
    Design a routing algorithm for a delivery fleet management system that determines the optimal delivery path with the lowest cumulative time. The path must consist of increasingly efficient route segments (where each subsequent segment takes less time than the previous one) from the start until a current checkpoint, and must include a specific future mandatory stop. The algorithm should evaluate all possible combinations of route segments up to the current checkpoint position, ensuring the mandatory future stop can be reached efficiently. The goal is to minimize the total delivery time while maintaining a progressively faster pace and incorporating the required future checkpoint in the final route plan.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design a routing algorithm for a delivery fleet management system that determines the optimal delivery path with the lowest cumulative time. The path must consist of increasingly efficient route segments (where each subsequent segment takes less time than the previous one) from the start until a current checkpoint, and must include a specific future mandatory stop. The algorithm should evaluate all possible combinations of route segments up to the current checkpoint position, ensuring the mandatory future stop can be reached efficiently. The goal is to minimize the total delivery time while maintaining a progressively faster pace and incorporating the required future checkpoint in the final route plan.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design a routing algorithm for a delivery fleet management system that determines the optimal delivery path with the lowest cumulative time. The path must consist of increasingly efficient route segments (where each subsequent segment takes less time than the previous one) from the start until a current checkpoint, and must include a specific future mandatory stop. The algorithm should evaluate all possible combinations of route segments up to the current checkpoint position, ensuring the mandatory future stop can be reached efficiently. The goal is to minimize the total delivery time while maintaining a progressively faster pace and incorporating the required future checkpoint in the final route plan.
    

    """
    dp = [[0 for i in range(n)] for i in range(n)]
    for i in range(n):
        if a[i] > a[0]:
            dp[0][i] = a[i] + a[0]
        else:
            dp[0][i] = a[i]
    for i in range(1, n):
        for j in range(n):
            if a[j] > a[i] and j > i:
                if dp[i - 1][i] + a[j] > dp[i - 1][j]:
                    dp[i][j] = dp[i - 1][i] + a[j]
                else:
                    dp[i][j] = dp[i - 1][j]
            else:
                dp[i][j] = dp[i - 1][j]
    return dp[index][k]