Scenario $3$: Resource Allocation for Employee Training Programs

A company has a $$600,000$ budget for employee training programs. The funds can be allocated among three training methods: online courses, in$-$person workshops, and leadership seminars. The expected improvement in employee productivity for each method is as follows:

Training Method

Productivity Improvement $(\%)$

Minimum Allocation $(\%)$

Maximum Allocation $($$$)$

Online Courses

$15$

$25$

$350,000$

In$-$Person Workshops

$12$

$20$

$250,000$

Leadership Seminars

$10$

$10$

$200,000$

The company wants to achieve an overall productivity improvement of at least $12\%.$ Goals include allocating at least $25\%$ of the total budget to online courses and ensuring that the budget for in$-$person workshops and leadership seminars combined does not exceed $60\%$ of the total budget.

$($a$)$ Formulate this problem as a goal programming model, assuming all goals are equally important.

$($b$)$ Solve this problem using available software. How much should be allocated to each training method? What is the total productivity improvement? Which goals are not met?