \downarrow $ $ \uparrow $
-10 18 1.848 1.774 1.867 2739.26 2538.11 2302.63 $ \downarrow $ $ \uparrow $
-30 14 1.639 1.640 1.790 2543.35 2326.83 2199.29 $ \downarrow $ $ \uparrow $
-50 10 1.325 1.573 1.615 2319.67 2100.00 1978.63 $ \downarrow $ $ \uparrow $
$ M $ +50 1.8 1.367 1.470 1.585 3141.47 2758.10 2890.05 $ \uparrow $ $ \downarrow $
+30 1.56 1.532 1.609 1.779 3250.26 2893.55 2918.62 $ \uparrow $ $ \downarrow $
+10 1.32 1.727 1.826 1.963 3319.51 2964.28 3022.37 $ \uparrow $ $ \downarrow $
-10 1.08 1.904 1.995 2.057 3471.43 3027.83 3126.29 $ \uparrow $ $ \downarrow $
-30 0.84 2.157 2.257 2.376 3520.79 3119.44 3257.81 $ \uparrow $ $ \downarrow $
-50 0.6 2.370 2.450 2.538 3752.00 3348.63 3486.90 $ \uparrow $ $ \downarrow $
$ \theta $ +50 0.6 2.752 2.568 2.330 4180.57 3042.78 3506.79 $ \downarrow $ $ \uparrow $
+30 0.52 2.695 2.418 2.257 3912.36 2965.03 3481.20 $ \downarrow $ $ \uparrow $
+10 0.44 2.516 2.343 2.194 3764.25 2860.24 3358.18 $ \downarrow $ $ \uparrow $
-10 0.36 2.404 2.267 2.210 3580.19 2725.57 3293.49 $ \downarrow $ $ \uparrow $
-30 0.28 2.279 2.001 1.935 3318.38 2530.29 3167.16 $ \downarrow $ $ \uparrow $
-50 0.20 2.011 1.883 1.520 3127.93 2350.03 2948.57 $ \downarrow $ $ \uparrow $
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