Problem $3$: Dynamic programming $(25$ points$)$

An IOE student has $4$ days remaining before final ex$-$ aminations begin in her four courses, and she wants to allocate this study time as effectively as possible. She needs at least $1$ day on each course, and she likes to concentrate on just one course each day, so she wants to allocate $\backslash (1,2,3\backslash ),$ or $4$ days to each course. Having recently taken the IOE $202,$ she decides to use dynamic programming to make these allocations to maximize the total grade points to be obtained from the four courses. She estimates that the alternative allocations for each course would yield the number of grade points shown in the following table:

Table $1$: Study Days vs Estimated Grade Points for Courses

$($a$)$ Formulate the given problem as an integer programming model. Clearly define the objective function to be maximized and specify all the necessary constraints. $(10$ points$)$

$($b$)$ Solve the problem using a dynamic programming approach. $($Hint: Define the value functions in a manner similar to the knapsack problem.$)(15$ points$)$