A small business –say, a photocopying service with a single large machine– faces the
following scheduling problem. Each morning they get a set of jobs from customers.
They want to do the jobs on their single machine in an order that keeps their customers
happiest. Customers i’s job will take ti time to complete. Given a schedule (i.e., an
ordering of the jobs), let Ci denote the finishing time of job i. For example, if job j is
the first to be done, we would have Cj = tj
; and if job j is done right after job i, we
would have Cj = Ci + tj
. Each customer i also has a given weight wi that represents
his or her importance to the business. The happiness of customer i is expected to be
dependent on the finishing time of i’s job. So the company decides that they want to
order the jobs to minimize the weight sum of the completion times, ∑n
i=1 wiCi
Design an efficient algorithm to solve this problem. That is, you are given a set of n
jobs with a processing time ti and a weight wi
for each job. You want to order the jobs
so as to minimize the weighted sum of the completion times, ∑n
i=1 wiCi
Example. Suppose there are two jobs: the first takes time t1 = 1 and has weight
w1 = 10, while the second job takes time t2 = 3 and has weight w2 = 2. Then doing
job 1 first would yield a weighted completion time of 10 · 1 + 2 · 4 = 18, while doing the
second job first would yield the larger weighted completion time of 10 · 4 + 2 · 3 = 46