# American Institute of Mathematical Sciences

January  2020, 7(1): 21-35. doi: 10.3934/jdg.2020002

## Sequencing grey games

 1 Isparta University of Applied Sciences, Faculty of Technology, Department of Computer Engineering, Isparta, Turkey 2 Süleyman Demirel University, Faculty of Economics and Business Administration, Department of Business Administration, Isparta, Turkey 3 Süleyman Demirel University, Faculty of Arts and Sciences, Department of Mathematics, Isparta, Turkey 4 Poznan University of Technology, Chair of Marketing and Economic Engineering, Poznan, Poland

* Corresponding author: zeynepalparslan@yahoo.com

Received  December 2018 Published  December 2019

The job scheduling problem is a notoriously difficult problem in combinatorial optimization and Operational Research. In this study, we handle the job scheduling problem by using a cooperative game theoretical approach. In the sequel, sequencing situations arising grom grey uncertainty are considered. Cooperative grey game theory is applied to analyze these situations. Further, grey sequencing games are constructed and grey equal gain splitting (GEGS) rule is introduced. It is shown that cooperative grey games are convex. An application is given based on Priority Based Scheduling Algorithm. The paper ends with a conclusion.

Citation: Serap Ergün, Osman Palanci, Sirma Zeynep Alparslan Gök, Şule Nizamoğlu, Gerhard Wilhelm Weber. Sequencing grey games. Journal of Dynamics & Games, 2020, 7 (1) : 21-35. doi: 10.3934/jdg.2020002
##### References:

