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Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers
An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment
Department of Industrial Engineering, Faculty of Engineering Kharazmi University, 15719-14911 Tehran, Iran |
In this paper investigate a production system with the defective quality process and preventive maintenance to establish the inspection policy and optimum inventory level for production items with considering uncertainty environment. The shortage occurs because of preventive maintenance and is considered as partial backlogging. Through the production process, at a random moment, the production of items from the state in-control turns into an out-of-control mode, so that parts of the defective product are manufactured an in-control state and outside of the control process mode. The online item inspection process begins after a time variable through the production period. The human inspection process has also been considered for the classify of defective goods. Uninspected products are accepted to the customer/buyer with minimal repair warranty and the defective items classify by the inspector at fixed cost before being shipped subject to salvaged items. Also, the inspection process of manufactured goods includes a human inspection error. Therefore, two types of classification errors (Type Ⅰ & Ⅱ) are considered to be more realistic than the proposed model. The input parameters of the model are considered as a triangular fuzzy environment, and the output parameters of the model are solved by the Zadeh's extension principle and nonlinear parametric programming. As a final point, a numerical example by graphical representations is obtainable to illustrate the proposed model.
References:
[1] |
M. Al-Salamah,
Economic production quantity in batch manufacturing with imperfect quality, imperfect inspection, and destructive and non-destructive acceptance sampling in a two-tier market, Comput. Ind. Eng., 93 (2016), 275-285.
doi: 10.1016/j.cie.2015.12.022. |
[2] |
A. Baykasoǧlu, K. Subulan and F. S. Karaslan,
A new fuzzy linear assignment method for multi-attribute decision making with an application to spare parts inventory classification, Appl. Soft Comput. J., 42 (2016), 1-17.
doi: 10.1016/j.asoc.2016.01.031. |
[3] |
B. Bouslah, A. Gharbi and R. Pellerin,
Integrated production, sampling quality control and maintenance of deteriorating production systems with AOQL constraint, Omega (United Kingdom), 61 (2016), 110-126.
doi: 10.1016/j.omega.2015.07.012. |
[4] |
S.-C. Chang, J.-S. Yao and H.-M. Lee,
Economic reorder point for fuzzy backorder quantity, Omega (United Kingdom), 109 (1998), 183-202.
doi: 10.1016/S0377-2217(97)00069-6. |
[5] |
Y. C. Chen,
An optimal production and inspection strategy with preventive maintenance error and rework, J. Manuf. Syst., 32 (2013), 99-106.
doi: 10.1016/j.jmsy.2012.07.010. |
[6] |
K. J. Chung, C. C. Her and S. D. Lin,
A two-warehouse inventory model with imperfect quality production processes, Comput. Ind. Eng., 56 (2009), 193-197.
doi: 10.1016/j.cie.2008.05.005. |
[7] |
S. K. De and S. S. Sana,
Fuzzy order quantity inventory model with fuzzy shortage quantity and fuzzy promotional index, Econ. Model., 31 (2013), 351-358.
doi: 10.1016/j.econmod.2012.11.046. |
[8] |
M. Farhangi, S. T. A. Niaki and B. Maleki Vishkaei, Closed-form equations for optimal lot sizing in deterministic EOQ models with exchangeable imperfect quality items, Sci. Iran., 22 (2015), 2621-2634. Google Scholar |
[9] |
Z. Hauck and J. Vörös,
Lot sizing in case of defective items with investments to increase the speed of quality control, Omega (United Kingdom), 52 (2015), 180-189.
doi: 10.1016/j.omega.2014.04.004. |
[10] |
J. T. Hsu and L. F. Hsu,
Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns, Comput. Ind. Eng., 64 (2013), 389-402.
doi: 10.1016/j.cie.2012.10.005. |
[11] |
J. T. Hsu and L. F. Hsu,
An EOQ model with imperfect quality items inspection errors shortage backordering and sales returns, Intern. J. Prod. Econ., 143 (2013), 162-170.
doi: 10.1016/j.ijpe.2012.12.025. |
[12] |
J. S. Hu, H. Zheng, C. Y. Guo and Y. P. Ji,
Optimal production run length with imperfect production processes and backorder in fuzzy random environment, Comput. Ind. Eng., 59 (2010), 9-15.
doi: 10.1016/j.cie.2010.01.012. |
[13] |
H. Ishii and T. Konno,
A stochastic inventory problem with fuzzy shortage cost, Eur. J. Oper. Res., 106 (1998), 90-94.
