# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020118

## Order allocation model in logistics service supply chain with demand updating and inequity aversion: A perspective of two option contracts comparison

 College of Management and Economics, Tianjin University, No.92, Weijin Road, Nankai District, Tianjin, 300072, China

* Corresponding author

** Co-Corresponding author

Received  August 2019 Revised  January 2020 Published  June 2020

This paper considers an logistics service supply chain consisting of a logistics service integrator (LSI) and a number of functional logistics service providers (FLSPs). In the environment of demand updating, we focus on the inequity aversion among the FLSPs and introduce two option contracts (the reservation option contract and the option guarantee contract), build the multi-objective programming models, to explore effects of the inequity aversion behavior on the order allocation, and whether the two option contracts can mitigate the impact of inequity aversion on order allocation. Three important conclusions are obtained after two option contracts comparisons: first, there is an optimal update time, at which point, the order allocation results reach the optimal value and tend to be stable. Second, two option contracts both can not only increase the total performance of the supply chain, but also mitigate the impact of inequity aversion on the allocation under certain conditions. Third, when demand decreases, the reservation option contract is better than option guarantee contract, in contrast, when demand increases, option guarantee contract is better.

Citation: Weihua Liu, Xinran Shen, Di Wang, Jingkun Wang. Order allocation model in logistics service supply chain with demand updating and inequity aversion: A perspective of two option contracts comparison. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020118
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##### References:
Utility of the LSI when $\xi<\mu$
Utility of the LSI when $\xi>\mu$
Utility of the FLSPs when $\xi<\mu$
Utility of the FLSPs when $\xi>\mu$
Total performance when $\xi<\mu$
Total performance when $\xi>\mu$
Utility of the LSI when $\xi<\mu$
Utility of the LSI when $\xi>\mu$
Utility of FLSPs when $\xi<\mu$
Utility of FLSPs when $\xi>\mu$
Total performance when $\xi<\mu$
Total performance when $\xi>\mu$
$\Delta\Pi_{LSI}^{1-2}$ in the case of decreased demand
$\Delta \Pi_{LSI}^{1-3}$ in the case of decreased demand
$\Delta U_{FLSP}^{1-2}$ in the case of decreased demand
$\Delta U_{FLSP}^{1-3}$ in the case of decreased demand
$\Delta TP^{1-2}$ in the case of decreased demand
$\Delta TP^{1-3}$ in the case of decreased demand
$\Delta\Pi_{LSI}^{1-2}$ in the case of increased demand
$\Delta\Pi_{LSI}^{1-3}$ in the case of increased demand
$\Delta\Pi_{FLSP}^{1-2}$ in the case of increased demand
$\Delta\Pi_{FLSP}^{1-3}$ in the case of increased demand
$\Delta TP^{1-2}$ in the case of increased demand
$\Delta TP^{1-3}$ in the case of increased demand
The differences between the relevant literatures and this paper
 Research content [21] [26] [6] This paper Supply chain structure A single provider and a single retailer Logistics service supply chain with single LSI and multiple FLSPs Agricultural product supply chain with a single manufacturer and a single retailer Logistics service supply chain with single LSI and multiple FLSPs Demand updating is considered $\times$ $\surd$ $\times$ $\surd$ Option types Reservation option $\times$ Derivative option Reservation option and derivative option Fairness concern is considered $\times$ $\surd$ $\times$ $\surd$ Research problem Supply chain coordination and performance management Behavioral management in order allocation Supply chain performance and risk management The effect of option and behavior on the performance of order allocation
 Research content [21] [26] [6] This paper Supply chain structure A single provider and a single retailer Logistics service supply chain with single LSI and multiple FLSPs Agricultural product supply chain with a single manufacturer and a single retailer Logistics service supply chain with single LSI and multiple FLSPs Demand updating is considered $\times$ $\surd$ $\times$ $\surd$ Option types Reservation option $\times$ Derivative option Reservation option and derivative option Fairness concern is considered $\times$ $\surd$ $\times$ $\surd$ Research problem Supply chain coordination and performance management Behavioral management in order allocation Supply chain performance and risk management The effect of option and behavior on the performance of order allocation
Notations
 Notation Description $c_{i, opp}$ Opportunity cost of FLSP $i$ at the time $t$ $C_I$ Demand updating cost of LSI $c_i$ The cost of unit logistics capacity for FLSP $i$ $d_{i, 0}$ Initial utility of FLSP $i$ $d_{i, 1}$ Satisfaction utility of FLSP $i$ with the reservation option in the second stage $d_{i, 2}$ Satisfaction utility of FLSP $i$ with the option guarantee in the second stage $D$ Demand in the first stage, which subjects to the normal distribution $D\thicksim N(\mu, \sigma^2)$ $e_i$ Option purchase price of unit logistics capacity for FLSP $i$ in reservation option contract $h_i$ Option executive price of unit logistics capacity for FLSP $i$ in reservation option contract $n$ The total number of FLSPs $p_i$ The market price at which the LSI buys the unit logistics capacity from the FLSP $i$ $q$ Option purchase quantity for LSI in option guarantee contract $d_{i, 2}$ Satisfaction utility of FLSP $i$ with the option