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July  2019, 15(3): 1317-1344. doi: 10.3934/jimo.2018097

An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment

Javad Taheri-Tolgari , Mohammad Mohammadi , , Bahman Naderi , Alireza Arshadi-Khamseh and  Abolfazl Mirzazadeh

Department of Industrial Engineering, Faculty of Engineering Kharazmi University, 15719-14911 Tehran, Iran

* Corresponding author: Mohammad Mohammadi

Received  September 2017 Revised  October 2017 Published  July 2019 Early access  July 2018

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.

Keywords: Inspection errors, Zadeh's extension principle, non-linear programming, shortage, imperfect items.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Javad Taheri-Tolgari, Mohammad Mohammadi, Bahman Naderi, Alireza Arshadi-Khamseh, Abolfazl Mirzazadeh. An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1317-1344. doi: 10.3934/jimo.2018097
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. 

[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. 

[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.

Figure 1.  Schematic representation of inventory system
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Figure 2.  Flowchart the solution procedure
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Figure 3.  Behavior of the target function at $\alpha = 0$
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Figure 4.  Behavior of the target function at $\alpha = 0.1$
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Figure 5.  Behavior of the target function at $\alpha = 0.2$
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Figure 6.  Behavior of the target function at $\alpha = 0.3$
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Figure 7.  Behavior of the target function at $\alpha = 0.4$
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Figure 8.  Behavior of the target function at $\alpha = 0.5$
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Figure 9.  Behavior of the target function at $\alpha = 0.6$
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Figure 10.  Behavior of the target function at $\alpha = 0.7$
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Figure 11.  Behavior of the target function at $\alpha = 0.8$
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Figure 12.  Behavior of the target function at $\alpha = 0.9$
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Figure 13.  Behavior of the target function at $\alpha = 1$
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Figure 14.  Membership function of output parameters-optimal solution
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Figure 15.  Membership function of output parameters-optimal solution
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Figure 16.  Membership function of output parameters-optimal solution
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Figure 17.  Membership function of output parameters-optimal solution
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Figure 18.  Membership function of Hessian matrix
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Table 1.  General data - input parameters
Triangular Fuzzy Number: [a, b, c]
General Data Symbol a b c
Production rate $P$ 300 500 700
Annual demand rate $D$ 250 450 650
Setup cost $k$ 400 600 800
Holding cost $c_h$ 1.5 2.5 4
Variable cost $c_m$ 75 100 125
Inspection cost $I_c$ 0.75 1 1.25
Repair cost for warranty $w$ 35 50 65
Salvage cost $s$ 10 15 20
Cost of falsely accepted imperfect item $c_a$ 23 28 33
Cost of falsely rejected perfect item $c_r$ 7 10 13
Maintenance cost $M_c $ 75 100 125
Shortage cost $s_c$ 5.5 6.5 7.5
Fraction of product shortage that is backorder $\gamma$ 0.6 0.75 0.9
Percentage of defective products/in control $\theta_1$ 0.05 0.15 0.25
Percentage of defective products/out-of-control $\theta_2$ 0.05 0.30 0.60
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Triangular Fuzzy Number: [a, b, c]
General Data Symbol a b c
Production rate $P$ 300 500 700
Annual demand rate $D$ 250 450 650
Setup cost $k$ 400 600 800
Holding cost $c_h$ 1.5 2.5 4
Variable cost $c_m$ 75 100 125
Inspection cost $I_c$ 0.75 1 1.25
Repair cost for warranty $w$ 35 50 65
Salvage cost $s$ 10 15 20
Cost of falsely accepted imperfect item $c_a$ 23 28 33
Cost of falsely rejected perfect item $c_r$ 7 10 13
Maintenance cost $M_c $ 75 100 125
Shortage cost $s_c$ 5.5 6.5 7.5
Fraction of product shortage that is backorder $\gamma$ 0.6 0.75 0.9
Percentage of defective products/in control $\theta_1$ 0.05 0.15 0.25
Percentage of defective products/out-of-control $\theta_2$ 0.05 0.30 0.60
Table 2.  General data - input parameters
General Data Symbol Probability Density Function
Type-Ⅰ error $\alpha$ $f(\alpha)=0.04e^{0.04\alpha}$
Type-Ⅱ error $\beta$ $f(\beta)=0.05e^{0.05\beta}$
Preventive maintenance time $PM_T$ $f(PM_T)=0.25e^{0.25PM_T}$
Time after which the production process shifts in control to out-of-control state $\sigma$ $f(\sigma)=0.33e^{0.33\sigma}$
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General Data Symbol Probability Density Function
Type-Ⅰ error $\alpha$ $f(\alpha)=0.04e^{0.04\alpha}$