show all references

##### References:
An illustration of our application
Gantt charts of D1
Gantt charts of D2
Gantt charts of D3
The properties of each jobs of D1
 Job Arrival Time Execute Time Priority Service Time J1 $\left[ 0, 1\right]$ $\left[ 2, 2\right]$ 1 $\left[ 95, 101\right]$ J2 $\left[ 1, 3\right]$ $\left[ 3, 3\right]$ 2 $\left[ 191, 198\right]$ J3 $\left[ 3, 4\right]$ $\left[ 5, 5\right]$ 3 $\left[ 288, 294\right]$
 Job Arrival Time Execute Time Priority Service Time J1 $\left[ 0, 1\right]$ $\left[ 2, 2\right]$ 1 $\left[ 95, 101\right]$ J2 $\left[ 1, 3\right]$ $\left[ 3, 3\right]$ 2 $\left[ 191, 198\right]$ J3 $\left[ 3, 4\right]$ $\left[ 5, 5\right]$ 3 $\left[ 288, 294\right]$
The properties of each jobs of D2
 Job Arrival Time Execute Time Priority Service Time J1 $\left[ 3, 5\right]$ $\left[ 3, 5\right]$ 2 $\left[ 153, 160\right]$ J2 $\left[ 0, 2\right]$ $\left[ 4, 6\right]$ 1 $\left[ 120, 127\right]$ J3 $\left[ 6, 8\right]$ $\left[ 7, 9\right]$ 3 $\left[ 186, 193\right]$
 Job Arrival Time Execute Time Priority Service Time J1 $\left[ 3, 5\right]$ $\left[ 3, 5\right]$ 2 $\left[ 153, 160\right]$ J2 $\left[ 0, 2\right]$ $\left[ 4, 6\right]$ 1 $\left[ 120, 127\right]$ J3 $\left[ 6, 8\right]$ $\left[ 7, 9\right]$ 3 $\left[ 186, 193\right]$
The properties of each jobs of D3
 Job Arrival Time Execute Time Priority Service Time J1 $\left[ 2, 4\right]$ $\left[ 2, 2\right]$ 2 $\left[ 124, 132\right]$ J2 $\left[ 4, 7\right]$ $\left[ 3, 3\right]$ 3 $\left[ 152, 160\right]$ J3 $\left[ 0, 3\right]$ $\left[ 4, 4\right]$ 1 $\left[ 90, 98\right]$
 Job Arrival Time Execute Time Priority Service Time J1 $\left[ 2, 4\right]$ $\left[ 2, 2\right]$ 2 $\left[ 124, 132\right]$ J2 $\left[ 4, 7\right]$ $\left[ 3, 3\right]$ 3 $\left[ 152, 160\right]$ J3 $\left[ 0, 3\right]$ $\left[ 4, 4\right]$ 1 $\left[ 90, 98\right]$
The wait time t of each jobs of D1, D2 and D3
 $\textbf{Job (Process)}$ $\textbf{Wait Time}$ J1 of D1 $t_{11} = \left[ 95, 100\right]$ J2 of D1 $t_{12} = \left[ 180, 195\right]$ J3 of D1 $t_{13} = \left[ 285, 290\right]$ J1 of D2 $t_{21} = \left[ 150, 155\right]$ J2 of D2 $t_{22} = \left[ 120, 125\right]$ J3 of D2 $t_{23} = \left[ 180, 185\right]$ J1 of D3 $t_{31} = \left[ 120, 125\right]$ J2 of D3 $t_{32} = \left[ 150, 155\right]$ J3 of D3 $t_{33} = \left[ 90, 95\right]$
 $\textbf{Job (Process)}$ $\textbf{Wait Time}$ J1 of D1 $t_{11} = \left[ 95, 100\right]$ J2 of D1 $t_{12} = \left[ 180, 195\right]$ J3 of D1 $t_{13} = \left[ 285, 290\right]$ J1 of D2 $t_{21} = \left[ 150, 155\right]$ J2 of D2 $t_{22} = \left[ 120, 125\right]$ J3 of D2 $t_{23} = \left[ 180, 185\right]$ J1 of D3 $t_{31} = \left[ 120, 125\right]$ J2 of D3 $t_{32} = \left[ 150, 155\right]$ J3 of D3 $t_{33} = \left[ 90, 95\right]$
The weights of c, d, n of J1 for D1, D2, D3
 $\textbf{Property of job}$ $\textbf{Compute Intensity}$ $\textbf{Data parsing}$ $\textbf{Network}$ cost $c$ $d$ $n$ J1D1 3 2 1 J2D1 2 3 1 J3D1 1 2 3 J1D2 3 2 1 J2D2 1 3 2 J3D2 1 2 3 J1D3 3 1 2 J2D3 2 3 1 J3D3 1 1 1
 $\textbf{Property of job}$ $\textbf{Compute Intensity}$ $\textbf{Data parsing}$ $\textbf{Network}$ cost $c$ $d$ $n$ J1D1 3 2 1 J2D1 2 3 1 J3D1 1 2 3 J1D2 3 2 1 J2D2 1 3 2 J3D2 1 2 3 J1D3 3 1 2 J2D3 2 3 1 J3D3 1 1 1
Grey marginal vectors
 $\sigma$ $m_{1}^{\sigma }\left( w^{\prime }\right)$ $m_{2}^{\sigma }\left( w^{\prime }\right)$ $m_{3}^{\sigma }\left( w^{\prime }\right)$ $\sigma _{1} = \left( 1, 2, 3\right)$ $m_{1}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{3}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $\sigma _{2} = \left( 1, 3, 2\right)$ $m_{1}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $m_{3}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $\sigma _{3} = \left( 2, 1, 3\right)$ $m_{1}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{3}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $\sigma _{4} = \left( 2, 3, 1\right)$ $m_{1}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 26500, 28050\right]$ $m_{2}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{3}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 42000, 44800\right]$ $\sigma _{5} = \left( 3, 1, 2\right)$ $m_{1}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $m_{3}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $\sigma _{6} = \left( 3, 2, 1\right)$ $m_{1}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 26500, 28050\right]$ $m_{2}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 42000, 44800\right]$ $m_{3}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$
