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You are a human resource executive in a large multinational company, ABC. ABC is experiencing challenges in effectively managing training and development programs for its employees. ABC has identified various training programs offered by specialized training consultants outside the company to enhance employee skills and competencies. However, ABC has limited resources and wants to optimize the allocation of training resources across departments to maximize the overall impact on performance/output. To address this challenge, you have been assigned to optimize the use of training and development funds. Your objective is to identify the optimal allocation of the employee time in different departments to various training programs to maximize the overall improvement in company output (in dollar terms) while considering budget constraints.

You have gathered the following information.

Departments: The company has four departments that have different training needs and restrictions of the number of hours for which employees can be made available to attend. For a department, what matters is the total hours for which its employees get training. Which individual gets training is not relevant in this case. Assume that the training can be for fraction of an hour and the cost and benefits would vary in that proportion. The departments and their respective training requirements are as follows:

Department 1: 30 employee-hours available (i.e., up to 30 hours are available for which employees could leave their regular work and attend training). Can be trained in Program A or Program B. (That is, some employee-hours could be in Training Program A, some employee-hours can be in Training Program B, but none should be trained in any other program as they dont address the needs of this department.)

Department 2: 40 employee-hours available. Can be trained in Training Program B, Training Program C, or Training Program E.

Department 3: 50 employee-hours available. Can be training in Training Program A, Training Program C, or Training Program D.

Department 4: 25 employee-hours available. Can be trained in Training Program B, Training Program D, or Training Program E.

Training Programs: ABC has identified five different training programs, labeled A, B, C, D, and E, offered by specialized training consultants outside the company. Each program has a different cost (per employee) and likely effectiveness level in improving employee output (in dollars per employee).

Program A: Cost - $1000 per employee-hour, Expected Improved output - $4000 per employee-hour

Program B: Cost - $1200 per employee-hour, Expected Improved output - $3000 per employee-hour

Program C: Cost -$800 per employee-hour, Expected Improved output - $2000 per employee-hour

Program D: Cost -$1500 per employee-hour, Expected Improved output - $7000 per employee-hour

Program E: Cost - $900 per employee-hour, Expected Improved output - $5000 per employee-hour

Budget Constraint: ABC has allocated a budget of $15,000 for various training and development programs offered by outside consultants.

Show your work in Excel with identification of decision variables, expression for objective function, equations/inequalities for constraints, and a solution through Solver. Interpret the results.