doi: 10.1016/S0377-2217(97)00173-2. |
[14] |
M. Y. Jaber and S. K. Goyal,
Economic production quantity model for items with imperfect quality subject to learning effects, Int. J. Prod. Econ., 115 (2008), 143-150.
doi: 10.1016/j.ijpe.2008.05.007. |
[15] |
D. K. Jana, B. Das and M. Maiti,
Multi-item partial backlogging inventory models over random planninghorizon in random fuzzy environment, Appl. Soft Comput. J., 21 (2014), 12-27.
doi: 10.1016/j.asoc.2014.02.021. |
[16] |
S. Karmakar, S. K. De and A. Goswami,
A pollution sensitive dense fuzzy economic production quantity model with cycle time dependent production rate, J. Clean. Prod., 154 (2017), 139-150.
doi: 10.1016/j.jclepro.2017.03.080. |
[17] |
N. Kazemi, E. Shekarian, L. E. Cárdenas-Barrón and E. U. Olugu,
Incorporating human learning into a fuzzy EOQ inventory model with backorders, Comput. Ind. Eng., 87 (2015), 540-542.
doi: 10.1016/j.cie.2015.05.014. |
[18] |
M. Khan, M. Y. Jaber and M. Bonney,
An economic order quantity (EOQ) for items with imperfect quality and inspection errors, Int. J. Prod. Econ., 133 (2011), 113-118.
doi: 10.1016/j.ijpe.2010.01.023. |
[19] |
M. Khan, M. Y. Jaber and M. I. M. Wahab,
Economic order quantity model for items with imperfect quality with learning in inspection, Int. J. Prod. Econ., 124 (2010), 87-96.
doi: 10.1016/j.ijpe.2009.10.011. |
[20] |
C. Krishnamoorthi,
An inventory model for product life cycle with maturity stage and defective items, Opsearch, 49 (2012), 209-222.
doi: 10.1007/s12597-012-0075-4. |
[21] |
T. Y. Lin,
Coordination policy for a two-stage supply chain considering quantity discounts and overlapped delivery with imperfect quality, Comput. Ind. Eng., 66 (2013), 53-62.
doi: 10.1016/j.cie.2013.06.012. |
[22] |
J. Liu and H. Zheng,
Fuzzy economic order quantity model with imperfect items, shortages and inspection errors, Syst. Eng. Procedia, 4 (2012), 282-289.
doi: 10.1016/j.sepro.2011.11.077. |
[23] |
W. N. Ma, D. C. Gong and G. C. Lin,
An optimal common production cycle time for imperfect production processes with scrap, Math. Comput. Model., 52 (2010), 724-737.
doi: 10.1016/j.mcm.2010.04.024. |
[24] |
B. Pal, S. S. Sana and K. Chaudhuri,
A mathematical model on EPQ for stochastic demand in an imperfect production system, J. Manuf. Syst., 32 (2013), 260-270.
doi: 10.1016/j.jmsy.2012.11.009. |
[25] |
S. Papachristos and I. Konstantaras,
Economic ordering quantity models for items with imperfect quality, Int. J. Prod. Econ., 100 (2006), 148-154.
doi: 10.1016/j.ijpe.2004.11.004. |
[26] |
M. A. Rahim and H. Ohta,
An integrated economic model for inventory and quality control problems, Eng. Optim., 37 (2005), 65-81.
doi: 10.1080/0305215042000268598. |
[27] |
J. Sadeghi, S. M. Mousavi and S. T. A. Niaki,
Optimizing an inventory model with fuzzy demand, backordering, and discount using a hybrid imperialist competitive algorithm, Appl. Math. Model., 40 (2016), 7318-7335.
doi: 10.1016/j.apm.2016.03.013. |
[28] |
M. K. Salameh and M. Y. Jaber,
Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ., 64 (2000), 59-64.
doi: 10.1016/S0925-5273(99)00044-4. |
[29] |
N. K. Samal and D. K. Pratihar,
Optimization of variable demand fuzzy economic order quantity inventory models without and with backordering, Comput. Ind. Eng., 78 (2014), 148-162.
doi: 10.1016/j.cie.2014.10.006. |
[30] |
S. S. Sana,
A production-inventory model of imperfect quality products in a three-layer supply chain, Decis. Support Syst., 50 (2011), 539-547.
doi: 10.1016/j.dss.2010.11.012. |
[31] |
J. Taheri-Tolgari, A. Mirzazadeh and F. Jolai,
An inventory model for imperfect items under inflationary conditions with considering inspection errors, Comput. Math. with Appl., 63 (2012), 1007-1019.
