# American Institute of Mathematical Sciences

July  2017, 13(3): 1511-1535. doi: 10.3934/jimo.2017005

## Stochastic machine breakdown and discrete delivery in an imperfect inventory-production system

 1 School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, 14155-6619, Iran 2 Department of Industrial Engineering, Iran University of Science and Technology, Tehran, 145888-9694, Iran 3 Department of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 15588, South Korea

* Corresponding author: bsbiswajitsarkar@gmail.com (Biswajit Sarkar), Phone Number-+82-10-7498-1981, Office Phone: +82-31-400-5259, Fax: +82-31-436-8146

Received  July 2015 Published  December 2016

In this paper, we develop an integrated inventory model to determine the optimal lot size and production uptime while considering stochastic machine breakdown and multiple shipments for a single-buyer and single-vendor. Machine breakdown cannot be controlled by the production house. Thus, we assume it as stochastic, not constant. Moreover, we assume that the manufacturing process produces defective items. When a breakdown takes place, the production system follows a no resumption policy. Some defective products cannot be reworked and are discarded from the system. To prevent shortages, we consider safety stock. The model assumes that both batch quantity and the distance between two shipments are identical and that the transportation cost is paid by the buyer. We prove the convexity of the total cost function and derive the closed-form solutions for decision variables analytically. To obtain the optimal production uptime, we determine both the lower and upper bounds for the optimal production uptime using a bisection searching algorithm. To illustrate the applicability of the proposed model, we provided a numerical example and sensitivity analysis.

