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# Sequencing grey games

• 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.

Mathematics Subject Classification: Primary: 91A12.

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• Figure 1.  An illustration of our application

Figure 2.  Gantt charts of D1

Figure 3.  Gantt charts of D2

Figure 4.  Gantt charts of D3

Table 1.  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 $\left[ 288, 294\right]$

Table 2.  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 $\left[ 186, 193\right]$

Table 3.  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 $\left[ 90, 98\right]$

Table 4.  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 $t_{33} = \left[ 90, 95\right]$

Table 5.  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

Table 6.  Grey marginal vectors

 $\sigma$ $m_{1}^{\sigma }\left( w^{\prime }\right)$ $m_{2}^{\sigma }\left( w^{\prime }\right)$ $m_{3}^{\sigma }\left( w^{\prime }\right)$ $\sigma _{1} = \left( 1, 2, 3\right)$ $m_{1}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{3}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $\sigma _{2} = \left( 1, 3, 2\right)$ $m_{1}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $m_{3}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $\sigma _{3} = \left( 2, 1, 3\right)$ $m_{1}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{3}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $\sigma _{4} = \left( 2, 3, 1\right)$ $m_{1}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 26500, 28050\right]$ $m_{2}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{3}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 42000, 44800\right]$ $\sigma _{5} = \left( 3, 1, 2\right)$ $m_{1}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $m_{2}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right]$ $m_{3}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$ $\sigma _{6} = \left( 3, 2, 1\right)$ $m_{1}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 26500, 28050\right]$ $m_{2}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 42000, 44800\right]$ $m_{3}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 0, 0\right]$
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