guarantee in the second stage $D$ Demand in the first stage, which subjects to the normal distribution $D\thicksim N(\mu, \sigma^2)$ $Q$ The updated demand in the second stage, which subjects to the normal distribution $(Q\mid\xi)\thicksim N(\mu(\xi), \nu^2)$ $r$ Option purchase price of unit logistics capacity for LSI in option guarantee contract $R$ Option compensation price of unit logistics capacity for LSI in option guarantee contract $v_{i, 1}$ In the first stage, profit utility of FLSP $i$ in reservation option contract $v_{i, 2}$ In the second stage, profit utility of FLSP $i$ in reservation option contract $w_{i1}$ The weight of satisfaction utility of FLSP $i$ $w_{i2}$ The weight of profit utility of FLSP $i$ $x_{i, 1}$ In the first stage, logistics service capacity provided by FLSP $i$ in reservation option contract $x_{i, 2}$ In the second stage, logistics service capacity provided by FLSP $i$ in reservation option contract $y_{i, 1}$ In the first stage, logistics service capacity provided by FLSP $i$ in option guarantee contract $y_{i, 2}$ In the second stage, logistics service capacity provided by FLSP $i$ in option guarantee contract $z_1$ The total profit of the LSI $z_2$ The total utility of the FLSP $\alpha_i$ Advantage unfair coefficient of FLSP $i$ $\beta_i$ Disadvantage unfair coefficient of FLSP $i$ $\lambda$ Unit logistics capacity income $\xi$ Based on the demanded sample information collected in the lead time, the estimated mean of the demand (Estimated Demand Average) $\tau(t)$ Demand forecast error, reflecting the degree of deviation between demand forecast and the actual needs $\theta_i^-$ Minimum logistics capacity provided by FLSP $i$ $\theta_i^+$ Maximum logistics capacity provided by FLSP $i$ $\eta$ Compensation threshold in option guarantee $TP$ Compensation threshold in option guarantee $\Delta LSI$ Change ratio in profit of LSI $\Delta FLSP$ Change ratio in profit of FLSP $\Delta TP$ Change ratio in total performance of supply chain
 Notation Description $c_{i, opp}$ Opportunity cost of FLSP $i$ at the time $t$ $C_I$ Demand updating cost of LSI $c_i$ The cost of unit logistics capacity for FLSP $i$ $d_{i, 0}$ Initial utility of FLSP $i$ $d_{i, 1}$ Satisfaction utility of FLSP $i$ with the reservation option in the second stage $d_{i, 2}$ Satisfaction utility of FLSP $i$ with the option guarantee in the second stage $D$ Demand in the first stage, which subjects to the normal distribution $D\thicksim N(\mu, \sigma^2)$ $e_i$ Option purchase price of unit logistics capacity for FLSP $i$ in reservation option contract $h_i$ Option executive price of unit logistics capacity for FLSP $i$ in reservation option contract $n$ The total number of FLSPs $p_i$ The market price at which the LSI buys the unit logistics capacity from the FLSP $i$ $q$ Option purchase quantity for LSI in option guarantee contract $d_{i, 2}$ Satisfaction utility of FLSP $i$ with the option guarantee in the second stage $D$ Demand in the first stage, which subjects to the normal distribution $D\thicksim N(\mu, \sigma^2)$ $Q$ The updated demand in the second stage, which subjects to the normal distribution $(Q\mid\xi)\thicksim N(\mu(\xi), \nu^2)$ $r$ Option purchase price of unit logistics capacity for LSI in option guarantee contract $R$ Option compensation price of unit logistics capacity for LSI in option guarantee contract $v_{i, 1}$ In the first stage, profit utility of FLSP $i$ in reservation option contract $v_{i, 2}$ In the second stage, profit utility of FLSP $i$ in reservation option contract $w_{i1}$ The weight of satisfaction utility of FLSP $i$ $w_{i2}$ The weight of profit utility of FLSP $i$ $x_{i, 1}$ In the first stage, logistics service capacity provided by FLSP $i$ in reservation option contract $x_{i, 2}$ In the second stage, logistics service capacity provided by FLSP $i$ in reservation option contract $y_{i, 1}$ In the first stage, logistics service capacity provided by FLSP $i$ in option guarantee contract $y_{i, 2}$ In the second stage, logistics service capacity provided by FLSP $i$ in option guarantee contract $z_1$ The total profit of the LSI $z_2$ The total utility of the FLSP $\alpha_i$ Advantage unfair coefficient of FLSP $i$ $\beta_i$ Disadvantage unfair coefficient of FLSP $i$ $\lambda$ Unit logistics capacity income $\xi$ Based on the demanded sample information collected in the lead time, the estimated mean of the demand (Estimated Demand Average) $\tau(t)$ Demand forecast error, reflecting the degree of deviation between demand forecast and the actual needs $\theta_i^-$ Minimum logistics capacity provided by FLSP $i$ $\theta_i^+$ Maximum logistics capacity provided by FLSP $i$ $\eta$ Compensation threshold in option guarantee $TP$ Compensation threshold in option guarantee $\Delta LSI$ Change ratio in profit of LSI $\Delta FLSP$ Change ratio in profit of FLSP $\Delta TP$ Change ratio in total performance of supply chain
Parameter seeting
 FLSP $p_i$ $c_i$ $e_i$ $h_i$ $[\theta_i^-, \theta_i^+]$ $w_{i1}$ $w_{i2}$ $r_i$ $d_{i, 0}$ $A_1$ 24 8 10 16 [15,24] 0.6 0.4 0.5 0.3 $A_2$ 15 5 6 10 [20,35] 0.6 0.4 0.4 0.35 $A_3$ 26 9 11 17 [25,45] 0.7 0.3 0.6 0.35 Note: (i) According to the Assumption1, if $\xi>\mu$, we set $\xi = 120$. If $\xi<\mu$, we set $\xi = 80$. (ii) in this paper, the demand $D$ before updating is a signal, the LSI would update the information and accordingly estimate the mean value as $\xi$. Therefore, the estimated value is effected by the updated information, because the actual demand often changes fiercely, the $\xi$ may be much larger or smaller $\mu$.