doi: 10.1016/j.camwa.2011.09.050. |
[32] |
J. Wu, K. Skouri, J. T. Teng and Y. Hu,
Two inventory systems with trapezoidal-type demand rate and time-dependent deterioration and backlogging, Expert Syst. Appl., 46 (2016), 367-379.
doi: 10.1016/j.eswa.2015.10.048. |
show all references
References:
[1] |
M. Al-Salamah,
Economic production quantity in batch manufacturing with imperfect quality, imperfect inspection, and destructive and non-destructive acceptance sampling in a two-tier market, Comput. Ind. Eng., 93 (2016), 275-285.
doi: 10.1016/j.cie.2015.12.022. |
[2] |
A. Baykasoǧlu, K. Subulan and F. S. Karaslan,
A new fuzzy linear assignment method for multi-attribute decision making with an application to spare parts inventory classification, Appl. Soft Comput. J., 42 (2016), 1-17.
doi: 10.1016/j.asoc.2016.01.031. |
[3] |
B. Bouslah, A. Gharbi and R. Pellerin,
Integrated production, sampling quality control and maintenance of deteriorating production systems with AOQL constraint, Omega (United Kingdom), 61 (2016), 110-126.
doi: 10.1016/j.omega.2015.07.012. |
[4] |
S.-C. Chang, J.-S. Yao and H.-M. Lee,
Economic reorder point for fuzzy backorder quantity, Omega (United Kingdom), 109 (1998), 183-202.
doi: 10.1016/S0377-2217(97)00069-6. |
[5] |
Y. C. Chen,
An optimal production and inspection strategy with preventive maintenance error and rework, J. Manuf. Syst., 32 (2013), 99-106.
doi: 10.1016/j.jmsy.2012.07.010. |
[6] |
K. J. Chung, C. C. Her and S. D. Lin,
A two-warehouse inventory model with imperfect quality production processes, Comput. Ind. Eng., 56 (2009), 193-197.
doi: 10.1016/j.cie.2008.05.005. |
[7] |
S. K. De and S. S. Sana,
Fuzzy order quantity inventory model with fuzzy shortage quantity and fuzzy promotional index, Econ. Model., 31 (2013), 351-358.
doi: 10.1016/j.econmod.2012.11.046. |
[8] |
M. Farhangi, S. T. A. Niaki and B. Maleki Vishkaei, Closed-form equations for optimal lot sizing in deterministic EOQ models with exchangeable imperfect quality items, Sci. Iran., 22 (2015), 2621-2634. Google Scholar |
[9] |
Z. Hauck and J. Vörös,
Lot sizing in case of defective items with investments to increase the speed of quality control, Omega (United Kingdom), 52 (2015), 180-189.
doi: 10.1016/j.omega.2014.04.004. |
[10] |
J. T. Hsu and L. F. Hsu,
Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns, Comput. Ind. Eng., 64 (2013), 389-402.
doi: 10.1016/j.cie.2012.10.005. |
[11] |
J. T. Hsu and L. F. Hsu,
An EOQ model with imperfect quality items inspection errors shortage backordering and sales returns, Intern. J. Prod. Econ., 143 (2013), 162-170.
doi: 10.1016/j.ijpe.2012.12.025. |
[12] |
J. S. Hu, H. Zheng, C. Y. Guo and Y. P. Ji,
Optimal production run length with imperfect production processes and backorder in fuzzy random environment, Comput. Ind. Eng., 59 (2010), 9-15.
doi: 10.1016/j.cie.2010.01.012. |
[13] |
H. Ishii and T. Konno,
A stochastic inventory problem with fuzzy shortage cost, Eur. J. Oper. Res., 106 (1998), 90-94.
doi: 10.1016/S0377-2217(97)00173-2. |
[14] |
M. Y. Jaber and S. K. Goyal,
Economic production quantity model for items with imperfect quality subject to learning effects, Int. J. Prod. Econ., 115 (2008), 143-150.
doi: 10.1016/j.ijpe.2008.05.007. |
[15] |
D. K. Jana, B. Das and M. Maiti,
Multi-item partial backlogging inventory models over random planninghorizon in random fuzzy environment, Appl. Soft Comput. J., 21 (2014), 12-27.
doi: 10.1016/j.asoc.2014.02.021. |
[16] |
S. Karmakar, S. K. De and A. Goswami,
A pollution sensitive dense fuzzy economic production quantity model with cycle time dependent production rate, J. Clean. Prod., 154 (2017), 139-150.
