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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

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

Received  September 2017 Revised  October 2017 Published  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.

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 & Management Optimization, 2019, 15 (3) : 1317-1344. doi: 10.3934/jimo.2018097
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##### References:
Schematic representation of inventory system
Flowchart the solution procedure
Behavior of the target function at $\alpha = 0$
Behavior of the target function at $\alpha = 0.1$
Behavior of the target function at $\alpha = 0.2$
Behavior of the target function at $\alpha = 0.3$
Behavior of the target function at $\alpha = 0.4$
Behavior of the target function at $\alpha = 0.5$
Behavior of the target function at $\alpha = 0.6$
Behavior of the target function at $\alpha = 0.7$
Behavior of the target function at $\alpha = 0.8$
Behavior of the target function at $\alpha = 0.9$
Behavior of the target function at $\alpha = 1$
Membership function of output parameters-optimal solution
Membership function of output parameters-optimal solution
Membership function of output parameters-optimal solution
Membership function of output parameters-optimal solution
Membership function of Hessian matrix
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
 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
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 $\sigma$ $f(\sigma)=0.33e^{0.33\sigma}$
 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 $\sigma$ $f(\sigma)=0.33e^{0.33\sigma}$
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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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