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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:
[1]

N. Abdul Hadi and A. Kamaluddin, Social collateral, repayment rates, and the creation of capital among the clients of microfinance, Procedia Economics and Finance, 31 (2015), 823-828.

[2]

A. Ahmadi-JavidP. Seyedi and S. Syam, A survey of healthcare facility location, Operations Research, 79 (2017), 223-263. doi: 10.1016/j.cor.2016.05.018.

[3]

J. AnS. H. Cho and C. S. Tang, Aggregating smallholder farmers in emerging economies, Production and Operations Management, 24 (2015), 1414-1429.

[4] B. Armendáriz de Aghion and J. Morduch, The Economics of Microfinance, MIT Press, Cambridge, 2005.
[5]

A. Ashta, Advanced Technologies for Microfinance: Solutions and Challenges, Idea Group Inc., PA, 2011.

[6]

C. Bartual SanfeliuR. Cervelló Royo and I. Moya Clemente, Measuring performance of social and non-profit microfinance institutions (MFIs): an application of multicriterion methodology, Mathematical and Computer Modelling, 57 (2013), 1671-1678. doi: 10.1016/j.mcm.2011.11.010.

[7]

S. Brenner, Location (Hotelling) games and applications, in Wiley Encyclopedia of Operations Research and Management Science (eds. J.J. Cochran), John Wiley & Sons, Inc, 2010.

[8]

G. BrutonS. KhavulD. Siegel and M. Wright, New financial alternatives in seeding entrepreneurship: Microfinance, crowdfunding, and peer-to-peer innovations, Entrepreneurship Theory and Practice, 39 (2015), 9-26.

[9]

J. Bruton and H. Min, Multiobjective design of transportation networks: Taxonomy and annotation, European Journal of Operational Research, 26 (1986), 187-201. doi: 10.1016/0377-2217(86)90180-3.

[10]

A. Drexl and K. Haase, Fast approximation methods for sales force deployment, Management Science, 45 (1999), 1307-1323.

[11]

Z. Drezner and H. W. Hamacher (Eds.), Facility Location: Applications and Theory, Springer Science & Business Media., Berlin, 2002. doi: 10.1007/978-3-642-56082-8.

[12]

J. R. Eastman, H. Jiang and J. Toledano, Multi-criteria and multi-objective decision making for land allocation using GIS, in Multicriteria analysis for land-use management (eds. E. Beinat and P. Nijkamp), Springer, (1998), 227-251.

[13]

M. ElliottB. Golub and M. O. Jackson, Financial networks and contagion, American Economic Review, 104 (2014), 3115-3153.

[14]

C. Expósito-IzquierdoA. Rossi and M. Sevaux, A two-level solution approach to solve the clustered capacitated vehicle routing problem, Industrial Engineering, 91 (2016), 274-289.

[15]

L. A. Greening and S. Bernow, Design of coordinated energy and environmental policies: Use of multi-criteria decision-making, Energy Policy, 32 (2004), 721-735.

[16]

B. Gutierrez-Nieto and C. Serrano-Cinca, Microfinance institutions and efficiency, Omega, 35 (2007), 131-142.

[17]

C. A. HaneC. BarnhartE. L. JohnsonR. E. MarstenG. L. Nemhauser and G. Sigismondi, The fleet assignment problem: Solving a large-scale integer program, Mathematical Programming, 70 (2007), 211-232. doi: 10.1007/BF01585938.

[18]

I. HeckmannT. Comes and S. Nickel, A critical review on supply chain risk: Definition, measure and modeling, Omega, 52 (2015), 119-132.

[19]

M. Herda and and M. Haviar, Hybrid genetic algorithms with selective crossover for the capacitated p-median problem, Central European Journal of Operations Research, 25 (2017), 651-664. doi: 10.1007/s10100-017-0471-1.

[20]

J. KalcsicsS. Nickel and M. Schroder, Towards a unified territorial design approach, applications, algorithms and GIS integration, Top, 13 (2005), 1-74. doi: 10.1007/BF02578982.