doi: 10.1016/j.jclepro.2017.03.080. |
[17] |
N. Kazemi, E. Shekarian, L. E. Cárdenas-Barrón and E. U. Olugu,
Incorporating human learning into a fuzzy EOQ inventory model with backorders, Comput. Ind. Eng., 87 (2015), 540-542.
doi: 10.1016/j.cie.2015.05.014. |
[18] |
M. Khan, M. Y. Jaber and M. Bonney,
An economic order quantity (EOQ) for items with imperfect quality and inspection errors, Int. J. Prod. Econ., 133 (2011), 113-118.
doi: 10.1016/j.ijpe.2010.01.023. |
[19] |
M. Khan, M. Y. Jaber and M. I. M. Wahab,
Economic order quantity model for items with imperfect quality with learning in inspection, Int. J. Prod. Econ., 124 (2010), 87-96.
doi: 10.1016/j.ijpe.2009.10.011. |
[20] |
C. Krishnamoorthi,
An inventory model for product life cycle with maturity stage and defective items, Opsearch, 49 (2012), 209-222.
doi: 10.1007/s12597-012-0075-4. |
[21] |
T. Y. Lin,
Coordination policy for a two-stage supply chain considering quantity discounts and overlapped delivery with imperfect quality, Comput. Ind. Eng., 66 (2013), 53-62.
doi: 10.1016/j.cie.2013.06.012. |
[22] |
J. Liu and H. Zheng,
Fuzzy economic order quantity model with imperfect items, shortages and inspection errors, Syst. Eng. Procedia, 4 (2012), 282-289.
doi: 10.1016/j.sepro.2011.11.077. |
[23] |
W. N. Ma, D. C. Gong and G. C. Lin,
An optimal common production cycle time for imperfect production processes with scrap, Math. Comput. Model., 52 (2010), 724-737.
doi: 10.1016/j.mcm.2010.04.024. |
[24] |
B. Pal, S. S. Sana and K. Chaudhuri,
A mathematical model on EPQ for stochastic demand in an imperfect production system, J. Manuf. Syst., 32 (2013), 260-270.
doi: 10.1016/j.jmsy.2012.11.009. |
[25] |
S. Papachristos and I. Konstantaras,
Economic ordering quantity models for items with imperfect quality, Int. J. Prod. Econ., 100 (2006), 148-154.
doi: 10.1016/j.ijpe.2004.11.004. |
[26] |
M. A. Rahim and H. Ohta,
An integrated economic model for inventory and quality control problems, Eng. Optim., 37 (2005), 65-81.
doi: 10.1080/0305215042000268598. |
[27] |
J. Sadeghi, S. M. Mousavi and S. T. A. Niaki,
Optimizing an inventory model with fuzzy demand, backordering, and discount using a hybrid imperialist competitive algorithm, Appl. Math. Model., 40 (2016), 7318-7335.
doi: 10.1016/j.apm.2016.03.013. |
[28] |
M. K. Salameh and M. Y. Jaber,
Economic production quantity model for items with imperfect quality, Int. J. Prod. Econ., 64 (2000), 59-64.
doi: 10.1016/S0925-5273(99)00044-4. |
[29] |
N. K. Samal and D. K. Pratihar,
Optimization of variable demand fuzzy economic order quantity inventory models without and with backordering, Comput. Ind. Eng., 78 (2014), 148-162.
doi: 10.1016/j.cie.2014.10.006. |
[30] |
S. S. Sana,
A production-inventory model of imperfect quality products in a three-layer supply chain, Decis. Support Syst., 50 (2011), 539-547.
doi: 10.1016/j.dss.2010.11.012. |
[31] |
J. Taheri-Tolgari, A. Mirzazadeh and F. Jolai,
An inventory model for imperfect items under inflationary conditions with considering inspection errors, Comput. Math. with Appl., 63 (2012), 1007-1019.
doi: 10.1016/j.camwa.2011.09.050. |
[32] |
J. Wu, K. Skouri, J. T. Teng and Y. Hu,
Two inventory systems with trapezoidal-type demand rate and time-dependent deterioration and backlogging, Expert Syst. Appl., 46 (2016), 367-379.
