American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021093
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Risk minimization inventory model with a profit target and option contracts under spot price uncertainty

 1 School of Economics and Management, Southwest University of Science and Technology, Mianyang, 621010, China 2 Zhongshan Institute, University of Electronic Science and Technology of China, Zhongshan, 528402, China 3 School of Marketing and Logistics Management, Nanjing University of Finance & Economics, Nanjing, 210023, China

*Corresponding author: Nana Wan

Received  June 2020 Revised  March 2021 Early access May 2021

This paper aims to analyze the inventory purchasing model for a manufacturer with an objective of minimizing risk and a constraint on profit target, where the manufacturer buys the components from the supplier or in the spot market and tailors them into the final products to meet a deterministic demand. This paper develops the mean-variance optimization models without and with option contracts, and conducts numerical examples to explore how the target profit level, the spot price uncertainty and option contracts affect the manufacturer's optimal solutions and the level of risk. It is shown that without and with option contracts the manufacturer's level of risk is non-decreasing in the target profit level. With (without) option contracts, the manufacturer suffers a zero risk from a higher spot price uncertainty if the profit target is low, whereas suffers a lower (higher) risk from a higher spot price uncertainty if the profit target is high. Finally, the level of risk faced by the manufacturer is not higher with option contracts than without them. This paper facilitates the application of option contracts in inventory purchasing management with a spot market for the risk minimization manufacturer with a profit target consideration. New insights are also provided for the manufacturer to set an appropriate profit target for an affordable level of risk, and establish the risk observation mechanism for hedging against the spot price volatility effectively.

Citation: Nana Wan, Li Li, Xiaozhi Wu, Jianchang Fan. Risk minimization inventory model with a profit target and option contracts under spot price uncertainty. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021093
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References:
The timeline of the event
The impact of the target profit level
The impact of the spot price uncertainty when $K>\left(r-w\right)D$
Notations
 Notations Descriptions $p_{s}$ Random spot price. $f(p_{s})$, $F(p_{s})$ PDF and CDF function of $p_{s}$. $\mu_{s}$, $\sigma_{s}$ Mean and standard deviation of $p_{s}$. $D$ Deterministic demand. $r$ Unit retail price of the final product. $\omega$ Unit wholesale price of the component. $o$ Unit option price. $e$ Unit exercise price. $K$ The target profit level. $q_{0}$ The firm order quantity without option contracts. $q_{1}$ The firm order quantity with option contracts. $q_{2}$ The options order quantity $H$ $\int_{e}^{+\infty}\left(p_{s}-e\right)f\left(p_{s}\right){\rm d}p_{s}$ $J$ $\int_{e}^{+\infty}\left(p_{s}-\mu_{s}\right)\left(p_{s}-e\right)f\left(p_{s}\right){\rm d}p_{s}$ $L$ $\int_{e}^{+\infty}\left(p_{s}-e\right)^{2}f\left(p_{s}\right){\rm d}p_{s}$ $E(\cdot)$ Expected value. $V(\cdot)$ Variance value.
 Notations Descriptions $p_{s}$ Random spot price. $f(p_{s})$, $F(p_{s})$ PDF and CDF function of $p_{s}$. $\mu_{s}$, $\sigma_{s}$ Mean and standard deviation of $p_{s}$. $D$ Deterministic demand. $r$ Unit retail price of the final product. $\omega$ Unit wholesale price of the component. $o$ Unit option price. $e$ Unit exercise price. $K$ The target profit level. $q_{0}$ The firm order quantity without option contracts. $q_{1}$ The firm order quantity with option contracts. $q_{2}$ The options order quantity $H$ $\int_{e}^{+\infty}\left(p_{s}-e\right)f\left(p_{s}\right){\rm d}p_{s}$ $J$ $\int_{e}^{+\infty}\left(p_{s}-\mu_{s}\right)\left(p_{s}-e\right)f\left(p_{s}\right){\rm d}p_{s}$ $L$ $\int_{e}^{+\infty}\left(p_{s}-e\right)^{2}f\left(p_{s}\right){\rm d}p_{s}$ $E(\cdot)$ Expected value. $V(\cdot)$ Variance value.
The impact of the wholesale price when $K>\left(r-w\right)D$
 $w$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 9.9 114.75 14222.00 86.66 30.37 12959.00 10.0 116.67 18148.10 75.91 43.33 15385.30 10.1 118.64 22709.90 64.74 56.36 17595.60 10.2 120.69 27966.70 53.77 68.78 19508.20 10.3 122.81 33983.80 43.51 80.10 21087.80 10.4 125.00 40833.30 34.24 96.07 22337.60
 $w$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 9.9 114.75 14222.00 86.66 30.37 12959.00 10.0 116.67 18148.10 75.91 43.33 15385.30 10.1 118.64 22709.90 64.74 56.36 17595.60 10.2 120.69 27966.70 53.77 68.78 19508.20 10.3 122.81 33983.80 43.51 80.10 21087.80 10.4 125.00 40833.30 34.24 96.07 22337.60
The impact of the option price when $K>\left(r-w\right)D$
 $o$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 2.9 116.67 18148.10 68.18 50.65 13999.33 3.0 116.67 18148.10 75.91 43.33 15385.30 3.1 116.67 18148.10 85.73 33.49 16620.79 3.2 116.67 18148.10 97.34 21.31 17563.07 3.3 116.67 18148.10 110.04 7.44 18078.89 3.4 116.67 18148.10 N.A. N.A. N.A.
 $o$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 2.9 116.67 18148.10 68.18 50.65 13999.33 3.0 116.67 18148.10 75.91 43.33 15385.30 3.1 116.67 18148.10 85.73 33.49 16620.79 3.2 116.67 18148.10 97.34 21.31 17563.07 3.3 116.67 18148.10 110.04 7.44 18078.89 3.4 116.67 18148.10 N.A. N.A. N.A.
The impact of the exercise price when $K>\left(r-w\right)D$
 $e$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 7.8 116.67 18148.10 63.96 54.52 13533.20 8.0 116.67 18148.10 75.91 43.33 15385.30 8.2 116.67 18148.10 89.21 30.03 16847.18 8.4 116.67 18148.10 102.50 15.95 17776.12 8.6 116.67 18148.10 114.54 2.47 18138.83 8.8 116.67 18148.10 N.A. N.A. N.A
 $e$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 7.8 116.67 18148.10 63.96 54.52 13533.20 8.0 116.67 18148.10 75.91 43.33 15385.30 8.2 116.67 18148.10 89.21 30.03 16847.18 8.4 116.67 18148.10 102.50 15.95 17776.12 8.6 116.67 18148.10 114.54 2.47 18138.83 8.8 116.67 18148.10 N.A. N.A. N.A
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