[21]

S. R. Khandker, Microfinance and poverty: Evidence using panel data from Bangladesh, The World Bank Economic Review, 19 (2005), 263-286.

[22]

J.-H. Lee, M. Jusup, B. Podobnik and Y. Iwasa, Agent-based mapping of credit risk for sustainable microfinance, PLoS ON, 10 (2015), e0126447.

[23]

A. Lockamy Ⅲ and K. McCormack, Analysing risks in supply networks to facilitate outsourcing decisions, International Journal of Production Research, 48 (2010), 593-611.

[24]

F. López, T. Ekin, F. Méndez and J. A. Jimenez, Hybrid Heuristic for dynamic location-allocation on micro-credit territory design, Computación y Sistemas, 19 (2015), 783-804.

[25]

H. Y. Mak and Z. J. Shen, Risk diversification and risk pooling in supply chain design, IIE Transactions, 44 (2012), 603-621.

[26]

J. G. March and Z. Shapira, Managerial perspectives on risk and risk taking, Management Science, 33 (1987), 1404-1418.

[27]

R. T. Marler and J. S. Arora, Survey of multi-objective optimization methods for engineering, Structural and Multidisciplinary Optimization, 26 (2004), 369-395. doi: 10.1007/s00158-003-0368-6.

[28]

R. T. Marler and J. S. Arora, The weighted sum method for multi-objective optimization: New insights, Structural and Multidisciplinary Optimization, 41 (2010), 853-862. doi: 10.1007/s00158-009-0460-7.

[29]

B. H. Massam, Multi-criteria decision making (MCDM) techniques in planning, Progress in Planning, 30 (1988), 1-84.

[30]

M. T. MeloS. Nickel and F. Saldanha-Da-Gama, Facility location and supply chain management-A review, European Journal of Operational Research, 196 (2009), 401-412. doi: 10.1016/j.ejor.2008.05.007.

[31]

M. S. R. Monteiro, Bank-branch Location and Sizing Under Economies of Scale, Ph.D thesis, Universidade de Minho, 2004.

[32]

M. S. R. Monteiro and D. B. Fontes, Locating and sizing bank-branches by opening, closing or maintaining facilities, in Operations Research Proceedings, Springer, (2006), 303-308.

[33]

A. NagurneyJ. CruzJ. Dong and D. Zhang, Supply chain networks, electronic commerce, and supply side and demand side risk, European Journal of Operational Research, 164 (2005), 120-142.

[34]

S. Nickel and J. Puerto, Location Theory: A Unified Approach, Springer Science & Business Media., Berlin, 2006.

[35]

J. PerezS. Maldonado and V. Marianov, A reconfiguration of fire station and fleet locations for the Santiago Fire Department, International Journal of Production Research, 54 (2016), 3170-3186.

[36]

A. A. Pinto and T. Parreira, A Hotelling-type network, in Dynamics, Games and Science I, Springer, Berlin, Heidelberg, 1 (2011), 709-720. doi: 10.1007/978-3-642-11456-4_45.

[37]

C. S. ReVelle and H. A. Eiselt, Location analysis: A synthesis and survey, European Journal of Operational Research, 165 (2005), 1-19. doi: 10.1016/j.ejor.2003.11.032.

[38]

C. S. ReVelleH. A. Eiselt and M. S. Daskin, A bibliography for some fundamental problem categories in discrete location science, European Journal of Operational Research, 184 (2008), 817-848. doi: 10.1016/j.ejor.2006.12.044.

[39]

R. Z. Rios-Mercado and E. Fernandez, A reactive grasp for a commercial territory design problem with multiple balancing requirements, Operations Research, 36 (2009), 755-776.

[40]

R. Z. Rios-Mercado and J. F. López-Pérez, Commercial territory design planning with realignment and disjoint assignment requirements, Omega, 41 (2013), 525-535.

[41]

D. Ronen, Sales territory alignment for sparse accounts, Omega, 11 (1983), 501-505.

[42]

M. A. Salazar-AguilarR. Z. Rios-Mercado and M. Cabrera-Rios, New models for commercial territory design, Networks and Spatial Economics, 11 (2011), 487-507. doi: 10.1007/s11067-010-9151-6.

