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# Management Science Question: CHB Inc.

CHB, Inc., is a bank holding company that is evaluating the potential for expanding into a 13-county region in the southwestern part of the state. State law permits establishing branches in any county that is adjacent to a county in which a PPB (principal place of business) is located. The following map shows the 13-county region with the population of each county indicated.

a. Assume that only one PPB can be established in the region. Where should it be located to maximize the population served? (Hint: Review the Ohio Trust formulation in Section 7.3. Consider minimizing the population not served, and introduce variable yi = 1 if it is not possible to establish a branch in county i, and yi = 0 otherwise.)

b. Suppose that two PPBs can be established in the region. Where should they be located to maximize the population served?

c. Management learned that a bank located in county 5 is considering selling. If CHB purchases this bank, the requisite PPB will be established in county 5, and a base for beginning expansion in the region will also be established. What advice would you give the management for CHB?

#### Solution Preview

see the attached file. Thanks

b. Suppose that two PPBs can be established in the region. Where should they be located to maximize the population served?
We can have only two PPB, thus constraint will change to
Sxi =2
Evertthing else will reamin same.

Let xi is a binary variable which takes a value of 1 if the PPB is established in county I and 0 if the PPB is not established in county i.
Let yi is a binary variable which takes a value of 1 if the population of county i is not served and 0 if the population of county i is served.

Decision variables: PPB Population Served County Population (pi)
County 1 x1 0 y1 0 195,000
County 2 x2 0 y2 0 96,000
County 3 x3 1 y3 0 87,000
County 4 x4 0 y4 0 52,000
County 5 x5 0 y5 0 233,000
County 6 x6 0 y6 0 57,000
County 7 x7 0 y7 0 117,000
County 8 x8 0 y8 0 88,000
County 9 x9 0 y9 0 106,000
County 10 x10 0 y10 1 76,000
County 11 x11 1 y11 0 95,000
County 12 x12 0 y12 0 323,000
County 13 x13 0 y13 0 175,000

Objective Function

Minimize the population not served 76000
Minimize z=Spiyi

Constraints

Each county cannot be served if it does not have a PPB in the county or in adjacent county.
LHS RHS
County 1: x1+x2+x3>=(1-y1) 1 >= 1
County 2: x1+x2+x3+x4+x6+x7>=(1-y2) 1 >= 1
County 3: x1+x2+x3+x4+x5>=(1-y3) 1 >= 1
County 4: ...

#### Solution Summary

This solution provides assistance with the management science question.

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