Case study $4$: GlobalTech's Expansion into New Markets $($Integer Programming$)$

GlobalTech, a rapidly growing technology consulting firm, is poised to expand its international presence. Currently, the company operates out of four major regional offices in London, Berlin, Toronto, and Dubai, and now seeks to establish additional offices in strategic global cities. After careful market analysis, GlobalTech has shortlisted seven cities for potential expansion: Sydney, Tokyo, Paris, Mumbai, Mexico City, Singapore, and New York.

To support this expansion, GlobalTech has allocated a budget of $\backslash (\backslash$$ $14\backslash )$ million for the setup of new offices and the transfer of key personnel from its existing offices. Due to the specialized nature of the firm's work, new offices must be staffed with employees from existing locations, but the total number of employees available for transfer is limited$.$ In addition, the cost of relocating employees varies depending on the distance between the existing and new offices, as well as living expenses and relocation costs.

GlobalTech needs to balance its available resources while ensuring that it meets market demands, geographical coverage, and profitability goals. GlobalTech faces several challenges and key requirements for this expansion:

$1.$ Budget Limitations: The total cost of opening new offices and transferring staff must not exceed the allocated $\backslash (\backslash$$ $14\backslash )$ million.

$2.$ Staffing Requirements: Each new office has a minimum staffing requirement, which must be fulfilled by transferring employees from existing offices.

$3.$ Relocation Costs: The cost of relocating employees varies depending on the city pairs, influenced by housing costs, relocation expenses, and cost of living adjustments.

$4.$ Geographic Expansion: GlobalTech aims to establish at least one new office in the AsiaPacific region $($either Sydney or Tokyo$)$ and at least one new office in the Americas $($either Mexico City or New York$).$

$5.$ Profit Potential: The firm has ranked the cities based on their projected profit potential, with New York ranked the highest, followed by Mexico City, Sydney, and others.

GlobalTech's expansion decisions must consider the following data:

$1.$ Setup Costs and Staffing Needs: Each prospective office has different setup costs and staffing needs, as shown in the table below:

$2.$ Available Employees for Transfer: GlobalTech has a limited number of employees available for transfer from its existing offices:

$3.$ Relocation Costs: The cost of relocating employees from existing offices to new locations varies according to the specific city pair, as shown below $($in thousands of dollars$)$:

$4.$ Profit Potential: Each prospective office has been ranked by GlobalTech based on its potential profitability:

Question:

$-$ Formulate and solve an integer programming model to help GlobalTech determine how many new offices it should open, where they should be located, and how to transfer employees.

$-$ Hint: Assign the value of $7$ to the most desired city and the value of $1$ to the least desired city in the objective function.