Article Contents
Article Contents

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

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.

Mathematics Subject Classification: Primary: 90B50; Secondary: 91A35, 91A80.

 Citation:

• Figure 1.  The retailer's order quantity changes with $\lambda_R$ under $U(0,250)$

Figure 2.  The retailer's order quantity changes with $\lambda_R$ under $N(100,60)$

Figure 3.  The retailer's order quantity changes with $\lambda_S$ under $U(0,250)$ with $\lambda_R = 2$

Figure 4.  The retailer's order quantity changes with $\lambda_S$ under $N(100,60)$ with $\lambda_R = 10$

Figure 5.  The difference of player's expected utility changes with $\lambda_R$ under $U(0,250)$

Figure 6.  The difference of player's expected utility changes with $\lambda_S$ under $U(0,250)$

Figure 7.  The supplier's expected utility changes with $w_j$ under $U(0,250)$

Figure 8.  The supplier's expected utility changes with $w_j$ under $N(100,60)$

Figure 9.  The retailer's order quantity changes with $\Omega$ under $U(0,250)$ with $\lambda_R = 2$, $\lambda_S = 1.5$

Figure 10.  The retailer's order quantity changes with $\Omega$ under $N(100,60)$with $\lambda_R = 10$, $\lambda_S = 2$

Figure 11.  The difference of player's expected utility changes with $\Omega$ under $U(0,250)$

Table 1.  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

Table 2.  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".

Table 3.  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

Table 4.  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

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Tables(4)