doi: 10.1016/j.eswa.2015.10.048. |


















Triangular Fuzzy Number: [a, b, c] | ||||
General Data | Symbol | a | b | c |
Production rate | 300 | 500 | 700 | |
Annual demand rate | 250 | 450 | 650 | |
Setup cost | 400 | 600 | 800 | |
Holding cost | 1.5 | 2.5 | 4 | |
Variable cost | 75 | 100 | 125 | |
Inspection cost | 0.75 | 1 | 1.25 | |
Repair cost for warranty | 35 | 50 | 65 | |
Salvage cost | 10 | 15 | 20 | |
Cost of falsely accepted imperfect item | 23 | 28 | 33 | |
Cost of falsely rejected perfect item | 7 | 10 | 13 | |
Maintenance cost | 75 | 100 | 125 | |
Shortage cost | 5.5 | 6.5 | 7.5 | |
Fraction of product shortage that is backorder | 0.6 | 0.75 | 0.9 | |
Percentage of defective products/in control | 0.05 | 0.15 | 0.25 | |
Percentage of defective products/out-of-control | 0.05 | 0.30 | 0.60 |
Triangular Fuzzy Number: [a, b, c] | ||||
General Data | Symbol | a | b | c |
Production rate | 300 | 500 | 700 | |
Annual demand rate | 250 | 450 | 650 | |
Setup cost | 400 | 600 | 800 | |
Holding cost | 1.5 | 2.5 | 4 | |
Variable cost | 75 | 100 | 125 | |
Inspection cost | 0.75 | 1 | 1.25 | |
Repair cost for warranty | 35 | 50 | 65 | |
Salvage cost | 10 | 15 | 20 | |
Cost of falsely accepted imperfect item | 23 | 28 | 33 | |
Cost of falsely rejected perfect item | 7 | 10 | 13 | |
Maintenance cost | 75 | 100 | 125 | |
Shortage cost | 5.5 | 6.5 | 7.5 | |
Fraction of product shortage that is backorder | 0.6 | 0.75 | 0.9 | |
Percentage of defective products/in control | 0.05 | 0.15 | 0.25 | |
Percentage of defective products/out-of-control | 0.05 | 0.30 | 0.60 |
General Data | Symbol | Probability Density Function |
Type-Ⅰ error | ||
Type-Ⅱ error | ||
Preventive maintenance time | ||
Time after which the production process shifts in control to out-of-control state |
General Data | Symbol | Probability Density Function |
Type-Ⅰ error | ||
Type-Ⅱ error | ||
Preventive maintenance time | ||
Time after which the production process shifts in control to out-of-control state |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of | |||||
0 | 300 | 700 | 250 | 650 | 400 | 800 | |
0.1 | 320 | 680 | 270 | 630 | 420 | 780 | |
0.2 | 340 | 660 | 290 | 610 | 440 | 760 | |
0.3 | 360 | 640 | 310 | 590 | 460 | 740 | |
0.4 | 380 | 620 | 330 | 570 | 480 | 720 | |
0.5 | 400 | 600 | 350 | 550 | 500 | 700 | |
0.6 | 420 | 580 | 370 | 530 | 520 | 680 | |
0.7 | 440 | 560 | 390 | 510 | 540 | 660 | |
0.8 | 460 | 540 | 410 | 490 | 560 | 640 | |
0.9 | 480 | 520 | 430 | 470 | 580 | 620 | |
1 | 500 | 500 | 450 | 450 | 600 | 600 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of | |||||
0 | 300 | 700 | 250 | 650 | 400 | 800 | |
0.1 | 320 | 680 | 270 | 630 | 420 | 780 | |
0.2 | 340 | 660 | 290 | 610 | 440 | 760 | |
0.3 | 360 | 640 | 310 | 590 | 460 | 740 | |
