# American Institute of Mathematical Sciences

March  2021, 17(2): 711-732. doi: 10.3934/jimo.2019131

## Loss-averse supply chain decisions with a capital constrained retailer

 1 School of Marketing and Logistics Management, Nanjing University of Finance and Economics, Nanjing, Jiangsu Province 210023, China 2 School of Business Administration, Hunan University, Changsha, Hunan Province 410082, China 3 Department of Industrial and Systems Engineering, School of Engineering and Sciences, Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo León, México 4 School of Business, State University of New York at Oswego, Oswego, NY 13126, USA

* Corresponding author: ottoyang@126.com (Honglin Yang)

Received  February 2019 Revised  July 2019 Published  October 2019

Fund Project: This research is supported by the National Natural Science Foundation of China under Grant Nos. 71571065, 71521061 and 71790593 and the Ministry of Education in China of Humanities and Social Science Project under Grant No. 19YJC630242

In real-world transactions, capital constraints restrict the rapid development of the enterprises in the supply chain. The loss aversion behaviors of enterprises directly affect the decision making. This paper investigates the optimal decisions of both the supplier and the capital constrained retailer being loss aversion decision makers under different financing strategies. The capital constrained retailer may borrow from a bank or use the supplier's trade credit to satisfy uncertain demand. With a wholesale price contract, we analytically solve the unique Stackelberg equilibrium under two financing schemes. We derive the critical wholesale price that determines the retailer's financing preference. We identify the impacts of the loss aversion coefficients and initial capital level on the operational and financing decisions. Numerical examples reveal that there exists a Pareto improvement zone regarding the retailer's loss aversion coefficient and initial capital level.

