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doi: 10.3934/jimo.2018176

## Risk-balanced territory design optimization for a Micro finance institution

 1 Universidad Autónoma de Nuevo León, Monterrey, Mexico 2 Texas State University, San Marcos, TX 78666, USA

* Corresponding author: Tahir Ekin

Received  August 2017 Revised  July 2018 Published  December 2018

Micro finance institutions (MFIs) play an important role in emerging economies as part of programs that aim to reduce income inequality and poverty. A territory design that balances the risk of branches is important for the profitability and long-term sustainability of a MFI. In order to address such particular business needs, this paper proposes a novel risk-balanced territory planning model for a MFI. The proposed mixed integer programming model lets the MFI choose the location of the branches to be designated as territory centers and allocate the customers to these centers with respect to planning criteria such as the total workload, monetary amount of loans and profit allocation while balancing the territory risk. This model is solved using a branch and cut based hybrid-heuristic framework. We discuss the impact of the risk balancing and merits of the proposed model.

Citation: Jesús Fabián López Pérez, Tahir Ekin, Jesus A. Jimenez, Francis A. Méndez Mediavilla. Risk-balanced territory design optimization for a Micro finance institution. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018176
##### References:

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##### References:
Illustration of branches and customers of the micro finance institution
Total Cuts and Number of Disconnected BUs versus Computational Time
Recenterings versus Computational Time
Objective Function Values of Risk and Distance for Model 1 (bold line) and Model 2 (dashed line)
Partial map of the implementation of the territory design model
Mathematical notation and description for sets
 Set Description I set of all branches V set of all BUs F set of existing (former) territory centers K union set of BUs that were assigned to each territory center from set F H set of pairs of BUs that must be assigned to different territories $\text{N}^i$ set of nodes which are adjacent to the $i^{th}$ branch; $i \in I$ C set of unconnected BUs assigned to each branch $\text{N}^C$ union set of all BUs that are adjacent to any member of C
 Set Description I set of all branches V set of all BUs F set of existing (former) territory centers K union set of BUs that were assigned to each territory center from set F H set of pairs of BUs that must be assigned to different territories $\text{N}^i$ set of nodes which are adjacent to the $i^{th}$ branch; $i \in I$ C set of unconnected BUs assigned to each branch $\text{N}^C$ union set of all BUs that are adjacent to any member of C
Mathematical notation and description for decision variables
 Decision Variable Description $X_{ij} \ \forall i \in I, j \in V$ set of all branches $Y_i \ \forall i \in I$ set of all BUs
 Decision Variable Description $X_{ij} \ \forall i \in I, j \in V$ set of all branches $Y_i \ \forall i \in I$ set of all BUs
Mathematical notation and description for parameters
 Parameter Description $d_{ij}$ Euclidean distance between nodes $i^{th}$ branch, $j^{th}$ BU; $i \in I,j \in V$ $w_1$ Weight of the importance of similarity with the existing design $M_{ij}$ Binary; if $j^{th}$ BU is assigned to $i^{th}$ branch in the existing plan, $i \in F$ $w_{2i}$ Weight of the risk function for each $i^{th}$ branch; $i \in I$ $PV_j$ Profit variance of $j^{th}$ BU; $j \in V$ $\gamma_i$ Threshold for total profit variance of $i^{th}$ branch; $i \in I$ p Number of territory centers $v_j^m$ Activity measure m for $j^{th}$ BU; $j \in V$, $m = 1,2,3$ $\mu_i^m$ Target level of activity measure m for $i^{th}$ branch; $i \in I$, $m = 1,2,3$ $t^m$ Territorial tolerance with respect to $m^{th}$ activity measure; $m = 1,2,3$ $\delta_i$ Maximum travel distance for BUs assigned to the $i^{th}$ branch; $i\in I$ $g_{ib}$ Binary; indicating if ith branch is of type $b$ or not; $b = 1,..,5$ $l_b$ Lower bound for the number of branches selected of type $b$; $b = 1,..,5$ $u_b$ Upper bound for the number of branches selected of type $b$; $b = 1,..,5$
 Parameter Description $d_{ij}$ Euclidean distance between nodes $i^{th}$ branch, $j^{th}$ BU; $i \in I,j \in V$ $w_1$ Weight of the importance of similarity with the existing design $M_{ij}$ Binary; if $j^{th}$ BU is assigned to $i^{th}$ branch in the existing plan, $i \in F$ $w_{2i}$ Weight of the risk function for each $i^{th}$ branch; $i \in I$ $PV_j$ Profit variance of $j^{th}$ BU; $j \in V$ $\gamma_i$ Threshold for total profit variance of $i^{th}$ branch; $i \in I$ p Number of territory centers $v_j^m$ Activity measure m for $j^{th}$ BU; $j \in V$, $m = 1,2,3$ $\mu_i^m$ Target level of activity measure m for $i^{th}$ branch; $i \in I$, $m = 1,2,3$ $t^m$ Territorial tolerance with respect to $m^{th}$ activity measure; $m = 1,2,3$ $\delta_i$ Maximum travel distance for BUs assigned to the $i^{th}$ branch; $i\in I$ $g_{ib}$ Binary; indicating if ith branch is of type $b$ or not; $b = 1,..,5$ $l_b$ Lower bound for the number of branches selected of type $b$; $b = 1,..,5$ $u_b$ Upper bound for the number of branches selected of type $b$; $b = 1,..,5$
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