[43]

Z. Shen, Integrated supply chain design models: A survey and future research directions, Journal of Industrial and Management Optimization, 3 (2007), 1-27. doi: 10.3934/jimo.2007.3.1.

[44]

M. Sodhi and C. S. Tang, Supply-chain research opportunities with the poor as suppliers or distributors in developing countries, Production and Operations Management, 23 (2013), 1483-1494.

[45]

B. C. TanselR. L. Francis and T. J. Lowe, State of the art-location on networks: A survey. Part Ⅰ: The p-Center and p-Median problems, Management Science, 29 (1983), 482-497. doi: 10.1287/mnsc.29.4.482.

[46]

E. Triantaphyllou, Multi-criteria decision making methods, in Multi-criteria Decision Making Methods: A Comparative Study, Springer, Boston, MA., (2000), 5-21.

[47]

M. Velasquez and P. T. Hester, An analysis of multi-criteria decision making methods, International Journal of Operations Research, 10 (2013), 56-66.

[48]

Q. WangR. BattaJ. Bhadury and C. M. Rump, Budget constrained location problem with opening and closing of facilities, Operations Research, 30 (2003), 2047-2069. doi: 10.1016/S0305-0548(02)00123-5.

[49]

E. K. Zavadskas and Z. Turskis, Multiple criteria decision making (MCDM) methods in economics: An overview, Technological and Economic Development of Economy, 17 (2011), 397-427.

[50]

A. A. Zoltners and P. Sinha, Sales territory alignment: A review and model, Management Science, 29 (1983), 1237-1256.

[51]

A. A. Zoltners and P. Sinha, The 2004 isms practice prize winner: Sales territory design: Thirty years of modeling and implementation, Marketing Science, 24 (2005), 313-331.

show all references

References:
[1]

N. Abdul Hadi and A. Kamaluddin, Social collateral, repayment rates, and the creation of capital among the clients of microfinance, Procedia Economics and Finance, 31 (2015), 823-828.

[2]

A. Ahmadi-JavidP. Seyedi and S. Syam, A survey of healthcare facility location, Operations Research, 79 (2017), 223-263. doi: 10.1016/j.cor.2016.05.018.

[3]

J. AnS. H. Cho and C. S. Tang, Aggregating smallholder farmers in emerging economies, Production and Operations Management, 24 (2015), 1414-1429.

[4] B. Armendáriz de Aghion and J. Morduch, The Economics of Microfinance, MIT Press, Cambridge, 2005.
[5]

A. Ashta, Advanced Technologies for Microfinance: Solutions and Challenges, Idea Group Inc., PA, 2011.

[6]

C. Bartual SanfeliuR. Cervelló Royo and I. Moya Clemente, Measuring performance of social and non-profit microfinance institutions (MFIs): an application of multicriterion methodology, Mathematical and Computer Modelling, 57 (2013), 1671-1678. doi: 10.1016/j.mcm.2011.11.010.

[7]

S. Brenner, Location (Hotelling) games and applications, in Wiley Encyclopedia of Operations Research and Management Science (eds. J.J. Cochran), John Wiley & Sons, Inc, 2010.

[8]

G. BrutonS. KhavulD. Siegel and M. Wright, New financial alternatives in seeding entrepreneurship: Microfinance, crowdfunding, and peer-to-peer innovations, Entrepreneurship Theory and Practice, 39 (2015), 9-26.

[9]

J. Bruton and H. Min, Multiobjective design of transportation networks: Taxonomy and annotation, European Journal of Operational Research, 26 (1986), 187-201. doi: 10.1016/0377-2217(86)90180-3.

[10]

A. Drexl and K. Haase, Fast approximation methods for sales force deployment, Management Science, 45 (1999), 1307-1323.

[11]

Z. Drezner and H. W. Hamacher (Eds.), Facility Location: Applications and Theory, Springer Science & Business Media., Berlin, 2002. doi: 10.1007/978-3-642-56082-8.