Citation: Ata Allah Taleizadeh, Hadi Samimi, Biswajit Sarkar, Babak Mohammadi. Stochastic machine breakdown and discrete delivery in an imperfect inventory-production system. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1511-1535. doi: 10.3934/jimo.2017005
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##### References:
The vendors on-hand inventory of perfect-quality items in our EPQ model when machine breakdown does not occur
The vendors on-hand inventory of defective items when machine breakdown does not occur
The buyers inventory level when machine breakdown does not occur
The vendors on-hand inventory of perfect-quality items in our EPQ model when machine breakdown occurs
The vendors on-hand inventory of defective items when machine breakdown occurs
The buyers inventory level when machine breakdown occurs
The behavior of $E[TCU(t_1,q)]$ with respect to $t_1$
The behavior of $E[TCU(t_1,q)]$ with respect to $\frac{1}{\beta}$
The behavior of $E[TCU(t_1,q)]$ with respect to $q$
Sensitivity analysis of $t^{*}_{1}$ for various parameter values
Sensitivity analysis of $q^*$ for various parameter values
 Decision variables $t_1$ production uptime when a breakdown does not occur (year). $q$ shipment quantity (units/delivery). Parameters $A$ setup cost of vendor (＄/setup). $C$ production cost of vendor (＄/unit). $C_s$ disposal cost of vendor (＄/unit). $h$ holding cost of vendor (＄/unit/year). $h_1$ holding cost for defective units of vendor (＄/defective unit/year). $C_t$ transportation cost of buyer (＄/delivery). $D$ demand rate of buyer (units/year). $A_1$ ordering cost of buyer (＄/order). $h_2$ holding cost of buyer (＄/unit/year). $H_1$ maximum level of on-hand inventory when machine breakdown does not occur (units). $H_2$ maximum level of on-hand inventory when machine breakdown occurs (units). $M$ machine repair time (time unit). $n$ number of shipments delivered during a cycle when machine breakdown does not occur. $P$ production rate (units/year). $T$ cycle length when breakdown does not occur (year). $T'$ cycle length when breakdown occurs (year). $T_U$ cycle length for integrated case (year). $t$ production time before a random breakdown occurs (year). $t_d$ time required to deplete all available perfect-quality items when machine breakdown does not occur (year). $t'_{d}$ time required to deplete all available perfect-quality items when machine breakdown occurs (year). $t_r$ machine repair time (year). $TC(t,q)$ total inventory costs per cycle when machine breakdown occurs (＄/cycle). $TC(t_1,q)$ total inventory costs per cycle when machine breakdown does not occur (＄/cycle). $TCU(t_1,q)$ total inventory costs per unit time for integrated case (＄/year).
 Decision variables $t_1$ production uptime when a breakdown does not occur (year). $q$ shipment quantity (units/delivery). Parameters $A$ setup cost of vendor (＄/setup). $C$ production cost of vendor (＄/unit). $C_s$ disposal cost of vendor (＄/unit). $h$ holding cost of vendor (＄/unit/year). $h_1$ holding cost for defective units of vendor (＄/defective unit/year). $C_t$ transportation cost of buyer (＄/delivery). $D$ demand rate of buyer (units/year). $A_1$ ordering cost of buyer (＄/order). $h_2$ holding cost of buyer (＄/unit/year). $H_1$ maximum level of on-hand inventory when machine breakdown does not occur (units). $H_2$ maximum level of on-hand inventory when machine breakdown occurs (units). $M$ machine repair time (time unit). $n$ number of shipments delivered during a cycle when machine breakdown does not occur. $P$ production rate (units/year). $T$ cycle length when breakdown does not occur (year). $T'$ cycle length when breakdown occurs (year). $T_U$ cycle length for integrated case (year). $t$ production time before a random breakdown occurs (year). $t_d$ time required to deplete all available perfect-quality items when machine breakdown does not occur (year). $t'_{d}$ time required to deplete all available perfect-quality items when machine breakdown occurs (year). $t_r$ machine repair time (year). $TC(t,q)$ total inventory costs per cycle when machine breakdown occurs (＄/cycle). $TC(t_1,q)$ total inventory costs per cycle when machine breakdown does not occur (＄/cycle). $TCU(t_1,q)$ total inventory costs per unit time for integrated case (＄/year).
Variations of $\beta$ effects on $t^{*}_{1L}$, $w(t^{*}_{1L})$, $t^{*}_{1U}$, and $w(t^{*}_{1U})$
 $\beta$ $\beta^{-1}$ $t^{*}_{1L}$ $w(t^{*}_{1L})$ $t^{*}_{1U}$ $w(t^{*}_{1U})$ 0.1 10 0.398635 0.406528006 0.402628 0.410519 0.2 5 0.398597 0.414274 0.40662 0.422284 0.3 3.33 0.398559 0.421913 0.410651 0.43396 0.4 2.5 0.398521 0.429443 0.41472 0.445535 0.5 2 0.398483 0.436868 0.418826 0.456999 1 1 0.398294 0.472429 0.439923 0.512281 1.1 0.909 0.398256 0.479241 0.444254 0.522854 1.2 0.833 0.398218 0.485956 0.448623 0.533239 1.3 0.769 0.39818 0.492577 0.453028 0.543428 1.4 0.714 0.398142 0.499106 0.45747 0.553411 1.5 0.667 0.398104 0.505546 0.461949 0.563177 2 0.5 0.397915 0.5365 0.484882 0.608482 3 0.333 0.397538 0.593332 0.533358 0.679759 4 0.25 0.397161 0.645899 0.585075 0.727634 5 0.2 0.396786 0.696877 0.639708 0.762221 6 0.167 0.396412 0.748207 0.69691 0.794729 7 0.142 0.396038 0.801053 0.756338 0.831822 8 0.125 0.395666 0.855961 0.817668 0.875487 9 0.111 0.395295 0.913066 0.880606 0.925223 10 0.1 0.394925 0.972266 0.944891 0.979797
