# A Project Scheduling Game Equilibrium Problem Based on Dynamic Resource Supply
## 3 Model Establishment
### Symbols and Assumptions  
1.1. Indicators  
1. Project activity index, \(j\in J=\{1,2,\ldots,J\}\)  
2. Material type index, \(k\in K=\{1,2,\ldots,K\}\)  
3. Time period index, \(t\in\{1,2,\ldots\}\)  
4. Activity mode index, \(m\in M=\{1,2,\ldots,M\}\)  
Figure 1: Structural model of the MRCPSP-MPSCP integrated system.  
\(i\): Activity mode index, \(i\in I=\{1,2,\ldots,I\}\)  
\(s\): Demand point index, \(s\in S=\{1,2,\ldots,S\}\)  
3.1.2. Parameters Related to Project Scheduling  
\(B\): Total available budget  
\(D\): Project planning cycle  
\(IC_{s}\): Inventory capacity at the demand point \(s\)  
\(P_{j}\): Set of predecessors of activity \(j\)  
\(c_{jm}\): Direct cost of activity \(j\) in mode \(m\)  
\(d_{jm}\): Operation time of activity \(j\) in \(m\) mode  
\(r_{jmk}\): The demand of \(m\) mode of activity \(j\) for resource \(k\) in each time period  
\(r_{k}\): Maximum supply capacity of resource \(k\) in each time period  
\(c_{0}\): Overhead cost per time period  
\(EF_{j}\): The earliest completion time of activity \(j\)  
\(LF_{j}\): The latest completion time of activity \(j\)  
\(R_{s}\): The set of activities for which demand point \(s\) is responsible  
\(Pc_{k}\): Unit purchase cost of resource \(k\)  
\(Oc_{k}\): Each order cost of resource \(k\)  
\(Ic_{k}\): Storage cost of resource \(k\) in each time period  
3.1.3. Parameters Related to Resource Supply  
\(T(t)\): Delivery date of time period \(t\)  
\(R_{k}\): Maximum amount of resource \(k\) transported each time  
\(P_{ik}\): Supply capacity of resource \(k\) at supply point \(i\)  
\(c_{isk}\): Unit transportation cost of resource \(k\) on the transportation path \((i,s)\)  
\(t_{isk}\): Unit transportation time of resource \(k\) on the transportation path \((i,s)\)  
3.1.4. Decision Variables  
\(v_{isk}(t)\): The allocation of resource \(k\) on transportation path \((i,s)\) in time period \(t\)  
\(x_{jmt}\ =\ \left\{\begin{array}{l}1,\text{ If activity $j$ executes mode $m$ in time period $t$}\\ 0,\text{Otherwise}\end{array}\right.\), represents the mode selection of activity \(j\)  
\(z_{kt}\ =\ \left\{\begin{array}{l}1,\text{ If resource $k$ transported at the beginning of time period $t$}\\ 0,\text{Otherwise}\end{array}\right.\), represents whether resource \(k\) is transported during time period \(t\)  
3.1.5. Intermediate Variables  
\(ST_{j}\): Starting time of activity \(j\)  
\(FT_{j}\): Completion time of activity \(j\)  
\(A_{t}\): Activity set of ongoing jobs in time period \(t\)  
\(S_{lst}\): The remaining amount of resource \(k\) at demand point \(s\) at the end of the time period \(t\)  
3.1.6. Assumptions  
1 The project contains \(j\) activities and two virtual activities, in which the two virtual activities represent the initial and final activities of the project, denoted as \(j=0\) and \(j=J+1\), respectively.  
2 Only when all the predecessor activities of the activity are completed can the activity begin.  
3 Each activity can only execute one mode without interruption.  
4 The supply capacity of the supply point and the inventory capacity of the demand point are limited and cannot be increased.  
5 The loading and unloading costs and time of the transport vehicles were included in the corresponding transport costs and time.  
C Uncertain parameters, such as resource demand, project activity time, and unit transportation cost, are random variables.  
2 Resources are consumed evenly in each time period.