 FLSP $p_i$ $c_i$ $e_i$ $h_i$ $[\theta_i^-, \theta_i^+]$ $w_{i1}$ $w_{i2}$ $r_i$ $d_{i, 0}$ $A_1$ 24 8 10 16 [15,24] 0.6 0.4 0.5 0.3 $A_2$ 15 5 6 10 [20,35] 0.6 0.4 0.4 0.35 $A_3$ 26 9 11 17 [25,45] 0.7 0.3 0.6 0.35 Note: (i) According to the Assumption1, if $\xi>\mu$, we set $\xi = 120$. If $\xi<\mu$, we set $\xi = 80$. (ii) in this paper, the demand $D$ before updating is a signal, the LSI would update the information and accordingly estimate the mean value as $\xi$. Therefore, the estimated value is effected by the updated information, because the actual demand often changes fiercely, the $\xi$ may be much larger or smaller $\mu$.
Summary of parameter influence laws
 Dependent variable Independent variable Demand update time $t$ Difference of the fairness Preference among the FLSPs $\xi<\mu$ $\xi>\mu$ $\xi<\mu$ $\xi>\mu$ Model 1 reservation option Utility of LSI $\searrow \longrightarrow$ $\nearrow \longrightarrow$ $\nearrow$ $\nearrow$ Utility of FLSPs $\nearrow \longrightarrow$ $\searrow \longrightarrow$ $\searrow$ $\searrow$ Total performance $\searrow \longrightarrow$ $\nearrow \longrightarrow$ $\nearrow$ $\nearrow$ Model 2 Option derivatives Utility of LSI $\uparrow \searrow \longrightarrow$ $\uparrow \searrow \longrightarrow$ $\rightarrow$ $\rightarrow$ Utility of FLSPs $\searrow \longrightarrow$ $\nearrow \longrightarrow$ $\searrow$ $\searrow$ Total performance $\searrow \longrightarrow$ $\nearrow \longrightarrow$ $\searrow$ $\searrow$ Note: $\nearrow$ indicates with the increase of independent variable, the value of dependent variable will increase, $\searrow$ indicates with the increase of independent variable, the value of dependent variable will decrease, $\longrightarrow$ indicates with the increase of independent variable, the value of dependent variable unchanged, $\uparrow$ indicates with the increase of independent variable, the value of dependent variable will increase suddenly.
 Dependent variable Independent variable Demand update time $t$ Difference of the fairness Preference among the FLSPs $\xi<\mu$ $\xi>\mu$ $\xi<\mu$ $\xi>\mu$ Model 1 reservation option Utility of LSI $\searrow \longrightarrow$ $\nearrow \longrightarrow$ $\nearrow$ $\nearrow$ Utility of FLSPs $\nearrow \longrightarrow$ $\searrow \longrightarrow$ $\searrow$ $\searrow$ Total performance $\searrow \longrightarrow$ $\nearrow \longrightarrow$ $\nearrow$ $\nearrow$ Model 2 Option derivatives Utility of LSI $\uparrow \searrow \longrightarrow$ $\uparrow \searrow \longrightarrow$ $\rightarrow$ $\rightarrow$ Utility of FLSPs $\searrow \longrightarrow$ $\nearrow \longrightarrow$ $\searrow$ $\searrow$ Total performance $\searrow \longrightarrow$ $\nearrow \longrightarrow$ $\searrow$ $\searrow$ Note: $\nearrow$ indicates with the increase of independent variable, the value of dependent variable will increase, $\searrow$ indicates with the increase of independent variable, the value of dependent variable will decrease, $\longrightarrow$ indicates with the increase of independent variable, the value of dependent variable unchanged, $\uparrow$ indicates with the increase of independent variable, the value of dependent variable will increase suddenly.
Application conditions that option can reduce the impact on allocation of inequity aversion
 $\xi<\mu$ Options stype Inequity aversion difference is small Inequity aversion difference is large $\Delta\Pi_{LSI}^{1-2}$ $\Delta U_{FLSP}^{1-2}$ $\Delta TP^{1-2}$ $\Delta\Pi_{LSI}^{1-3}$ $\Delta\ U_{FLSPI}^{1-3}$ $\Delta\ TP^{1-3}$ Option1 $Y$ $Y$ $Y^*$ $Y$ $Y$ $Y^*$ Option2 $Y^*$ $Y^*$ $Y$ $Y^*$ $Y^*$ $Y$ $\xi>\mu$ Option1 $N$ $Y$ $Y$ $N$ $Y$ $N$ Option2 $Y$ $Y^*$ $N$ $Y$ $Y^*$ $Y$ Note: $Y$ indicates that the option contract can weaken the impact of the inequity aversion; $Y^*$ indicates that the weakening effect is better; $N$ indicates that option contract cannot diminish the impact of the inequity aversion.
 $\xi<\mu$ Options stype Inequity aversion difference is small Inequity aversion difference is large $\Delta\Pi_{LSI}^{1-2}$ $\Delta U_{FLSP}^{1-2}$ $\Delta TP^{1-2}$ $\Delta\Pi_{LSI}^{1-3}$ $\Delta\ U_{FLSPI}^{1-3}$ $\Delta\ TP^{1-3}$ Option1 $Y$ $Y$ $Y^*$ $Y$ $Y$ $Y^*$ Option2 $Y^*$ $Y^*$ $Y$ $Y^*$ $Y^*$ $Y$ $\xi>\mu$ Option1 $N$ $Y$ $Y$ $N$ $Y$ $N$ Option2 $Y$ $Y^*$ $N$ $Y$ $Y^*$ $Y$ Note: $Y$ indicates that the option contract can weaken the impact of the inequity aversion; $Y^*$ indicates that the weakening effect is better; $N$ indicates that option contract cannot diminish the impact of the inequity aversion.