Citation: Wenyan Zhuo, Honglin Yang, Leopoldo Eduardo Cárdenas-Barrón, Hong Wan. Loss-averse supply chain decisions with a capital constrained retailer. Journal of Industrial & Management Optimization, 2021, 17 (2) : 711-732. doi: 10.3934/jimo.2019131
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##### References:
The retailer's order quantity changes with $\lambda_R$ under $U(0,250)$
The retailer's order quantity changes with $\lambda_R$ under $N(100,60)$
The retailer's order quantity changes with $\lambda_S$ under $U(0,250)$ with $\lambda_R = 2$
The retailer's order quantity changes with $\lambda_S$ under $N(100,60)$ with $\lambda_R = 10$
The difference of player's expected utility changes with $\lambda_R$ under $U(0,250)$
The difference of player's expected utility changes with $\lambda_S$ under $U(0,250)$
The supplier's expected utility changes with $w_j$ under $U(0,250)$
The supplier's expected utility changes with $w_j$ under $N(100,60)$
The retailer's order quantity changes with $\Omega$ under $U(0,250)$ with $\lambda_R = 2$, $\lambda_S = 1.5$
The retailer's order quantity changes with $\Omega$ under $N(100,60)$with $\lambda_R = 10$, $\lambda_S = 2$
The difference of player's expected utility changes with $\Omega$ under $U(0,250)$
Comparisons of three models with loss aversion
 Literature Financing scheme Loss-averse player Decision objective Upstream Downstream Zhang et al.(2016) BCF Retailer EPM EUM Yan et al.(2018) TCF or SI Retailer EPM EUM This paper BCF or TCF Retailer and Supplier EUM EUM
 Literature Financing scheme Loss-averse player Decision objective Upstream Downstream Zhang et al.(2016) BCF Retailer EPM EUM Yan et al.(2018) TCF or SI Retailer EPM EUM This paper BCF or TCF Retailer and Supplier EUM EUM
Notation
 Notation Definition $p$ Retail price $c$ Production cost $w_j$ Wholesale price, where $j=B,T$ denotes bank credit financing or trade credit financing, respectively (the supplier's decision variable) $X$ Random demand, defined over continuous interval$[0,+∞)$ $f(X)$ Probability density function of $X$ $F(X)$ Cumulative distribution function of $X$ $z(X)$ Failure rate of the demand distribution, $z(x)=\frac{f(x)}{\bar{F}(x)}$ $q_j$ Order quantity, where $j=B,T$ (the retailer's decision variable) $\lambda_i$ Loss-aversion coefficient, where $i=S,R$ denotes the supplier or retailer, respectively $\pi_{ij}$ Profit of player $i$ under financing scheme $j$, where $i=S,R$ and $j=B,T$ $U(\pi)$ Utility function $EU(\pi_{ij})$ Expected utility function $\Omega$ Retailer's initial capital level $r_f$ Risk-free interest rate $r_B$ Interest rate of bank loans $r_T$ Interest rate of trade credit (the supplier's decision variable) For notation purposes, we use the symbols "BCF" and "TCF" to represent "bank credit financing" and "trade credit financing", respectively. In addition, for convenience, we refer to the supplier as "she" and the retailer as "he".
 Notation Definition $p$ Retail price $c$ Production cost $w_j$ Wholesale price, where $j=B,T$ denotes bank credit financing or trade credit financing, respectively (the supplier's decision variable) $X$ Random demand, defined over continuous interval$[0,+∞)$ $f(X)$ Probability density function of $X$ $F(X)$ Cumulative distribution function of $X$ $z(X)$ Failure rate of the demand distribution, $z(x)=\frac{f(x)}{\bar{F}(x)}$ $q_j$ Order quantity, where $j=B,T$ (the retailer's decision variable) $\lambda_i$ Loss-aversion coefficient, where $i=S,R$ denotes the supplier or retailer, respectively $\pi_{ij}$ Profit of player $i$ under financing scheme $j$, where $i=S,R$ and $j=B,T$ $U(\pi)$ Utility function $EU(\pi_{ij})$ Expected utility function $\Omega$ Retailer's initial capital level $r_f$ Risk-free interest rate $r_B$ Interest rate of bank loans $r_T$ Interest rate of trade credit (the supplier's decision variable) For notation purposes, we use the symbols "BCF" and "TCF" to represent "bank credit financing" and "trade credit financing", respectively. In addition, for convenience, we refer to the supplier as "she" and the retailer as "he".