0.4 | 380 | 620 | 330 | 570 | 480 | 720 | |
0.5 | 400 | 600 | 350 | 550 | 500 | 700 | |
0.6 | 420 | 580 | 370 | 530 | 520 | 680 | |
0.7 | 440 | 560 | 390 | 510 | 540 | 660 | |
0.8 | 460 | 540 | 410 | 490 | 560 | 640 | |
0.9 | 480 | 520 | 430 | 470 | 580 | 620 | |
1 | 500 | 500 | 450 | 450 | 600 | 600 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of | |||||
0 | 1.5 | 4 | 75 | 125 | 0.75 | 1.25 | |
0.1 | 1.6 | 3.85 | 77.5 | 122.5 | 0.775 | 1.225 | |
0.2 | 1.7 | 3.7 | 80 | 120 | 0.8 | 1.2 | |
0.3 | 1.8 | 3.55 | 82.5 | 117.5 | 0.825 | 1.175 | |
0.4 | 1.9 | 3.4 | 85 | 115 | 0.85 | 1.15 | |
0.5 | 2 | 3.25 | 87.5 | 112.5 | 0.875 | 1.125 | |
0.6 | 2.1 | 3.1 | 90 | 110 | 0.9 | 1.1 | |
0.7 | 2.2 | 2.95 | 92.5 | 107.5 | 0.925 | 1.075 | |
0.8 | 2.3 | 2.8 | 95 | 105 | 0.95 | 1.05 | |
0.9 | 2.4 | 2.65 | 97.5 | 102.5 | 0.975 | 1.025 | |
1 | 2.5 | 2.5 | 100 | 100 | 1 | 1 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of | |||||
0 | 1.5 | 4 | 75 | 125 | 0.75 | 1.25 | |
0.1 | 1.6 | 3.85 | 77.5 | 122.5 | 0.775 | 1.225 | |
0.2 | 1.7 | 3.7 | 80 | 120 | 0.8 | 1.2 | |
0.3 | 1.8 | 3.55 | 82.5 | 117.5 | 0.825 | 1.175 | |
0.4 | 1.9 | 3.4 | 85 | 115 | 0.85 | 1.15 | |
0.5 | 2 | 3.25 | 87.5 | 112.5 | 0.875 | 1.125 | |
0.6 | 2.1 | 3.1 | 90 | 110 | 0.9 | 1.1 | |
0.7 | 2.2 | 2.95 | 92.5 | 107.5 | 0.925 | 1.075 | |
0.8 | 2.3 | 2.8 | 95 | 105 | 0.95 | 1.05 | |
0.9 | 2.4 | 2.65 | 97.5 | 102.5 | 0.975 | 1.025 | |
1 | 2.5 | 2.5 | 100 | 100 | 1 | 1 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of | |||||
0 | 35 | 65 | 10 | 20 | 23 | 33 | |
0.1 | 36.5 | 63.5 | 10.5 | 19.5 | 23.5 | 32.5 | |
0.2 | 38 | 62 | 11 | 19 | 24 | 32 | |
0.3 | 39.5 | 60.5 | 11.5 | 18.5 | 24.5 | 31.5 | |
0.4 | 41 | 59 | 12 | 18 | 25 | 31 | |
0.5 | 42.5 | 57.5 | 12.5 | 17.5 | 25.5 | 30.5 | |
0.6 | 44 | 56 | 13 | 17 | 26 | 30 | |
0.7 | 45.5 | 54.5 | 13.5 | 16.5 | 26.5 | 29.5 | |
0.8 | 47 | 53 | 14 | 16 | 27 | 29 | |
0.9 | 48.5 | 51.5 | 14.5 | 15.5 | 27.5 | 28.5 | |
1 | 50 | 50 | 15 | 15 | 28 | 28 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of | |||||
0 | 35 | 65 | 10 | 20 | 23 | 33 | |
0.1 | 36.5 | 63.5 | 10.5 | 19.5 | 23.5 | 32.5 | |
0.2 | 38 | 62 | 11 | 19 | 24 | 32 | |
0.3 | 39.5 | 60.5 | 11.5 | 18.5 | 24.5 | 31.5 | |
0.4 | 41 | 59 | 12 | 18 | 25 | 31 | |
0.5 | 42.5 | 57.5 | 12.5 | 17.5 | 25.5 | 30.5 | |
0.6 | 44 | 56 | 13 | 17 | 26 | 30 | |
0.7 | 45.5 | 54.5 | 13.5 | 16.5 | 26.5 | 29.5 | |
0.8 | 47 | 53 | 14 | 16 | 27 | 29 | |
0.9 | 48.5 | 51.5 | 14.5 | 15.5 | 27.5 | 28.5 | |
1 | 50 | 50 | 15 | 15 | 28 | 28 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of |
|||||
0 | 7 | 13 | 75 | 125 | 5.5 | 7.5 | |
0.1 | 7.3 | 12.7 | 77.5 | 122.5 | 5.6 | 7.4 | |
0.2 | 7.6 | 12.4 | 80 | 120 | 5.7 | 7.3 | |
0.3 | 7.9 | 12.1 | 82.5 | 117.5 | 5.8 | 7.2 | |