Allocation data of none option contract ([26])
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 10 528.280 564.390 641.890 0.305 0.306 0.292 0.191 0.188 0.182 15 561.290 596.290 676.780 0.344 0.340 0.319 0.217 0.214 0.207 20 604.580 642.280 748.980 0.356 0.355 0.347 0.224 0.222 0.215 25 621.360 660.270 746.250 0.362 0.359 0.351 0.228 0.225 0.218 30 628.930 668.260 752.570 0.364 0.359 0.354 0.229 0.225 0.219 35 631.240 671.020 753.180 0.364 0.359 0.354 0.229 0.225 0.219 40 631.240 671.850 754.020 0.364 0.363 0.354 0.229 0.225 0.219 45 631.240 671.980 754.380 0.364 0.363 0.354 0.229 0.227 0.219 50 632.240 672.980 755.380 0.364 0.363 0.354 0.229 0.227 0.219
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 10 528.280 564.390 641.890 0.305 0.306 0.292 0.191 0.188 0.182 15 561.290 596.290 676.780 0.344 0.340 0.319 0.217 0.214 0.207 20 604.580 642.280 748.980 0.356 0.355 0.347 0.224 0.222 0.215 25 621.360 660.270 746.250 0.362 0.359 0.351 0.228 0.225 0.218 30 628.930 668.260 752.570 0.364 0.359 0.354 0.229 0.225 0.219 35 631.240 671.020 753.180 0.364 0.359 0.354 0.229 0.225 0.219 40 631.240 671.850 754.020 0.364 0.363 0.354 0.229 0.225 0.219 45 631.240 671.980 754.380 0.364 0.363 0.354 0.229 0.227 0.219 50 632.240 672.980 755.380 0.364 0.363 0.354 0.229 0.227 0.219
Allocation data of option1 (reservation option contract)
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 5 5212.604 5214.035 5219.711 0.390 0.390 0.390 0.999 1.000 1.000 10 4989.574 4991.008 4996.698 0.385 0.384 0.384 0.999 1.000 1.000 15 4820.468 4821.904 4827.605 0.380 0.380 0.380 0.999 0.999 1.000 20 4731.125 4732.563 4738.269 0.378 0.378 0.378 0.999 0.999 1.000 25 4692.747 4694.186 4699.894 0.377 0.377 0.377 0.999 0.999 1.000 30 4677.743 4679.182 4684.891 0.376 0.376 0.376 0.999 0.999 1.000 35 4672.095 4673.534 4679.243 0.376 0.376 0.376 0.999 0.999 1.000 40 4669.999 4671.438 4677.148 0.376 0.376 0.376 0.999 0.999 1.000 45 4669.226 4670.665 4676.374 0.376 0.376 0.376 0.999 0.999 1.000 50 4668.941 4670.380 4676.090 0.376 0.376 0.376 0.999 0.999 1.000
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 5 5212.604 5214.035 5219.711 0.390 0.390 0.390 0.999 1.000 1.000 10 4989.574 4991.008 4996.698 0.385 0.384 0.384 0.999 1.000 1.000 15 4820.468 4821.904 4827.605 0.380 0.380 0.380 0.999 0.999 1.000 20 4731.125 4732.563 4738.269 0.378 0.378 0.378 0.999 0.999 1.000 25 4692.747 4694.186 4699.894 0.377 0.377 0.377 0.999 0.999 1.000 30 4677.743 4679.182 4684.891 0.376 0.376 0.376 0.999 0.999 1.000 35 4672.095 4673.534 4679.243 0.376 0.376 0.376 0.999 0.999 1.000 40 4669.999 4671.438 4677.148 0.376 0.376 0.376 0.999 0.999 1.000 45 4669.226 4670.665 4676.374 0.376 0.376 0.376 0.999 0.999 1.000 50 4668.941 4670.380 4676.090 0.376 0.376 0.376 0.999 0.999 1.000
Allocation data of option2 (option guarantee contract)
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 5 3513.128 3513.128 3513.127 1.423 1.423 1.422 0.994 0.995 0.996 10 4473.498 4473.498 4473.498 1.422 1.422 1.422 0.992 0.994 0.995 15 4533.986 4533.986 4533.986 1.422 1.422 1.422 0.991 0.994 0.995 20 4458.996 4458.996 4458.996 1.422 1.422 1.422 0.991 0.993 0.994 25 4426.783 4426.783 4426.783 1.422 1.422 1.421 0.991 0.993 0.994 30 4414.189 4414.189 4414.189 1.422 1.422 1.421 0.991 0.993 0.994 35 4409.448 4409.448 4409.448 1.422 1.422 1.421 0.991 0.993 0.994 40 4407.690 4407.690 4407.689 1.422 1.422 1.421 0.991 0.993 0.994 45 4407.040 4407.040 4407.040 1.422 1.422 1.421 0.991 0.993 0.994 50 4406.801 4406.801 4406.801 1.422 1.422 1.421 0.991 0.993 0.994
