Dynamic Programming Part 1: intro and the assembly-line scheduling problem

What is dynamic programming Is an algorithm method that solves a problem by combining solutions of subproblems This sounds similar to divide-and-conquer However, there’s a difference between the two: –In divide-and-conquer, the subproblems don’t overlap –In dynamic programming, the subproblems overlap Subproblems share sub-subproblems!

Dynamic programming may be good for these problems Dynamic programming is typically applied to optimization problems These optimization problems may have many possible solutions Each solution has a value associated with it We want a solution with the optimal (min or max) value –We say “a solution,” not “the solution,” because there may be more than 1 solution with the same, best value

Sequence in the development of a dynamic programming algorithm 1.Characterize the structure of an optimal solution 2.Recursively define the value of an optimal solution 3.Compute the value of an optimal solution in a bottom-up fashion (smaller subproblems first) 4.Construct an optimal solution from computed information Step 4 omitted if only the value of the best solutions is required

Assembly line scheduling 2 assembly lines at an auto assembly plant Normally they operate independently and concurrently But when there is a rush job the manager halts the assembly lines and use stations in both assembly lines in an optimal way, to be explained next

Step 1: structure of fastest way: What’s the fastest way through station S(1,j)? First suppose the fastest way through station S(1,j) is though S(1,j-1) Key observation: the chassis must have taken a fastest way from the starting point through station S(1,j-1) Why? If there had been a faster way to get through station S(1,j-1), we could have substituted this faster way to yield a faster way through station S(1,j): a contradiction

Dynamic optimality or Optimal substructure property An optimal solution to a problem (finding the fastest way through station S(i,j) contians within it an optimal solution to subproblems (finding the fastest way through either S(1,j-1) or s(2,j-1)).

Step 2: a recursive solution Let f i [j] = fastest possible time to get a chassis from the starting point through station S(i,j) Ultimate goal: compute fastest time to get a chassis all the way through the factory Denote this time by f* We have…

Equations for the first stations on each assembly line

Step 3: computing the fastest times

Step 4: constructing the fastest way