# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021158
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## Outsourcing contract design for the green transformation of manufacturing systems under asymmetric information

 Business School, SiChuan University, Chengdu 610065, China

* Corresponding author: Dong Cai

Received  March 2021 Revised  June 2021 Early access September 2021

Fund Project: This research was funded by [National Natural Science Foundation of China] grant number [71871150] and [Project of Sichuan University Innovation Spark Project Library] grant number [2021CXC22]

Accepting that suppliers' capability for green transformation of manufacturing systems is private information and that the efforts made in the process of transformation invisibly involve private actions of buyers, we construct an outsourcing model including transformation services before project delivery and maintenance services within the warranty period after project delivery and research the optimal outsourcing contract design for buyers. We find that the buyer can design a set of contract menus, including fixed compensation and variable compensation related to the quantity of energy conservation and emission reduction (ECER), to identify suppliers with different transformation capabilities and encourage them to make the best efforts under asymmetric information. Second, to identify the suppliers' transformation capability, the buyer needs to pay information rent to the supplier with high transformation ability. Meanwhile, the existence of asymmetric information will make the supplier with low transformation ability exert insufficient effort, and the existence of asymmetric information will always reduce the buyer's expected utility. In addition, the example analysis shows that asymmetric information always reduces the expected number of ECERs of the buyer. Therefore, it is suggested that the government should consider screening the transformation ability of green technology suppliers, disclose to the market and recommend suppliers with high transformation ability to reduce the negative impact caused by asymmetric information.