[12]

J. R. Eastman, H. Jiang and J. Toledano, Multi-criteria and multi-objective decision making for land allocation using GIS, in Multicriteria analysis for land-use management (eds. E. Beinat and P. Nijkamp), Springer, (1998), 227-251.

[13]

M. ElliottB. Golub and M. O. Jackson, Financial networks and contagion, American Economic Review, 104 (2014), 3115-3153.

[14]

C. Expósito-IzquierdoA. Rossi and M. Sevaux, A two-level solution approach to solve the clustered capacitated vehicle routing problem, Industrial Engineering, 91 (2016), 274-289.

[15]

L. A. Greening and S. Bernow, Design of coordinated energy and environmental policies: Use of multi-criteria decision-making, Energy Policy, 32 (2004), 721-735.

[16]

B. Gutierrez-Nieto and C. Serrano-Cinca, Microfinance institutions and efficiency, Omega, 35 (2007), 131-142.

[17]

C. A. HaneC. BarnhartE. L. JohnsonR. E. MarstenG. L. Nemhauser and G. Sigismondi, The fleet assignment problem: Solving a large-scale integer program, Mathematical Programming, 70 (2007), 211-232. doi: 10.1007/BF01585938.

[18]

I. HeckmannT. Comes and S. Nickel, A critical review on supply chain risk: Definition, measure and modeling, Omega, 52 (2015), 119-132.

[19]

M. Herda and and M. Haviar, Hybrid genetic algorithms with selective crossover for the capacitated p-median problem, Central European Journal of Operations Research, 25 (2017), 651-664. doi: 10.1007/s10100-017-0471-1.

[20]

J. KalcsicsS. Nickel and M. Schroder, Towards a unified territorial design approach, applications, algorithms and GIS integration, Top, 13 (2005), 1-74. doi: 10.1007/BF02578982.

[21]

S. R. Khandker, Microfinance and poverty: Evidence using panel data from Bangladesh, The World Bank Economic Review, 19 (2005), 263-286.

[22]

J.-H. Lee, M. Jusup, B. Podobnik and Y. Iwasa, Agent-based mapping of credit risk for sustainable microfinance, PLoS ON, 10 (2015), e0126447.

[23]

A. Lockamy Ⅲ and K. McCormack, Analysing risks in supply networks to facilitate outsourcing decisions, International Journal of Production Research, 48 (2010), 593-611.

[24]

F. López, T. Ekin, F. Méndez and J. A. Jimenez, Hybrid Heuristic for dynamic location-allocation on micro-credit territory design, Computación y Sistemas, 19 (2015), 783-804.

[25]

H. Y. Mak and Z. J. Shen, Risk diversification and risk pooling in supply chain design, IIE Transactions, 44 (2012), 603-621.

[26]

J. G. March and Z. Shapira, Managerial perspectives on risk and risk taking, Management Science, 33 (1987), 1404-1418.

[27]

R. T. Marler and J. S. Arora, Survey of multi-objective optimization methods for engineering, Structural and Multidisciplinary Optimization, 26 (2004), 369-395. doi: 10.1007/s00158-003-0368-6.

[28]

R. T. Marler and J. S. Arora, The weighted sum method for multi-objective optimization: New insights, Structural and Multidisciplinary Optimization, 41 (2010), 853-862. doi: 10.1007/s00158-009-0460-7.

[29]

B. H. Massam, Multi-criteria decision making (MCDM) techniques in planning, Progress in Planning, 30 (1988), 1-84.

[30]

M. T. MeloS. Nickel and F. Saldanha-Da-Gama, Facility location and supply chain management-A review, European Journal of Operational Research, 196 (2009), 401-412. doi: 10.1016/j.ejor.2008.05.007.

[31]

M. S. R. Monteiro, Bank-branch Location and Sizing Under Economies of Scale, Ph.D thesis, Universidade de Minho, 2004.

[32]

M. S. R. Monteiro and D. B. Fontes, Locating and sizing bank-branches by opening, closing or maintaining facilities, in Operations Research Proceedings, Springer, (2006), 303-308.