Type-Ⅱ error $\beta$ $f(\beta)=0.05e^{0.05\beta}$
Preventive maintenance time $PM_T$ $f(PM_T)=0.25e^{0.25PM_T}$
Time after which the production process shifts in control to out-of-control state $\sigma$ $f(\sigma)=0.33e^{0.33\sigma}$
Table 3.  General data - $\alpha$ cuts input parameters
Lower and upper bound of $P$ Lower and upper bound of $D$ Lower and upper bound of $k$
$P_{\alpha}^L$ $P_{\alpha}^U$ $D_{\alpha}^L$ $D_{\alpha}^U$ $k_{\alpha}^L$ $k_{\alpha}^U$
$\alpha~ level$ 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
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Lower and upper bound of $P$ Lower and upper bound of $D$ Lower and upper bound of $k$
$P_{\alpha}^L$ $P_{\alpha}^U$ $D_{\alpha}^L$ $D_{\alpha}^U$ $k_{\alpha}^L$ $k_{\alpha}^U$
$\alpha~ level$ 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
Table 4.  General data - $\alpha$ cuts input
Lower and upper bound of $c_h$ Lower and upper bound of $c_m$ Lower and upper bound of $I_c$
${c_h}_{\alpha}^L$ ${c_h}_{\alpha}^U$ ${c_m}_{\alpha}^L$ ${c_m}_{\alpha}^U$ ${I_c}_{\alpha}^L$ ${I_c}_{\alpha}^U$
$\alpha~ level$ 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
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Lower and upper bound of $c_h$ Lower and upper bound of $c_m$ Lower and upper bound of $I_c$
${c_h}_{\alpha}^L$ ${c_h}_{\alpha}^U$ ${c_m}_{\alpha}^L$ ${c_m}_{\alpha}^U$ ${I_c}_{\alpha}^L$ ${I_c}_{\alpha}^U$
$\alpha~ level$ 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
Table 5.  General data - $\alpha$ cuts input parameters
Lower and upper bound of $w$ Lower and upper bound of $s$ Lower and upper bound of $c_a$
$w_{\alpha}^L$ $w_{\alpha}^U$ $s_{\alpha}^L$ $s_{\alpha}^U$ ${c_a}_{\alpha}^L$ ${c_a}_{\alpha}^U$
$\alpha~ level$ 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
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Lower and upper bound of $w$ Lower and upper bound of $s$ Lower and upper bound of $c_a$
$w_{\alpha}^L$ $w_{\alpha}^U$ $s_{\alpha}^L$ $s_{\alpha}^U$ ${c_a}_{\alpha}^L$ ${c_a}_{\alpha}^U$
$\alpha~ level$ 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
Table 6.  General data - $\alpha$ cuts input parameters
Lower and upper bound of $c_r$ Lower and upper bound of $M_c$ Lower and upper bound of $s_c$
${c_r}_{\alpha}^L$ ${c_r}_{\alpha}^U$ ${M_c}_{\alpha}^L$ ${M_c}_{\alpha}^U$ ${s_c}_{\alpha}^L$ ${s_c}_{\alpha}^U$
$\alpha~ level$ 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
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Lower and upper bound of $c_r$ Lower and upper bound of $M_c$ Lower and upper bound of $s_c$
${c_r}_{\alpha}^L$ ${c_r}_{\alpha}^U$ ${M_c}_{\alpha}^L$ ${M_c}_{\alpha}^U$ ${s_c}_{\alpha}^L$ ${s_c}_{\alpha}^U$
$\alpha~ level$ 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
Table 7.  General data - $\alpha$ cuts input parameters
Lower and upper bound of $\theta_1$ Lower and upper bound of $\theta_2$ Lower and upper bound of $\gamma$
${\theta_1}_{\alpha}^L$ ${\theta_1}_{\alpha}^U$ ${\theta_2}_{\alpha}^L$ ${\theta_2}_{\alpha}^U$ $\gamma_{\alpha}^L$ $\gamma_{\alpha}^U$
$\alpha~ level$ 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
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Lower and upper bound of $\theta_1$ Lower and upper bound of $\theta_2$ Lower and upper bound of $\gamma$
${\theta_1}_{\alpha}^L$ ${\theta_1}_{\alpha}^U$ ${\theta_2}_{\alpha}^L$ ${\theta_2}_{\alpha}^U$ $\gamma_{\alpha}^L$ $\gamma_{\alpha}^U$
$\alpha~ level$ 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
Table 8.  General data - $\alpha$ cuts input parameters
Lower and upper bound of $E[TCU]$ Lower and upper bound of $I_m$ Lower and upper bound of $\delta$
$E[TCU]_{\alpha}^L$ $E[TCU]_{\alpha}^U$ ${I_m}_{\alpha}^L$ ${I_m}_{\alpha}^U$ $\delta_{\alpha}^L$ $\delta_{\alpha}^U$
$\alpha~ level$ 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
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Lower and upper bound of $E[TCU]$ Lower and upper bound of $I_m$ Lower and upper bound of $\delta$
$E[TCU]_{\alpha}^L$ $E[TCU]_{\alpha}^U$ ${I_m}_{\alpha}^L$ ${I_m}_{\alpha}^U$ $\delta_{\alpha}^L$ $\delta_{\alpha}^U$
$\alpha~ level$ 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
Table 9.  General data - $\alpha$ cuts Hessian matrix
Lower and upper bound of $H_1$ Lower and upper bound of $H_2$
$H_1^L$ $H_1^U$ $H_2^L$ $H_2^U$
$\alpha~ level$ 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
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Lower and upper bound of $H_1$ Lower and upper bound of $H_2$
$H_1^L$ $H_1^U$ $H_2^L$ $H_2^U$
$\alpha~ level$ 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
Table 10.  General data - optimal output parameters in crisp state
Defuzzification method
Variable Centroid Signed distance
$E[TCU]$ 116.07 116.08
$I_m$ 302.38 304.56
$\delta$ 0.3413 0.262
Hessian matrix $H_1$ 3.1193 3.1426
Hessian matrix $H_2$ 0.0001 0.0001
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Defuzzification method
Variable Centroid Signed distance
$E[TCU]$ 116.07 116.08
$I_m$ 302.38 304.56
$\delta$ 0.3413 0.262
Hessian matrix $H_1$ 3.1193 3.1426
Hessian matrix $H_2$ 0.0001 0.0001
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