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 5 3513.128 3513.128 3513.127 1.423 1.423 1.422 0.994 0.995 0.996 10 4473.498 4473.498 4473.498 1.422 1.422 1.422 0.992 0.994 0.995 15 4533.986 4533.986 4533.986 1.422 1.422 1.422 0.991 0.994 0.995 20 4458.996 4458.996 4458.996 1.422 1.422 1.422 0.991 0.993 0.994 25 4426.783 4426.783 4426.783 1.422 1.422 1.421 0.991 0.993 0.994 30 4414.189 4414.189 4414.189 1.422 1.422 1.421 0.991 0.993 0.994 35 4409.448 4409.448 4409.448 1.422 1.422 1.421 0.991 0.993 0.994 40 4407.690 4407.690 4407.689 1.422 1.422 1.421 0.991 0.993 0.994 45 4407.040 4407.040 4407.040 1.422 1.422 1.421 0.991 0.993 0.994 50 4406.801 4406.801 4406.801 1.422 1.422 1.421 0.991 0.993 0.994
Changes in order allocation when inequity aversion difference is small
 t $\Delta\prod_{LSI}^{1-2}$ $\Delta U_{FLSP}^{1-2}$ $\Delta TP^{1-2}$ None option Option1 Option2 None option Option1 Option2 None option Option1 Option2 10 6.835$\%$ 0.029$\%$ 0.000$\%$ 1.553$\%$ 0.019$\%$ 0.005$\%$ 0.624$\%$ 0.012$\%$ 0.254$\%$ 15 6.236$\%$ 0.030$\%$ 0.000$\%$ 1.553$\%$ 0.020$\%$ 0.005$\%$ 1.279$\%$ 0.013$\%$ 0.264$\%$ 20 6.236$\%$ 0.030$\%$ 0.000$\%$ 1.213$\%$ 0.021$\%$ 0.005$\%$ 0.449$\%$ 0.013$\%$ 0.238$\%$ 25 6.262$\%$ 0.031$\%$ 0.000$\%$ 1.413$\%$ 0.021$\%$ 0.005$\%$ 0.750$\%$ 0.013$\%$ 0.226$\%$ 30 6.253$\%$ 0.031$\%$ 0.000$\%$ 1.625$\%$ 0.021$\%$ 0.005$\%$ 1.442$\%$ 0.013$\%$ 0.232$\%$ 35 6.302$\%$ 0.031$\%$ 0.000$\%$ 1.848$\%$ 0.021$\%$ 0.005$\%$ 1.442$\%$ 0.013$\%$ 0.230$\%$ 40 6.433$\%$ 0.031$\%$ 0.000$\%$ 1.848$\%$ 0.021$\%$ 0.005$\%$ 0.371$\%$ 0.013$\%$ 0.232$\%$ 45 6.454$\%$ 0.031$\%$ 0.000$\%$ 0.909$\%$ 0.021$\%$ 0.005$\%$ 0.371$\%$ 0.013$\%$ 0.234$\%$ 50 6.444$\%$ 0.031$\%$ 0.000$\%$ 0.909$\%$ 0.021$\%$ 0.005$\%$ 0.371$\%$ 0.013$\%$ 0.225$\%$
 t $\Delta\prod_{LSI}^{1-2}$ $\Delta U_{FLSP}^{1-2}$ $\Delta TP^{1-2}$ None option Option1 Option2 None option Option1 Option2 None option Option1 Option2 10 6.835$\%$ 0.029$\%$ 0.000$\%$ 1.553$\%$ 0.019$\%$ 0.005$\%$ 0.624$\%$ 0.012$\%$ 0.254$\%$ 15 6.236$\%$ 0.030$\%$ 0.000$\%$ 1.553$\%$ 0.020$\%$ 0.005$\%$ 1.279$\%$ 0.013$\%$ 0.264$\%$ 20 6.236$\%$ 0.030$\%$ 0.000$\%$ 1.213$\%$ 0.021$\%$ 0.005$\%$ 0.449$\%$ 0.013$\%$ 0.238$\%$ 25 6.262$\%$ 0.031$\%$ 0.000$\%$ 1.413$\%$ 0.021$\%$ 0.005$\%$ 0.750$\%$ 0.013$\%$ 0.226$\%$ 30 6.253$\%$ 0.031$\%$ 0.000$\%$ 1.625$\%$ 0.021$\%$ 0.005$\%$ 1.442$\%$ 0.013$\%$ 0.232$\%$ 35 6.302$\%$ 0.031$\%$ 0.000$\%$ 1.848$\%$ 0.021$\%$ 0.005$\%$ 1.442$\%$ 0.013$\%$ 0.230$\%$ 40 6.433$\%$ 0.031$\%$ 0.000$\%$ 1.848$\%$ 0.021$\%$ 0.005$\%$ 0.371$\%$ 0.013$\%$ 0.232$\%$ 45 6.454$\%$ 0.031$\%$ 0.000$\%$ 0.909$\%$ 0.021$\%$ 0.005$\%$ 0.371$\%$ 0.013$\%$ 0.234$\%$ 50 6.444$\%$ 0.031$\%$ 0.000$\%$ 0.909$\%$ 0.021$\%$ 0.005$\%$ 0.371$\%$ 0.013$\%$ 0.225$\%$
Changes in order allocation when inequity aversion difference is large