 $\sigma$ $m_{1}^{\sigma }\left( w^{\prime }\right)$ $m_{2}^{\sigma }\left( w^{\prime }\right)$ $m_{3}^{\sigma }\left( w^{\prime }\right)$ $\sigma _{1} = \left( 1, 2, 3\right)$ $m_{1}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{3}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $\sigma _{2} = \left( 1, 3, 2\right)$ $m_{1}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $m_{3}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $\sigma _{3} = \left( 2, 1, 3\right)$ $m_{1}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{3}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $\sigma _{4} = \left( 2, 3, 1\right)$ $m_{1}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 26500, 28050\right]$ $m_{2}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{3}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 42000, 44800\right]$ $\sigma _{5} = \left( 3, 1, 2\right)$ $m_{1}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $m_{3}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $\sigma _{6} = \left( 3, 2, 1\right)$ $m_{1}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 26500, 28050\right]$ $m_{2}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 42000, 44800\right]$ $m_{3}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$
 [1] Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics & Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013 [2] Deng-Feng Li, Yin-Fang Ye, Wei Fei. Extension of generalized solidarity values to interval-valued cooperative games. Journal of Industrial & Management Optimization, 2020, 16 (2) : 919-931. doi: 10.3934/jimo.2018185 [3] J-F. Clouët, R. Sentis. Milne problem for non-grey radiative transfer. Kinetic & Related Models, 2009, 2 (2) : 345-362. doi: 10.3934/krm.2009.2.345 [4] Fabián Crocce, Ernesto Mordecki. A non-iterative algorithm for generalized pig games. Journal of Dynamics & Games, 2018, 5 (4) : 331-341. doi: 10.3934/jdg.2018020 [5] Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-15. doi: 10.3934/jimo.2019036 [6] Serap Ergün, Sirma Zeynep Alparslan Gök, Tuncay Aydoǧan, Gerhard Wilhelm Weber. Performance analysis of a cooperative flow game algorithm in ad hoc networks and a comparison to Dijkstra's algorithm. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1085-1100. doi: 10.3934/jimo.2018086 [7] Mohamed A. Tawhid, Ahmed F. Ali. A simplex grey wolf optimizer for solving integer programming and minimax problems. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 301-323. doi: 10.3934/naco.2017020 [8] Jiahua Zhang, Shu-Cherng Fang, Yifan Xu, Ziteng Wang. A cooperative game with envy. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2049-2066. doi: 10.3934/jimo.2017031 [9] Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics & Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117 [10] Konstantin Avrachenkov, Giovanni Neglia, Vikas Vikram Singh. Network formation games with teams. Journal of Dynamics & Games, 2016, 3 (4) : 303-318. doi: 10.3934/jdg.2016016 [11] Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33 [12] Yonghui Zhou, Jian Yu, Long Wang. Topological essentiality in infinite games. Journal of Industrial & Management Optimization, 2012, 8 (1) : 179-187. doi: 10.3934/jimo.2012.8.179 [13] Carlos Hervés-Beloso, Emma Moreno-García. Market games and walrasian equilibria. Journal of Dynamics & Games, 2020, 7 (1) : 65-77. doi: 10.3934/jdg.2020004 [14] Weifu Sun, Xu Yang, Yijun Chen. Elimination algorithm of complex network redundant data stream based on information theory. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020256 [15] Elena Travaglia, Valentina La Morgia, Ezio Venturino. Poxvirus, red and grey squirrel dynamics: Is the recovery of a common predator affecting system equilibria? Insights from a predator-prey ecoepidemic model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2023-2040. doi: 10.3934/dcdsb.2019200 [16] Daniel Brinkman, Christian Ringhofer. A kinetic games framework for insurance plans. Kinetic & Related Models, 2017, 10 (1) : 93-116. doi: 10.3934/krm.2017004 [17] Oliver Juarez-Romero, William Olvera-Lopez, Francisco Sanchez-Sanchez. A simple family of solutions for forest games. Journal of Dynamics & Games, 2017, 4 (2) : 87-96. doi: 10.3934/jdg.2017006 [18] Dmitry Kleinbock, Barak Weiss. Modified Schmidt games and a conjecture of Margulis. Journal of Modern Dynamics, 2013, 7 (3) : 429-460. doi: 10.3934/jmd.2013.7.429 [19] Sylvain Sorin, Cheng Wan. Finite composite games: Equilibria and dynamics. Journal of Dynamics & Games, 2016, 3 (1) : 101-120. doi: 10.3934/jdg.2016005 [20] Andrzej Swierniak, Michal Krzeslak. Application of evolutionary games to modeling carcinogenesis. Mathematical Biosciences & Engineering, 2013, 10 (3) : 873-911. doi: 10.3934/mbe.2013.10.873

Impact Factor:

## Tools

Article outline

Figures and Tables