0.4 | 8.2 | 11.8 | 85 | 115 | 5.9 | 7.1 | |
0.5 | 8.5 | 11.5 | 87.5 | 112.5 | 6 | 7 | |
0.6 | 8.8 | 11.2 | 90 | 110 | 6.1 | 6.9 | |
0.7 | 9.1 | 10.9 | 92.5 | 107.5 | 6.2 | 6.8 | |
0.8 | 9.4 | 10.6 | 95 | 105 | 6.3 | 6.7 | |
0.9 | 9.7 | 10.3 | 97.5 | 102.5 | 6.4 | 6.6 | |
1 | 10 | 10 | 100 | 100 | 6.5 | 6.5 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of |
|||||
0 | 7 | 13 | 75 | 125 | 5.5 | 7.5 | |
0.1 | 7.3 | 12.7 | 77.5 | 122.5 | 5.6 | 7.4 | |
0.2 | 7.6 | 12.4 | 80 | 120 | 5.7 | 7.3 | |
0.3 | 7.9 | 12.1 | 82.5 | 117.5 | 5.8 | 7.2 | |
0.4 | 8.2 | 11.8 | 85 | 115 | 5.9 | 7.1 | |
0.5 | 8.5 | 11.5 | 87.5 | 112.5 | 6 | 7 | |
0.6 | 8.8 | 11.2 | 90 | 110 | 6.1 | 6.9 | |
0.7 | 9.1 | 10.9 | 92.5 | 107.5 | 6.2 | 6.8 | |
0.8 | 9.4 | 10.6 | 95 | 105 | 6.3 | 6.7 | |
0.9 | 9.7 | 10.3 | 97.5 | 102.5 | 6.4 | 6.6 | |
1 | 10 | 10 | 100 | 100 | 6.5 | 6.5 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of |
|||||
0 | 0.6 | 0.9 | 0.05 | 0.25 | 0.05 | 0.6 | |
0.1 | 0.615 | 0.885 | 0.06 | 0.24 | 0.075 | 0.57 | |
0.2 | 0.63 | 0.87 | 0.07 | 0.23 | 0.1 | 0.54 | |
0.3 | 0.645 | 0.855 | 0.08 | 0.22 | 0.125 | 0.51 | |
0.4 | 0.66 | 0.84 | 0.09 | 0.21 | 0.15 | 0.48 | |
0.5 | 0.675 | 0.825 | 0.1 | 0.2 | 0.175 | 0.45 | |
0.6 | 0.69 | 0.81 | 0.11 | 0.19 | 0.2 | 0.42 | |
0.7 | 0.705 | 0.795 | 0.12 | 0.18 | 0.225 | 0.39 | |
0.8 | 0.72 | 0.78 | 0.13 | 0.17 | 0.25 | 0.36 | |
0.9 | 0.735 | 0.765 | 0.14 | 0.16 | 0.275 | 0.33 | |
1 | 0.75 | 0.75 | 0.15 | 0.15 | 0.3 | 0.3 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of |
|||||
0 | 0.6 | 0.9 | 0.05 | 0.25 | 0.05 | 0.6 | |
0.1 | 0.615 | 0.885 | 0.06 | 0.24 | 0.075 | 0.57 | |
0.2 | 0.63 | 0.87 | 0.07 | 0.23 | 0.1 | 0.54 | |
0.3 | 0.645 | 0.855 | 0.08 | 0.22 | 0.125 | 0.51 | |
0.4 | 0.66 | 0.84 | 0.09 | 0.21 | 0.15 | 0.48 | |
0.5 | 0.675 | 0.825 | 0.1 | 0.2 | 0.175 | 0.45 | |
0.6 | 0.69 | 0.81 | 0.11 | 0.19 | 0.2 | 0.42 | |
0.7 | 0.705 | 0.795 | 0.12 | 0.18 | 0.225 | 0.39 | |
0.8 | 0.72 | 0.78 | 0.13 | 0.17 | 0.25 | 0.36 | |
0.9 | 0.735 | 0.765 | 0.14 | 0.16 | 0.275 | 0.33 | |
1 | 0.75 | 0.75 | 0.15 | 0.15 | 0.3 | 0.3 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of |
|||||
0 | 87.0133 | 145.1221 | 261.5060 | 334.5721 | 0 | 1 | |
0.1 | 89.9842 | 142.218 | 268.8547 | 332.0059 | 0 | 0.6745 | |
0.2 | 92.9146 | 139.3142 | 276.4437 | 329.4848 | 0 | 0.4356 | |
0.3 | 95.8269 | 136.4106 | 282.4803 | 327.0103 | 0 | 0.3054 | |
0.4 | 98.7300 | 133.5071 | 287.6358 | 324.5838 | 0 | 0.2221 | |
0.5 | 101.627 | 130.6038 | 292.2257 | 322.2072 | 0 | 0.1640 | |
0.6 | 104.522 | 127.7005 | 296.4265 | 319.8825 | 0 | 0.1211 | |
0.7 | 107.414 | 124.7973 | 300.3485 | 317.6118 | 0 | 0.0882 | |
0.8 | 110.305 | 121.8939 | 304.0651 | 315.3975 | 0 | 0.0622 | |
0.9 | 113.195 | 118.9902 | 307.6279 | 313.2330 | 0.0096 | 0.0412 | |