 t $\Delta\prod_{LSI}^{1-3}$ $\Delta U_{FLSP}^{1-3}$ $\Delta TP^{1-3}$ None option Option1 Option2 None option Option1 Option2 None option Option1 Option2 10 21.506$\%$ 0.143$\%$ 0.000$\%$ 4.643$\%$ 0.078$\%$ 0.023$\%$ 4.007$\%$ 0.062$\%$ 0.348$\%$ 15 20.576$\%$ 0.148$\%$ 0.000$\%$ 4.484$\%$ 0.085$\%$ 0.023$\%$ 7.386$\%$ 0.064$\%$ 0.363$\%$ 20 23.884$\%$ 0.151$\%$ 0.000$\%$ 4.017$\%$ 0.088$\%$ 0.023$\%$ 2.570$\%$ 0.065$\%$ 0.350$\%$ 25 20.099$\%$ 0.152$\%$ 0.000$\%$ 4.295$\%$ 0.090$\%$ 0.023$\%$ 2.774$\%$ 0.066$\%$ 0.343$\%$ 30 19.659$\%$ 0.153$\%$ 0.000$\%$ 4.356$\%$ 0.091$\%$ 0.023$\%$ 2.791$\%$ 0.066$\%$ 0.348$\%$ 35 19.318$\%$ 0.153$\%$ 0.000$\%$ 4.573$\%$ 0.091$\%$ 0.023$\%$ 2.791$\%$ 0.066$\%$ 0.349$\%$ 40 19.451$\%$ 0.153$\%$ 0.000$\%$ 4.573$\%$ 0.091$\%$ 0.023$\%$ 2.791$\%$ 0.066$\%$ 0.350$\%$ 45 19.508$\%$ 0.153$\%$ 0.000$\%$ 4.573$\%$ 0.091$\%$ 0.023$\%$ 2.791$\%$ 0.066$\%$ 0.350$\%$ 50 19.477$\%$ 0.153$\%$ 0.000$\%$ 4.573$\%$ 0.091$\%$ 0.023$\%$ 2.791$\%$ 0.066$\%$ 0.350$\%$
 t $\Delta\prod_{LSI}^{1-3}$ $\Delta U_{FLSP}^{1-3}$ $\Delta TP^{1-3}$ None option Option1 Option2 None option Option1 Option2 None option Option1 Option2 10 21.506$\%$ 0.143$\%$ 0.000$\%$ 4.643$\%$ 0.078$\%$ 0.023$\%$ 4.007$\%$ 0.062$\%$ 0.348$\%$ 15 20.576$\%$ 0.148$\%$ 0.000$\%$ 4.484$\%$ 0.085$\%$ 0.023$\%$ 7.386$\%$ 0.064$\%$ 0.363$\%$ 20 23.884$\%$ 0.151$\%$ 0.000$\%$ 4.017$\%$ 0.088$\%$ 0.023$\%$ 2.570$\%$ 0.065$\%$ 0.350$\%$ 25 20.099$\%$ 0.152$\%$ 0.000$\%$ 4.295$\%$ 0.090$\%$ 0.023$\%$ 2.774$\%$ 0.066$\%$ 0.343$\%$ 30 19.659$\%$ 0.153$\%$ 0.000$\%$ 4.356$\%$ 0.091$\%$ 0.023$\%$ 2.791$\%$ 0.066$\%$ 0.348$\%$ 35 19.318$\%$ 0.153$\%$ 0.000$\%$ 4.573$\%$ 0.091$\%$ 0.023$\%$ 2.791$\%$ 0.066$\%$ 0.349$\%$ 40 19.451$\%$ 0.153$\%$ 0.000$\%$ 4.573$\%$ 0.091$\%$ 0.023$\%$ 2.791$\%$ 0.066$\%$ 0.350$\%$ 45 19.508$\%$ 0.153$\%$ 0.000$\%$ 4.573$\%$ 0.091$\%$ 0.023$\%$ 2.791$\%$ 0.066$\%$ 0.350$\%$ 50 19.477$\%$ 0.153$\%$ 0.000$\%$ 4.573$\%$ 0.091$\%$ 0.023$\%$ 2.791$\%$ 0.066$\%$ 0.350$\%$
Allocation data of none option contract ([26])
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 10 5043.467 5042.428 4890.278 0.141 0.148 0.137 0.505 0.505 0.505 15 5208.289 5190.688 5212.365 0.268 0.267 0.272 0.734 0.734 0.734 20 5305.389 5306.389 5304.389 0.297 0.296 0.291 0.786 0.786 0.786 25 5321.357 5320.357 5325.357 0.311 0.314 0.320 0.804 0.804 0.803 30 5322.357 5321.357 5326.357 0.319 0.325 0.319 0.807 0.806 0.806 35 5323.357 5322.357 5327.357 0.326 0.327 0.327 0.807 0.807 0.807 40 5324.357 5323.357 5328.357 0.328 0.328 0.327 0.807 0.807 0.807 45 5325.357 5324.357 5329.357 0.328 0.328 0.328 0.807 0.807 0.807 50 5326.357 5325.357 5330.357 0.328 0.328 0.328 0.807 0.807 0.807
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 10 5043.467 5042.428 4890.278 0.141 0.148 0.137 0.505 0.505 0.505 15 5208.289 5190.688 5212.365 0.268 0.267 0.272 0.734 0.734 0.734 20 5305.389 5306.389 5304.389 0.297 0.296 0.291 0.786 0.786 0.786 25 5321.357 5320.357 5325.357 0.311 0.314 0.320 0.804 0.804 0.803 30 5322.357 5321.357 5326.357 0.319 0.325 0.319 0.807 0.806 0.806 35 5323.357 5322.357 5327.357 0.326 0.327 0.327 0.807 0.807 0.807 40 5324.357 5323.357 5328.357 0.328 0.328 0.327 0.807 0.807 0.807 45 5325.357 5324.357 5329.357 0.328 0.328 0.328 0.807 0.807 0.807 50 5326.357 5325.357 5330.357 0.328 0.328 0.328 0.807 0.807 0.807
Allocation data of option1 (reservation option contract)
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 5 5951.744 5953.379 5959.872 0.410 0.410 0.410 0.999 1.000 1.000 10 6174.774 6176.406 6182.885 0.417 0.416 0.416 1.000 1.000 1.000 15 6343.880 6345.510 6351.978 0.421 0.421 0.421 1.000 1.000 1.000 20 6433.223 6434.851 6441.314 0.424 0.424 0.424 1.000 1.000 1.000 25 6471.601 6473.229 6479.689 0.425 0.425 0.425 1.000 1.000 1.000 30 6486.605 6488.233 6494.693 0.425 0.425 0.425 1.000 1.000 1.000 35 6492.253 6493.880 6500.340 0.425 0.425 0.425 1.000 1.000 1.000 40 6494.349 6495.976 6502.435 0.426 0.425 0.425 1.000 1.000 1.000 45 6495.122 6496.749 6503.209 0.426 0.426 0.425 1.000 1.000 1.000 50 6495.407 6497.034 6503.493 0.426 0.426 0.425 1.000 1.000 1.000
