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# System prepended metadata

title: DFCS Zoning Optimization Model

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# DFCS Zoning Optimization Model


## Data Variables
1) $n_{i,i_2}$ County i neighbors county i_2 (0 if not, 1 if adjacent?) $i,i_2 \ \ \ \epsilon \ \  \{1, \dots, 159\}$
2) $r_{i,i_2}$ 1 if county i in region with county i_2, 0 otherwise
3) $w_i$ - workload in county i
4) $s_i$ - number of social workers in county i
5) $m$ - max workload


## Decision Variables
1) $Z_j$ = 1 if zone j exists 0 otherwise
2) $X_{i,j}$ = 1 if county i in zone j 0 otherwise
3) $W_j$ = total workload of in zone j
4) $\lambda_j = \sum_{i=1}^{159} r_{i, j} \cdot X_{i,j} \forall j, k$ redefine r_ij to county i in region j


## Objective Components
1) Minimize max(workload - full load, 0)
2) Prioritize zones that can share directors
3) Prioritize counties in zones with less resources (Not fulton & dekalb)
4) Later (prioritize zones that are closer geographically)
## Constraints
1) Every county must be in exactly 1 zone
    1.1) $\sum_{j=1}^{159} X_{i_j} = 1 \ \ \forall \ \ i$
2) Zones do not overlap with region borders
    2.1) $\lambda_j = \sum_{i=1}^{159} \sum_{i_2=1}^{159} r_{i,i_2} \cdot X_{i,j} \cdot X_{i_2, j} \ \ \forall \ \ j$
    2.2) $\lambda_j =  \sum_{i=1}^{159} X_{i,j} \ \ \forall \ \ j$
    
> [name=Carl]
2.4) $\lambda_j = \sum_{i=1}^{159}  r_{i,k} \cdot X_{i,j} \sum_{i_2=1}^{159}  r_{i_2,k} \cdot X_{i_2, j} \ \ \forall \ \ j, k$
    2.5) $\lambda_j =  \sum_{i=1}^{159} X_{i,j} \ \ \forall \ \ j$    2.5) $\lambda_j =  \sum_{i=1}^{159} X_{i,j} \ \ \forall \ \ j$
3) Crude workload limit for zones
    3.1) $\frac{W_j}{\sum_{i=1}^{159} X_{i,j}} \le m \ \ \forall \ \ j$
4) All cases are preserved
    4.1) $W_j = \sum_{i = 1}^{159}w_i \cdot X_{i,j} \ \ \forall \ \ j$
5) No more than 3 counties in the zone (for now all counties need not to be adjacent)
    6.1) $\sum_{i = 1}^{159}X_{i,j} \leq 3 \ \ \forall \ \ j$
6) Define workload of zones given counties in zone
    7.1) $W_j = \sum_{i=1}^{159} X_{i,j} \cdot w_i \ \ \forall \ \ j$
7) Define existing zones (and probably don’t modify them)