1 | 116.0856 | 116.0856 | 311.0745 | 311.0745 | 0.0239 | 0.0239 |
Lower and upper bound of |
Lower and upper bound of |
Lower and upper bound of |
|||||
0 | 87.0133 | 145.1221 | 261.5060 | 334.5721 | 0 | 1 | |
0.1 | 89.9842 | 142.218 | 268.8547 | 332.0059 | 0 | 0.6745 | |
0.2 | 92.9146 | 139.3142 | 276.4437 | 329.4848 | 0 | 0.4356 | |
0.3 | 95.8269 | 136.4106 | 282.4803 | 327.0103 | 0 | 0.3054 | |
0.4 | 98.7300 | 133.5071 | 287.6358 | 324.5838 | 0 | 0.2221 | |
0.5 | 101.627 | 130.6038 | 292.2257 | 322.2072 | 0 | 0.1640 | |
0.6 | 104.522 | 127.7005 | 296.4265 | 319.8825 | 0 | 0.1211 | |
0.7 | 107.414 | 124.7973 | 300.3485 | 317.6118 | 0 | 0.0882 | |
0.8 | 110.305 | 121.8939 | 304.0651 | 315.3975 | 0 | 0.0622 | |
0.9 | 113.195 | 118.9902 | 307.6279 | 313.2330 | 0.0096 | 0.0412 | |
1 | 116.0856 | 116.0856 | 311.0745 | 311.0745 | 0.0239 | 0.0239 |
Lower and upper bound of |
Lower and upper bound of |
||||
0 | 0.176880867 | 5.9687669 | 9.23117E-06 | 0.000239 | |
0.1 | 0.366359442 | 5.691024185 | 1.82229E-05 | 0.000228 | |
0.2 | 0.652038831 | 5.417609624 | 3.09138E-05 | 0.000218 | |
0.3 | 0.947627216 | 5.148450421 | 4.33739E-05 | 0.000208 | |
0.4 | 1.25140767 | 4.883472592 | 5.57048E-05 | 0.000197 | |
0.5 | 1.562433439 | 4.622600569 | 6.79622E-05 | 0.000187 | |
0.6 | 1.880135101 | 4.365756781 | 8.01786E-05 | 0.000177 | |
0.7 | 2.204158168 | 4.112861206 | 9.23724E-05 | 0.000166 | |
0.8 | 2.534281476 | 3.863830732 | 0.000104554 | 0.000156 | |
0.9 | 2.870373498 | 3.546574322 | 0.000116726 | 0.000143 | |
1 | 3.212363938 | 3.212363938 | 0.000128889 | 0.000129 |
Lower and upper bound of |
Lower and upper bound of |
||||
0 | 0.176880867 | 5.9687669 | 9.23117E-06 | 0.000239 | |
0.1 | 0.366359442 | 5.691024185 | 1.82229E-05 | 0.000228 | |
0.2 | 0.652038831 | 5.417609624 | 3.09138E-05 | 0.000218 | |
0.3 | 0.947627216 | 5.148450421 | 4.33739E-05 | 0.000208 | |
0.4 | 1.25140767 | 4.883472592 | 5.57048E-05 | 0.000197 | |
0.5 | 1.562433439 | 4.622600569 | 6.79622E-05 | 0.000187 | |
0.6 | 1.880135101 | 4.365756781 | 8.01786E-05 | 0.000177 | |
0.7 | 2.204158168 | 4.112861206 | 9.23724E-05 | 0.000166 | |
0.8 | 2.534281476 | 3.863830732 | 0.000104554 | 0.000156 | |
0.9 | 2.870373498 | 3.546574322 | 0.000116726 | 0.000143 | |
1 | 3.212363938 | 3.212363938 | 0.000128889 | 0.000129 |
Defuzzification method | ||
Variable | Centroid | Signed distance |
116.07 | 116.08 | |
302.38 | 304.56 | |
0.3413 | 0.262 | |
Hessian matrix |
3.1193 | 3.1426 |
Hessian matrix |
0.0001 | 0.0001 |
Defuzzification method | ||
Variable | Centroid | Signed distance |
116.07 | 116.08 | |
302.38 | 304.56 | |
0.3413 | 0.262 | |
Hessian matrix |
3.1193 | 3.1426 |
Hessian matrix |
0.0001 | 0.0001 |
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