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 5 5951.744 5953.379 5959.872 0.410 0.410 0.410 0.999 1.000 1.000 10 6174.774 6176.406 6182.885 0.417 0.416 0.416 1.000 1.000 1.000 15 6343.880 6345.510 6351.978 0.421 0.421 0.421 1.000 1.000 1.000 20 6433.223 6434.851 6441.314 0.424 0.424 0.424 1.000 1.000 1.000 25 6471.601 6473.229 6479.689 0.425 0.425 0.425 1.000 1.000 1.000 30 6486.605 6488.233 6494.693 0.425 0.425 0.425 1.000 1.000 1.000 35 6492.253 6493.880 6500.340 0.425 0.425 0.425 1.000 1.000 1.000 40 6494.349 6495.976 6502.435 0.426 0.425 0.425 1.000 1.000 1.000 45 6495.122 6496.749 6503.209 0.426 0.426 0.425 1.000 1.000 1.000 50 6495.407 6497.034 6503.493 0.426 0.426 0.425 1.000 1.000 1.000
Allocation data of option2 (option guarantee contract)
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 5 4123.226 4123.226 4123.226 1.424 1.424 1.423 0.999 0.999 0.999 10 5441.720 5441.720 5441.720 1.424 1.424 1.424 0.999 0.999 0.999 15 5775.426 5775.426 5775.426 1.424 1.424 1.424 1.000 1.000 1.000 20 5845.852 5845.852 5845.852 1.425 1.424 1.424 1.000 1.000 1.000 25 5876.104 5876.104 5876.104 1.425 1.425 1.424 1.000 1.000 1.000 30 5887.931 5887.931 5887.931 1.425 1.425 1.424 1.000 1.000 1.000 35 5892.383 5892.383 5892.383 1.425 1.425 1.424 1.000 1.000 1.000 40 5894.035 5894.035 5894.035 1.425 1.425 1.424 1.000 1.000 1.000 45 5894.645 5894.645 5894.645 1.425 1.425 1.424 1.000 1.000 1.000 50 5894.869 5894.869 5894.869 1.425 1.425 1.424 1.000 1.000 1.000
 t $\prod_{LSI}$ $U_{FLSP}$ $TP$ Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 Scenario1 Scenario2 Scenario3 5 4123.226 4123.226 4123.226 1.424 1.424 1.423 0.999 0.999 0.999 10 5441.720 5441.720 5441.720 1.424 1.424 1.424 0.999 0.999 0.999 15 5775.426 5775.426 5775.426 1.424 1.424 1.424 1.000 1.000 1.000 20 5845.852 5845.852 5845.852 1.425 1.424 1.424 1.000 1.000 1.000 25 5876.104 5876.104 5876.104 1.425 1.425 1.424 1.000 1.000 1.000 30 5887.931 5887.931 5887.931 1.425 1.425 1.424 1.000 1.000 1.000 35 5892.383 5892.383 5892.383 1.425 1.425 1.424 1.000 1.000 1.000 40 5894.035 5894.035 5894.035 1.425 1.425 1.424 1.000 1.000 1.000 45 5894.645 5894.645 5894.645 1.425 1.425 1.424 1.000 1.000 1.000 50 5894.869 5894.869 5894.869 1.425 1.425 1.424 1.000 1.000 1.000
Changes in order allocation when inequity aversion difference is small
 t $\Delta\prod_{LSI}^{1-2}$ $\Delta U_{FLSP}^{1-2}$ $\Delta TP^{1-2}$ None option Option1 Option2 None option Option1 Option2 None option Option1 Option2 10 0.020$\%$ 0.027$\%$ 0.000$\%$ 4.977$\%$ 0.010$\%$ 0.007$\%$ 0.020$\%$ 0.010$\%$ 0.066$\%$ 15 0.021$\%$ 0.026$\%$ 0.000$\%$ 0.234$\%$ 0.009$\%$ 0.006$\%$ 0.060$\%$ 0.010$\%$ 0.034$\%$ 20 0.024$\%$ 0.026$\%$ 0.000$\%$ 0.448$\%$ 0.008$\%$ 0.004$\%$ 0.013$\%$ 0.009$\%$ 0.014$\%$ 25 0.019$\%$ 0.025$\%$ 0.000$\%$ 0.964$\%$ 0.008$\%$ 0.003$\%$ 0.025$\%$ 0.009$\%$ 0.005$\%$ 30 0.019$\%$ 0.025$\%$ 0.000$\%$ 1.970$\%$ 0.008$\%$ 0.002$\%$ 0.011$\%$ 0.009$\%$ 0.002$\%$ 35 0.019$\%$ 0.025$\%$ 0.000$\%$ 0.233$\%$ 0.008$\%$ 0.002$\%$ 0.015$\%$ 0.009$\%$ 0.001$\%$ 40 0.019$\%$ 0.025$\%$ 0.000$\%$ 0.153$\%$ 0.008$\%$ 0.002$\%$ 0.011$\%$ 0.009$\%$ 0.000$\%$ 45 0.019$\%$ 0.025$\%$ 0.000$\%$ 0.139$\%$ 0.008$\%$ 0.002$\%$ 0.011$\%$ 0.009$\%$ 0.000$\%$ 50 0.019$\%$ 0.025$\%$ 0.000$\%$ 0.128$\%$ 0.008$\%$ 0.002$\%$ 0.011$\%$ 0.009$\%$ 0.000$\%$
