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

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doi: 10.3934/naco.2019032

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

Sankar Kumar Roy ^1,, , Magfura Pervin ^1, and Gerhard Wilhelm Weber ^2,

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.

Keywords: Inventory, Variable deterioration, Imperfect quality items, Time-dependent demand, Inspection policy, Trade-credit, Optimization.

Mathematics Subject Classification: Primary: 90B05, 91A10; Secondary: 90C26.

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. 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
[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. Jalan, R. 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. Khan, M. 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. Mishra, L. E. Cárdenas-Barrón, S. Tiwari, A. 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. Ouyang, J. 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. Ouyang, L. 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. 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. Google Scholar
[27]	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
[28]	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
[29]	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
[30]	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, 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. Sana, S. 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. Sarkar, S. 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. Teng, H. J. Chang, C. 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. 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
[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. Jalan, R. 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. Khan, M. 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. Mishra, L. E. Cárdenas-Barrón, S. Tiwari, A. 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. Ouyang, J. 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. Ouyang, L. 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. 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. Google Scholar
[27]	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
[28]	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
[29]	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
[30]	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, 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. Sana, S. 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. Sarkar, S. 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. Teng, H. J. Chang, C. 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

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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

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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

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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

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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 $

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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

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$ 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 $

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Download as excel

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 $

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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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American Institute of Mathematical Sciences