DP34 流水线调度问题 Assembly Line Scheduling @geeksforgeeks

A car factory has two assembly lines, each with n stations. A station is denoted by S_i,j where i is either 1 or 2 and indicates the assembly line the station is on, and j indicates the number of the station. The time taken per station is denoted by a_i,j. Each station is dedicated to some sort of work like engine fitting, body fitting, painting and so on. So, a car chassis must pass through each of the n stations in order before exiting the factory. The parallel stations of the two assembly lines perform the same task. After it passes through station S_i,j, it will continue to station S_i,j+1 unless it decides to transfer to the other line. Continuing on the same line incurs no extra cost, but transferring from line i at station j – 1 to station j on the other line takes time t_i,j. Each assembly line takes an entry time e_i and exit time x_i which may be different for the two lines. Give an algorithm for computing the minimum time it will take to build a car chassis.

The below figure presents the problem in a clear picture:

The following information can be extracted from the problem statement to make it simpler:

• Two assembly lines, 1 and 2, each with stations from 1 to n.
• A car chassis must pass through all stations from 1 to n in order(in any of the two assembly lines). i.e. it cannot jump from station i to station j if they are not at one move distance.
• The car chassis can move one station forward in the same line, or one station diagonally in the other line. It incurs an extra cost ti, j to move to station j from line i. No cost is incurred for movement in same line.
• The time taken in station j on line i is a_{i, j}.
• S_{i, j} represents a station j on line i.

Breaking the problem into smaller