 t $\Delta\prod_{LSI}^{1-2}$ $\Delta U_{FLSP}^{1-2}$ $\Delta TP^{1-2}$ None option Option1 Option2 None option Option1 Option2 None option Option1 Option2 10 0.020$\%$ 0.027$\%$ 0.000$\%$ 4.977$\%$ 0.010$\%$ 0.007$\%$ 0.020$\%$ 0.010$\%$ 0.066$\%$ 15 0.021$\%$ 0.026$\%$ 0.000$\%$ 0.234$\%$ 0.009$\%$ 0.006$\%$ 0.060$\%$ 0.010$\%$ 0.034$\%$ 20 0.024$\%$ 0.026$\%$ 0.000$\%$ 0.448$\%$ 0.008$\%$ 0.004$\%$ 0.013$\%$ 0.009$\%$ 0.014$\%$ 25 0.019$\%$ 0.025$\%$ 0.000$\%$ 0.964$\%$ 0.008$\%$ 0.003$\%$ 0.025$\%$ 0.009$\%$ 0.005$\%$ 30 0.019$\%$ 0.025$\%$ 0.000$\%$ 1.970$\%$ 0.008$\%$ 0.002$\%$ 0.011$\%$ 0.009$\%$ 0.002$\%$ 35 0.019$\%$ 0.025$\%$ 0.000$\%$ 0.233$\%$ 0.008$\%$ 0.002$\%$ 0.015$\%$ 0.009$\%$ 0.001$\%$ 40 0.019$\%$ 0.025$\%$ 0.000$\%$ 0.153$\%$ 0.008$\%$ 0.002$\%$ 0.011$\%$ 0.009$\%$ 0.000$\%$ 45 0.019$\%$ 0.025$\%$ 0.000$\%$ 0.139$\%$ 0.008$\%$ 0.002$\%$ 0.011$\%$ 0.009$\%$ 0.000$\%$ 50 0.019$\%$ 0.025$\%$ 0.000$\%$ 0.128$\%$ 0.008$\%$ 0.002$\%$ 0.011$\%$ 0.009$\%$ 0.000$\%$
Changes in order allocation when inequity aversion difference is large
 t $\Delta\prod_{LSI}^{1-3}$ $\Delta U_{FLSP}^{1-3}$ $\Delta TP^{1-3}$ None option Option1 Option2 None option Option1 Option2 None option Option1 Option2 10 0.037$\%$ 0.131$\%$ 0.000$\%$ 2.841$\%$ 0.037$\%$ 0.036$\%$ 0.040$\%$ 0.049$\%$ 0.000$\%$ 15 0.078$\%$ 0.127$\%$ 0.000$\%$ 1.593$\%$ 0.031$\%$ 0.029$\%$ 0.046$\%$ 0.048$\%$ 0.000$\%$ 20 0.019$\%$ 0.126$\%$ 0.000$\%$ 2.087$\%$ 0.028$\%$ 0.018$\%$ 0.045$\%$ 0.047$\%$ 0.000$\%$ 25 0.075$\%$ 0.125$\%$ 0.000$\%$ 2.785$\%$ 0.026$\%$ 0.013$\%$ 0.037$\%$ 0.046$\%$ 0.000$\%$ 30 0.075$\%$ 0.125$\%$ 0.000$\%$ 0.199$\%$ 0.026$\%$ 0.011$\%$ 0.010$\%$ 0.046$\%$ 0.000$\%$ 35 0.075$\%$ 0.124$\%$ 0.000$\%$ 0.156$\%$ 0.026$\%$ 0.011$\%$ 0.011$\%$ 0.046$\%$ 0.000$\%$ 40 0.075$\%$ 0.124$\%$ 0.000$\%$ 0.109$\%$ 0.026$\%$ 0.010$\%$ 0.006$\%$ 0.046$\%$ 0.000$\%$ 45 0.075$\%$ 0.124$\%$ 0.000$\%$ 0.106$\%$ 0.026$\%$ 0.010$\%$ 0.006$\%$ 0.046$\%$ 0.000$\%$ 50 0.075$\%$ 0.124$\%$ 0.000$\%$ 0.106$\%$ 0.026$\%$ 0.010$\%$ 0.006$\%$ 0.046$\%$ 0.000$\%$
 t $\Delta\prod_{LSI}^{1-3}$ $\Delta U_{FLSP}^{1-3}$ $\Delta TP^{1-3}$ None option Option1 Option2 None option Option1 Option2 None option Option1 Option2 10 0.037$\%$ 0.131$\%$ 0.000$\%$ 2.841$\%$ 0.037$\%$ 0.036$\%$ 0.040$\%$ 0.049$\%$ 0.000$\%$ 15 0.078$\%$ 0.127$\%$ 0.000$\%$ 1.593$\%$ 0.031$\%$ 0.029$\%$ 0.046$\%$ 0.048$\%$ 0.000$\%$ 20 0.019$\%$ 0.126$\%$ 0.000$\%$ 2.087$\%$ 0.028$\%$ 0.018$\%$ 0.045$\%$ 0.047$\%$ 0.000$\%$ 25 0.075$\%$ 0.125$\%$ 0.000$\%$ 2.785$\%$ 0.026$\%$ 0.013$\%$ 0.037$\%$ 0.046$\%$ 0.000$\%$ 30 0.075$\%$ 0.125$\%$ 0.000$\%$ 0.199$\%$ 0.026$\%$ 0.011$\%$ 0.010$\%$ 0.046$\%$ 0.000$\%$ 35 0.075$\%$ 0.124$\%$ 0.000$\%$ 0.156$\%$ 0.026$\%$ 0.011$\%$ 0.011$\%$ 0.046$\%$ 0.000$\%$ 40 0.075$\%$ 0.124$\%$ 0.000$\%$ 0.109$\%$ 0.026$\%$ 0.010$\%$ 0.006$\%$ 0.046$\%$ 0.000$\%$ 45 0.075$\%$ 0.124$\%$ 0.000$\%$ 0.106$\%$ 0.026$\%$ 0.010$\%$ 0.006$\%$ 0.046$\%$ 0.000$\%$ 50 0.075$\%$ 0.124$\%$ 0.000$\%$ 0.106$\%$ 0.026$\%$ 0.010$\%$ 0.006$\%$ 0.046$\%$ 0.000$\%$
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