[33]

A. NagurneyJ. CruzJ. Dong and D. Zhang, Supply chain networks, electronic commerce, and supply side and demand side risk, European Journal of Operational Research, 164 (2005), 120-142.

[34]

S. Nickel and J. Puerto, Location Theory: A Unified Approach, Springer Science & Business Media., Berlin, 2006.

[35]

J. PerezS. Maldonado and V. Marianov, A reconfiguration of fire station and fleet locations for the Santiago Fire Department, International Journal of Production Research, 54 (2016), 3170-3186.

[36]

A. A. Pinto and T. Parreira, A Hotelling-type network, in Dynamics, Games and Science I, Springer, Berlin, Heidelberg, 1 (2011), 709-720. doi: 10.1007/978-3-642-11456-4_45.

[37]

C. S. ReVelle and H. A. Eiselt, Location analysis: A synthesis and survey, European Journal of Operational Research, 165 (2005), 1-19. doi: 10.1016/j.ejor.2003.11.032.

[38]

C. S. ReVelleH. A. Eiselt and M. S. Daskin, A bibliography for some fundamental problem categories in discrete location science, European Journal of Operational Research, 184 (2008), 817-848. doi: 10.1016/j.ejor.2006.12.044.

[39]

R. Z. Rios-Mercado and E. Fernandez, A reactive grasp for a commercial territory design problem with multiple balancing requirements, Operations Research, 36 (2009), 755-776.

[40]

R. Z. Rios-Mercado and J. F. López-Pérez, Commercial territory design planning with realignment and disjoint assignment requirements, Omega, 41 (2013), 525-535.

[41]

D. Ronen, Sales territory alignment for sparse accounts, Omega, 11 (1983), 501-505.

[42]

M. A. Salazar-AguilarR. Z. Rios-Mercado and M. Cabrera-Rios, New models for commercial territory design, Networks and Spatial Economics, 11 (2011), 487-507. doi: 10.1007/s11067-010-9151-6.

[43]

Z. Shen, Integrated supply chain design models: A survey and future research directions, Journal of Industrial and Management Optimization, 3 (2007), 1-27. doi: 10.3934/jimo.2007.3.1.

[44]

M. Sodhi and C. S. Tang, Supply-chain research opportunities with the poor as suppliers or distributors in developing countries, Production and Operations Management, 23 (2013), 1483-1494.

[45]

B. C. TanselR. L. Francis and T. J. Lowe, State of the art-location on networks: A survey. Part Ⅰ: The p-Center and p-Median problems, Management Science, 29 (1983), 482-497. doi: 10.1287/mnsc.29.4.482.

[46]

E. Triantaphyllou, Multi-criteria decision making methods, in Multi-criteria Decision Making Methods: A Comparative Study, Springer, Boston, MA., (2000), 5-21.

[47]

M. Velasquez and P. T. Hester, An analysis of multi-criteria decision making methods, International Journal of Operations Research, 10 (2013), 56-66.

[48]

Q. WangR. BattaJ. Bhadury and C. M. Rump, Budget constrained location problem with opening and closing of facilities, Operations Research, 30 (2003), 2047-2069. doi: 10.1016/S0305-0548(02)00123-5.

[49]

E. K. Zavadskas and Z. Turskis, Multiple criteria decision making (MCDM) methods in economics: An overview, Technological and Economic Development of Economy, 17 (2011), 397-427.

[50]

A. A. Zoltners and P. Sinha, Sales territory alignment: A review and model, Management Science, 29 (1983), 1237-1256.

[51]

A. A. Zoltners and P. Sinha, The 2004 isms practice prize winner: Sales territory design: Thirty years of modeling and implementation, Marketing Science, 24 (2005), 313-331.

Figure 1.  Illustration of branches and customers of the micro finance institution
Figure 2.  Total Cuts and Number of Disconnected BUs versus Computational Time
Figure 3.  Recenterings versus Computational Time
Figure 4.  Objective Function Values of Risk and Distance for Model 1 (bold line) and Model 2 (dashed line)
Figure 5.  Partial map of the implementation of the territory design model
Table 1.  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
Table 2.  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
Table 3.  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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