 $\beta$ $\beta^{-1}$ $t^{*}_{1L}$ $w(t^{*}_{1L})$ $t^{*}_{1U}$ $w(t^{*}_{1U})$ 0.1 10 0.398635 0.406528006 0.402628 0.410519 0.2 5 0.398597 0.414274 0.40662 0.422284 0.3 3.33 0.398559 0.421913 0.410651 0.43396 0.4 2.5 0.398521 0.429443 0.41472 0.445535 0.5 2 0.398483 0.436868 0.418826 0.456999 1 1 0.398294 0.472429 0.439923 0.512281 1.1 0.909 0.398256 0.479241 0.444254 0.522854 1.2 0.833 0.398218 0.485956 0.448623 0.533239 1.3 0.769 0.39818 0.492577 0.453028 0.543428 1.4 0.714 0.398142 0.499106 0.45747 0.553411 1.5 0.667 0.398104 0.505546 0.461949 0.563177 2 0.5 0.397915 0.5365 0.484882 0.608482 3 0.333 0.397538 0.593332 0.533358 0.679759 4 0.25 0.397161 0.645899 0.585075 0.727634 5 0.2 0.396786 0.696877 0.639708 0.762221 6 0.167 0.396412 0.748207 0.69691 0.794729 7 0.142 0.396038 0.801053 0.756338 0.831822 8 0.125 0.395666 0.855961 0.817668 0.875487 9 0.111 0.395295 0.913066 0.880606 0.925223 10 0.1 0.394925 0.972266 0.944891 0.979797
Sensitivity analysis of $t^{*}_{1}$, $q^{*}$, and $E[TCU(t^{*}_{1},q^{*})]$ for various parameter values
 Rate of $q^{*}$ $t^{*}_{1}$ $E[TCU(t^{*}_{1},q^{*})]$ Rate of $q^{*}$ $t^{*}_{1}$ $E[TCU(t^{*}_{1},q^{*})]$ A change h change -0.3 730.29 0.3613 11485.71 -0.3 778.49 0.4845 11382.55 -0.2 730.29 0.3789 11544.89 -0.2 761.38 0.4564 11477.72 -0.1 730.29 0.3985 11601.68 -0.1 745.35 0.4326 11568.80 0 730.29 0.4121 11656.35 0 730.29 0.4121 11656.35 0.1 730.29 0.4278 11709.14 0.1 716.11 0.3943 11740.79 0.2 730.29 0.4431 11760.24 0.2 702.72 0.3786 11822.47 0.3 730.29 0.4578 11809.83 0.3 690.06 0.3646 11901.69 $h_2$ -0.3 806.47 0.4121 11552.87 $\beta$ -0.3 730.29 0.40802 11608.71 -0.2 778.49 0.4121 11588.52 -0.2 730.29 0.40939 11624.56 -0.1 753.24 0.4121 11622.97 -0.1 730.29 0.41077 11640.44 0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35 0.1 709.32 0.4121 11688.73 0.1 730.29 0.41356 11672.29 0.2 690.06 0.4121 11720.21 0.2 730.29 0.41497 11688.26 0.3 672.29 0.4121 11750.86 0.3 730.29 0.41639 11704.27 $C_t$ -0.3 730.29 0.4121 11477.42 $t_r$ -0.3 730.29 0.41222 11643.20 -0.2 730.29 0.4121 11540.70 -0.2 730.29 0.41220 11647.58 -0.1 730.29 0.4121 11600.13 -0.1 730.29 0.41218 11651.96 0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35 0.1 730.29 0.4121 11709.81 0.1 730.29 0.41214 11660.73 0.2 730.29 0.4121 11760.90 0.2 730.29 0.41212 11665.11 0.3 730.29 0.4121 11809.90 0.3 730.29 0.41210 11669.50 P -0.3 730.29 0.7005 11590.45 D -0.3 611.01 0.31465 8589.613 -0.2 730.29 0.5664 11537.73 -0.2 653.19 0.34625 9625.374 -0.1 730.29 0.4767 11628.08 -0.1 692.82 0.37862 10647.10 0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35 0.1 730.29 0.3633 11678.38 0.1 765.94 0.44731 12654.21 0.2 730.29 0.3249 11696.05 0.2 800.00 0.48453 13641.47 0.3 730.29 0.2940 11710.54 0.3 832.66 0.52439 14618.68
 Rate of $q^{*}$ $t^{*}_{1}$ $E[TCU(t^{*}_{1},q^{*})]$ Rate of $q^{*}$ $t^{*}_{1}$ $E[TCU(t^{*}_{1},q^{*})]$ A change h change -0.3 730.29 0.3613 11485.71 -0.3 778.49 0.4845 11382.55 -0.2 730.29 0.3789 11544.89 -0.2 761.38 0.4564 11477.72 -0.1 730.29 0.3985 11601.68 -0.1 745.35 0.4326 11568.80 0 730.29 0.4121 11656.35 0 730.29 0.4121 11656.35 0.1 730.29 0.4278 11709.14 0.1 716.11 0.3943 11740.79 0.2 730.29 0.4431 11760.24 0.2 702.72 0.3786 11822.47 0.3 730.29 0.4578 11809.83 0.3 690.06 0.3646 11901.69 $h_2$ -0.3 806.47 0.4121 11552.87 $\beta$ -0.3 730.29 0.40802 11608.71 -0.2 778.49 0.4121 11588.52 -0.2 730.29 0.40939 11624.56 -0.1 753.24 0.4121 11622.97 -0.1 730.29 0.41077 11640.44 0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35 0.1 709.32 0.4121 11688.73 0.1 730.29 0.41356 11672.29 0.2 690.06 0.4121 11720.21 0.2 730.29 0.41497 11688.26 0.3 672.29 0.4121 11750.86 0.3 730.29 0.41639 11704.27 $C_t$ -0.3 730.29 0.4121 11477.42 $t_r$ -0.3 730.29 0.41222 11643.20 -0.2 730.29 0.4121 11540.70 -0.2 730.29 0.41220 11647.58 -0.1 730.29 0.4121 11600.13 -0.1 730.29 0.41218 11651.96 0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35 0.1 730.29 0.4121 11709.81 0.1 730.29 0.41214 11660.73 0.2 730.29 0.4121 11760.90 0.2 730.29 0.41212 11665.11 0.3 730.29 0.4121 11809.90 0.3 730.29 0.41210 11669.50 P -0.3 730.29 0.7005 11590.45 D -0.3 611.01 0.31465 8589.613 -0.2 730.29 0.5664 11537.73 -0.2 653.19 0.34625 9625.374 -0.1 730.29 0.4767 11628.08 -0.1 692.82 0.37862 10647.10 0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35 0.1 730.29 0.3633 11678.38 0.1 765.94 0.44731 12654.21 0.2 730.29 0.3249 11696.05 0.2 800.00 0.48453 13641.47 0.3 730.29 0.2940 11710.54 0.3 832.66 0.52439 14618.68
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