# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2019027

## Optimal credit periods under two-level trade credit

 1 School of Business Administration, Hunan University, Changsha, Hunan Province 410082, China 2 School of Business, State University of New York at Oswego, Oswego, NY 13126, USA

* Corresponding author: ottoyang@126.com(HonglinYang)

Received  January 2018 Revised  September 2018 Published  May 2019

Fund Project: This work was supported by the National Natural Science Foundation of China under Grant Nos. 71571065, 71790593 and 71521061

In a two-echelon single-supplier and single-retailer supply chain with permissible delay in payment, we investigate the two-level trade credit policy in which the supplier offers the retailer with limited capital a credit period and in turn the retailer also provides a credit period to customers. The demand rate is sensitive to both retail price and the customerso credit period. By using the backward induction method, we analytically derive the unique equilibrium of both credit periods in the Stackelberg game to determine the retaileros pricing strategy. We find that the optimal retail price is not always decreasing in the credit period offered by the supplier to the retailer. In addition, we characterize the conditions under which the retailer is willing to voluntarily provide customers a credit period. Numerical examples and sensitivity analysis of key parameters are presented to illustrate the theoretical results and managerial insights.

Citation: Honglin Yang, Heping Dai, Hong Wan, Lingling Chu. Optimal credit periods under two-level trade credit. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019027
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The case of two-level trade credit
$T+N\leq M$
$M<T+N$
$M, N$ and $p$ in case 1
$N, D$ and $TPR$ in case 1
$M, TPR$ and $TPS$ in case 1
M, N and p in case 1
N, D and TPR in case 2
M, TPR and TPS in case 2
M, N and p in case 3
N, D and TPR in case 2
The case of two-level trade credit
The comparisons of three models with permissible delay in payment
 Trade credit policy Demand Decision variable Optimal delay period one-level price-and-time $M$ obtain the analytic solution of (Yangos et al., 2017) dependent optimal $M$ two-level price-and-credit constant give a solution algorithm (Shahos et al., 2015) dependent of optimal two-level price-and-credit $M,N$ obtain the analytic solutions (our paper) dependent of optimal $M$ and $N$ Note:M represents the credit period offered by the supplier to the retailer, and N represents the credit period provided by the retailer to customers.
 Trade credit policy Demand Decision variable Optimal delay period one-level price-and-time $M$ obtain the analytic solution of (Yangos et al., 2017) dependent optimal $M$ two-level price-and-credit constant give a solution algorithm (Shahos et al., 2015) dependent of optimal two-level price-and-credit $M,N$ obtain the analytic solutions (our paper) dependent of optimal $M$ and $N$ Note:M represents the credit period offered by the supplier to the retailer, and N represents the credit period provided by the retailer to customers.
Notations and explanations
 Notation Definition $w$ wholesale price per unit. $p$ retail price per unit (decision variable). $M$ credit period offered by the supplier (decision variable). $N$ credit period provided by the retailer (decision variable) $T$ ordering cycle. $D(p,N)$ customers$'$ annual demand rate depending on $p$ and $N$. $I_c$ retailer$'$s interest charged per dollar per year. $I_e$ retailer$'$s interest earned per dollar per year. $c$ supplier$'$s procurement cost per unit. $h$ retailer$'$s holding cost per unit. $A$ ordering cost per order. $TPS$ supplier$'$s annual total profit. $TPR$ retailer$'$s annual total profit.
 Notation Definition $w$ wholesale price per unit. $p$ retail price per unit (decision variable). $M$ credit period offered by the supplier (decision variable). $N$ credit period provided by the retailer (decision variable) $T$ ordering cycle. $D(p,N)$ customers$'$ annual demand rate depending on $p$ and $N$. $I_c$ retailer$'$s interest charged per dollar per year. $I_e$ retailer$'$s interest earned per dollar per year. $c$ supplier$'$s procurement cost per unit. $h$ retailer$'$s holding cost per unit. $A$ ordering cost per order. $TPS$ supplier$'$s annual total profit. $TPR$ retailer$'$s annual total profit.
Sensitivity analysis with respect to γ
 Parameter Value $M^*$ $N^*$ $p^*$ $\pi_r^*$ $\pi_s^*$ $\gamma$ 4500 0.5938 0.0000 2185.6 2295832 4595664 5000 0.5938 0.0000 2185.6 2295832 4595664 5500 0.5938 0.0000 2185.6 2295832 4595664 6000 0.5938 0.0000 2185.6 2295832 4595664 6500 0.7252 0.3502 2221.8 2500038 4619147 7000 0.8333 0.5833 2248.3 2720222 4666667 7500 0.9229 0.7979 2276.4 2957288 4734861 8000 0.9974 0.9974 2306.0 3212169 4821253 8500 1.0613 1.0613 2316.5 3481700 4918165 9000 1.1181 1.1181 2326.9 3762082 5018776
 Parameter Value $M^*$ $N^*$ $p^*$ $\pi_r^*$ $\pi_s^*$ $\gamma$ 4500 0.5938 0.0000 2185.6 2295832 4595664 5000 0.5938 0.0000 2185.6 2295832 4595664 5500 0.5938 0.0000 2185.6 2295832 4595664 6000 0.5938 0.0000 2185.6 2295832 4595664 6500 0.7252 0.3502 2221.8 2500038 4619147 7000 0.8333 0.5833 2248.3 2720222 4666667 7500 0.9229 0.7979 2276.4 2957288 4734861 8000 0.9974 0.9974 2306.0 3212169 4821253 8500 1.0613 1.0613 2316.5 3481700 4918165 9000 1.1181 1.1181 2326.9 3762082 5018776
Sensitivity analysis with respect to Ie and Ic
 Parameter Value $M^*$ $N^*$ $p^*$ $\pi_r^*$ $\pi_s^*$ $I_e$ 0.050 1.2530 1.0863 2289.8 3437846 4914066 0.055 1.0245 0.8245 2269.0 3035272 4772856 0.060 0.8333 0.5833 2248.3 2720222 4666667 0.065 0.6701 0.3368 2224.8 2469392 4589728 0.070 0.5268 0.0000 2186.9 2269113 4542227 $I_c$ 0.07 0.8244 0.8244 2281.5 2734825 4691700 0.075 0.8304 0.6637 2259.4 2725086 4675004 0.080 0.8333 0.5833 2248.3 2720222 4666667 0.085 0.8351 0.5351 2241.7 2717306 4661668 0.090 0.8363 0.5030 2237.3 2715363 4658337
 Parameter Value $M^*$ $N^*$ $p^*$ $\pi_r^*$ $\pi_s^*$ $I_e$ 0.050 1.2530 1.0863 2289.8 3437846 4914066 0.055 1.0245 0.8245 2269.0 3035272 4772856 0.060 0.8333 0.5833 2248.3 2720222 4666667 0.065 0.6701 0.3368 2224.8 2469392 4589728 0.070 0.5268 0.0000 2186.9 2269113 4542227 $I_c$ 0.07 0.8244 0.8244 2281.5 2734825 4691700 0.075 0.8304 0.6637 2259.4 2725086 4675004 0.080 0.8333 0.5833 2248.3 2720222 4666667 0.085 0.8351 0.5351 2241.7 2717306 4661668 0.090 0.8363 0.5030 2237.3 2715363 4658337
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