Citation: Chun-xiang Guo, Dong Cai, Yu-yang Tan. Outsourcing contract design for the green transformation of manufacturing systems under asymmetric information. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021158
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##### References:
Sequence of outsourcing events
The influence of the probability that the contractor is a high type on the employer's expected number of ECERs
The influence of the probability of the high-type contractor on the optimal incentive intensity and effort of the low-type contractor
The influence of the gap of the contractor's transformation ability on the expected number of ECERs of the employer
The influence of the gap of the transformation ability of the contractor on the optimal incentive intensity and optimal effort of the low-type contractor
Overview of the literature
 Reference Objective Solution Conclusion Deif [4] Design and improve green manufacturing systems A new model A comprehensive qualitative answer to the question. Tsai et al. [26] Assist the justification of capital investments for green manufacturing systems (GMSs) Activity-based costing Provide insight into the value of capital investments for a GMS based ABC Chuang et al. [2] Evaluate the performance of a green manufacturing system A new model The three-layer assessment model is practical Li et al. [11] Scheduling problems that arise in green manufacturing companies Efficient heuristics Find a schedule that minimizes the total completion time Li et al. [12] Enhance environmental awareness and green manufacturing practices Questionnaire investigation The importance of corporate stakeholders should be promoted Lotfi et al. [16,17,18,19] Considers a closed-loop supply chain by taking into account sustainability, resilience, robustness, and risk aversion. Two stage mixed-integer linear programming model The robust counterpart provides a better estimation of related factors. Goli et al. [5,6] Design a flexible-responsive manufacturing system with automatic material handling systems. A fuzzy mixed integer linear programming model Proposed algorithms have a high performance compared to CPLEX and other two well-known algorithms Paksoy et al. [23,24] The optimization of supply chain structures considering both economic and environmental performances An integer nonlinear programming model Help decision makers find the optimal solution Tirkolaee et al. [27,28] Try to explain and formulate the sustainable medical waste management problem for pandemics Proposed a biobjective MILP model Discuss the practical implications of the results Cao et al. [3] Solve asymmetry information in outsourcing of production Contract theory Design a quality incentive contract Wang et al. [29] Solve asymmetry information in outsourcing of R & D Stage-gate contracts The stage-gate contract can help offset the information asymmetry Hui et al. [7] Solve asymmetry information in outsourcing of managed security services Bilateral liability-based contracts Bilateral liability-based contracts can work in the real world Huang et al. [8] Contract design problem of IT service outsourcing Contract theory The optimal contract is designed to regulate IT suppliers effectively This research Contract design problem of green transformation of manufacturing system outsourcing Outsourcing contract design Design a set of contract menus to resolve asymmetric information
 Reference Objective Solution Conclusion Deif [4] Design and improve green manufacturing systems A new model A comprehensive qualitative answer to the question. Tsai et al. [26] Assist the justification of capital investments for green manufacturing systems (GMSs) Activity-based costing Provide insight into the value of capital investments for a GMS based ABC Chuang et al. [2] Evaluate the performance of a green manufacturing system A new model The three-layer assessment model is practical Li et al. [11] Scheduling problems that arise in green manufacturing companies Efficient heuristics Find a schedule that minimizes the total completion time Li et al. [12] Enhance environmental awareness and green manufacturing practices Questionnaire investigation The importance of corporate stakeholders should be promoted Lotfi et al. [16,17,18,19] Considers a closed-loop supply chain by taking into account sustainability, resilience, robustness, and risk aversion. Two stage mixed-integer linear programming model The robust counterpart provides a better estimation of related factors. Goli et al. [5,6] Design a flexible-responsive manufacturing system with automatic material handling systems. A fuzzy mixed integer linear programming model Proposed algorithms have a high performance compared to CPLEX and other two well-known algorithms Paksoy et al. [23,24] The optimization of supply chain structures considering both economic and environmental performances An integer nonlinear programming model Help decision makers find the optimal solution Tirkolaee et al. [27,28] Try to explain and formulate the sustainable medical waste management problem for pandemics Proposed a biobjective MILP model Discuss the practical implications of the results Cao et al. [3] Solve asymmetry information in outsourcing of production Contract theory Design a quality incentive contract Wang et al. [29] Solve asymmetry information in outsourcing of R & D Stage-gate contracts The stage-gate contract can help offset the information asymmetry Hui et al. [7] Solve asymmetry information in outsourcing of managed security services Bilateral liability-based contracts Bilateral liability-based contracts can work in the real world Huang et al. [8] Contract design problem of IT service outsourcing Contract theory The optimal contract is designed to regulate IT suppliers effectively This research Contract design problem of green transformation of manufacturing system outsourcing Outsourcing contract design Design a set of contract menus to resolve asymmetric information
The symbols involved
 Symbols Definition Decision variables $F$ Fixed remuneration provided by the employer to the contractor $f$ Unit reward of ECER exceeding the benchmark after system transformation Parameters $\beta$ Capability parameters of the contractor's transformation system $i$ The types of contractors: high type (with higher transformation capacity ${\beta _H}$) and low type (with low transformation capacity ${\beta _L}$), $i = H\;or\;L$ $\rho$ The probability that the contractor is high $e$ Contractor's efforts to transform the system $r$ Number of ECERs achieved after system transformation ${r_0}$ The benchmark number of ECERs required by the employer according to relevant government or industry standards $n$ Failure number in warranty period after system transformation $\varepsilon$ The maximum number of failures in the warranty period after the completion of system transformation $c$ Total cost of the contractor's modification of the system $w$ The single maintenance cost $u$ Value coefficient of ECER quantity to the employer $v$ Sensitivity coefficient of the employer to the number of failures in the warranty period
 Symbols Definition Decision variables $F$ Fixed remuneration provided by the employer to the contractor $f$ Unit reward of ECER exceeding the benchmark after system transformation Parameters $\beta$ Capability parameters of the contractor's transformation system $i$ The types of contractors: high type (with higher transformation capacity ${\beta _H}$) and low type (with low transformation capacity ${\beta _L}$), $i = H\;or\;L$ $\rho$ The probability that the contractor is high $e$ Contractor's efforts to transform the system $r$ Number of ECERs achieved after system transformation ${r_0}$ The benchmark number of ECERs required by the employer according to relevant government or industry standards $n$ Failure number in warranty period after system transformation $\varepsilon$ The maximum number of failures in the warranty period after the completion of system transformation $c$ Total cost of the contractor's modification of the system $w$ The single maintenance cost $u$ Value coefficient of ECER quantity to the employer $v$ Sensitivity coefficient of the employer to the number of failures in the warranty period
Optimal outsourcing contract of the employer in different situations
 Situations High type Low type Symmetric information $\left\{ {F_H^{S{\rm{*}}}, f_H^{S{\rm{*}}}} \right\}$ $\left\{ {F_L^{S*}, f_L^{S*}} \right\}$ Asymmetric information $0< u + v \le x$ $\left\{ {F_H^{A{\rm{*}}}, f_H^{A{\rm{*}}}} \right\}$ $\left\{ {F_L^{A{\rm{*}}}, f_{L1}^{A{\rm{*}}}} \right\}$ $x< u + v$ $\left\{ {F_L^{A{\rm{*}}}, f_{L2}^{A{\rm{*}}}} \right\}$ 1 Among them, $f_{L1}^{A{\rm{*}}}{\rm{ = }}0$, $f_{L2}^{A{\rm{*}}}{\rm{ = }}\frac{{\left({1 - \rho } \right){\beta _L}\left({u{\rm{ + }}v} \right) - \rho \left({{\beta _H}{\rm{ + }}w{\beta _H} - {\beta _L} - w{\beta _L}} \right)}}{{\left({1 - \rho } \right){\beta _L}{\rm{ + }}\rho {\beta _H} - \rho {\beta _L}}}$, $x = \frac{{\rho {\beta _L}\left({{\beta _H}{\rm{ + }}w{\beta _H} - {\beta _L} - w{\beta _L}} \right)}}{{1 - \rho }}$
 Situations High type Low type Symmetric information $\left\{ {F_H^{S{\rm{*}}}, f_H^{S{\rm{*}}}} \right\}$ $\left\{ {F_L^{S*}, f_L^{S*}} \right\}$ Asymmetric information $0< u + v \le x$ $\left\{ {F_H^{A{\rm{*}}}, f_H^{A{\rm{*}}}} \right\}$ $\left\{ {F_L^{A{\rm{*}}}, f_{L1}^{A{\rm{*}}}} \right\}$ $x< u + v$ $\left\{ {F_L^{A{\rm{*}}}, f_{L2}^{A{\rm{*}}}} \right\}$ 1 Among them, $f_{L1}^{A{\rm{*}}}{\rm{ = }}0$, $f_{L2}^{A{\rm{*}}}{\rm{ = }}\frac{{\left({1 - \rho } \right){\beta _L}\left({u{\rm{ + }}v} \right) - \rho \left({{\beta _H}{\rm{ + }}w{\beta _H} - {\beta _L} - w{\beta _L}} \right)}}{{\left({1 - \rho } \right){\beta _L}{\rm{ + }}\rho {\beta _H} - \rho {\beta _L}}}$, $x = \frac{{\rho {\beta _L}\left({{\beta _H}{\rm{ + }}w{\beta _H} - {\beta _L} - w{\beta _L}} \right)}}{{1 - \rho }}$
Optimal strategies of the employer under different circumstances
 Circumstances Strategies The information of contractor's transformation capability is symmetric Make the optimal outsourcing contract according to the type of contractor The information of contractor's transformation capability is asymmetric Make a set of optimal outsourcing contract menu for contractors to choose from
 Circumstances Strategies The information of contractor's transformation capability is symmetric Make the optimal outsourcing contract according to the type of contractor The information of contractor's transformation capability is asymmetric Make a set of optimal outsourcing contract menu for contractors to choose from
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