Sensitivity analysis with respect to ${\lambda _R}$
 $\lambda_R$ $w_B$ $w_T$ $q_B$ $q_T$ $\Delta EU(\pi_{R})$ $\Delta EU(\pi_{S})$ $\Delta EU(\pi_{SC})$ 1.0 56.0 64.7 105.8 112.9 -681.9 921.6 239.7 1.5 53.1 59.3 104.3 110.4 -512.6 690.1 177.5 2.0 50.5 55.1 102.7 108.5 -379.6 536.9 157.3 2.5 48.3 51.7 101.2 106.7 -280.4 426.7 146.3 3.0 46.3 48.8 99.9 105.2 -207.1 343.6 136.5 3.5 44.5 46.4 98.8 103.6 -149.3 277.3 128.0 4.0 43.0 44.2 97.3 102.4 -75.8 224.0 148.2 4.5 41.6 42.4 95.9 100.7 -37.0 180.0 143.0 5.0 40.3 40.7 94.6 99.0 -8.5 142.9 134.3 5.5 39.1 39.2 93.4 98.1 38.4 111.3 149.7 6.0 38.0 37.8 92.1 96.9 73.4 84.1 157.5 6.5 37.0 36.6 90.8 95.5 95.0 60.3 155.3 7.0 36.0 35.5 89.9 94.0 118.3 39.9 158.2 7.5 35.1 34.4 88.7 93.1 135.3 21.2 156.5 8.0 34.3 33.4 87.3 91.9 153.1 4.6 157.7 8.5 33.5 32.5 86.4 90.8 155.4 -10.4 145.0 9.0 32.8 31.7 85.2 89.5 157.0 -23.8 133.2 9.5 32.2 30.9 83.6 88.4 175.9 -35.9 140.0 10.0 31.6 30.2 82.3 87.3 187.2 -47.1 140.1
 $\lambda_R$ $w_B$ $w_T$ $q_B$ $q_T$ $\Delta EU(\pi_{R})$ $\Delta EU(\pi_{S})$ $\Delta EU(\pi_{SC})$ 1.0 56.0 64.7 105.8 112.9 -681.9 921.6 239.7 1.5 53.1 59.3 104.3 110.4 -512.6 690.1 177.5 2.0 50.5 55.1 102.7 108.5 -379.6 536.9 157.3 2.5 48.3 51.7 101.2 106.7 -280.4 426.7 146.3 3.0 46.3 48.8 99.9 105.2 -207.1 343.6 136.5 3.5 44.5 46.4 98.8 103.6 -149.3 277.3 128.0 4.0 43.0 44.2 97.3 102.4 -75.8 224.0 148.2 4.5 41.6 42.4 95.9 100.7 -37.0 180.0 143.0 5.0 40.3 40.7 94.6 99.0 -8.5 142.9 134.3 5.5 39.1 39.2 93.4 98.1 38.4 111.3 149.7 6.0 38.0 37.8 92.1 96.9 73.4 84.1 157.5 6.5 37.0 36.6 90.8 95.5 95.0 60.3 155.3 7.0 36.0 35.5 89.9 94.0 118.3 39.9 158.2 7.5 35.1 34.4 88.7 93.1 135.3 21.2 156.5 8.0 34.3 33.4 87.3 91.9 153.1 4.6 157.7 8.5 33.5 32.5 86.4 90.8 155.4 -10.4 145.0 9.0 32.8 31.7 85.2 89.5 157.0 -23.8 133.2 9.5 32.2 30.9 83.6 88.4 175.9 -35.9 140.0 10.0 31.6 30.2 82.3 87.3 187.2 -47.1 140.1
Sensitivity analysis with respect to ${\lambda _S}$
 $\lambda_S$ $w_B$ $w_T$ $q_B$ $q_T$ $\Delta EU(\pi_{R})$ $\Delta EU(\pi_{S})$ $\Delta EU(\pi_{SC})$ 1 65.2 77.8 104.7 116.2 -1642.9 1649.4 6.5 5 65.2 78.6 104.7 114.3 -1711.1 1521.3 -189.8 10 65.2 79.6 104.7 112.0 -1788.8 1369.2 -419.6 15 65.2 80.5 104.7 109.7 -1859.1 1225.7 -633.4 20 65.2 81.3 104.7 107.5 -1922.8 1090.5 -832.3 25 65.2 82.1 104.7 105.3 -1980.4 963.1 -1017.3 30 65.2 82.8 104.7 103.1 -2032.5 843.3 -1189.2 35 65.2 83.4 104.7 100.9 -2079.6 730.6 -1349.0 40 65.2 84.1 104.7 98.9 -2122.0 624.8 -1497.2 45 65.2 84.6 104.7 96.8 -2160.3 525.5 -1634.8 50 65.2 85.1 104.7 94.8 -2194.6 432.3 -1762.3 55 65.2 85.6 104.7 92.9 -2225.5 344.8 -1880.7 60 65.2 86.0 104.7 91.0 -2253.2 262.8 -1990.4 65 65.2 86.3 104.7 89.2 -2278.1 185.9 -2092.2 70 65.2 86.7 104.7 87.5 -2300.4 113.6 -2186.8 75 65.2 87.0 104.7 85.8 -2320.5 45.8 -2274.7 80 65.2 87.3 104.7 84.2 -2338.6 -17.9 -2356.5 85 65.2 87.5 104.7 82.7 -2354.9 -77.9 -2432.8 90 65.2 87.7 104.7 81.2 -2369.6 -134.3 -2503.9 95 65.2 88.0 104.7 79.8 -2382.9 -187.5 -2570.4 100 65.2 88.2 104.7 78.5 -2394.9 -237.7 -2632.6
 $\lambda_S$ $w_B$ $w_T$ $q_B$ $q_T$ $\Delta EU(\pi_{R})$ $\Delta EU(\pi_{S})$ $\Delta EU(\pi_{SC})$ 1 65.2 77.8 104.7 116.2 -1642.9 1649.4 6.5 5 65.2 78.6 104.7 114.3 -1711.1 1521.3 -189.8 10 65.2 79.6 104.7 112.0 -1788.8 1369.2 -419.6 15 65.2 80.5 104.7 109.7 -1859.1 1225.7 -633.4 20 65.2 81.3 104.7 107.5 -1922.8 1090.5 -832.3 25 65.2 82.1 104.7 105.3 -1980.4 963.1 -1017.3 30 65.2 82.8 104.7 103.1 -2032.5 843.3 -1189.2 35 65.2 83.4 104.7 100.9 -2079.6 730.6 -1349.0 40 65.2 84.1 104.7 98.9 -2122.0 624.8 -1497.2 45 65.2 84.6 104.7 96.8 -2160.3 525.5 -1634.8 50 65.2 85.1 104.7 94.8 -2194.6 432.3 -1762.3 55 65.2 85.6 104.7 92.9 -2225.5 344.8 -1880.7 60 65.2 86.0 104.7 91.0 -2253.2 262.8 -1990.4 65 65.2 86.3 104.7 89.2 -2278.1 185.9 -2092.2 70 65.2 86.7 104.7 87.5 -2300.4 113.6 -2186.8 75 65.2 87.0 104.7 85.8 -2320.5 45.8 -2274.7 80 65.2 87.3 104.7 84.2 -2338.6 -17.9 -2356.5 85 65.2 87.5 104.7 82.7 -2354.9 -77.9 -2432.8 90 65.2 87.7 104.7 81.2 -2369.6 -134.3 -2503.9 95 65.2 88.0 104.7 79.8 -2382.9 -187.5 -2570.4 100 65.2 88.2 104.7 78.5 -2394.9 -237.7 -2632.6