A group of 𝑛 people are to be assigned to a set of 𝑛 jobs, with one person assigned to each job.For a given set of 𝑛2 values 𝐶𝑖𝑗, 𝑖, 𝑗 = 1, ⋯ , 𝑛, a cost 𝐶𝑖𝑗 is incurred when person 𝑖 is assignedto job 𝑗. Suppose that all 𝐶𝑖𝑗 constitute a set of 𝑛2 independent exponential random variables,each having mean 1. Two algorithms for the assignment problem are given below. Which of themresults in a smaller expected total cost? Why?
(a) Algorithm 1: Start with the first person. Assign the job with the smallest cost, i.e., 𝑗1 ∈arg min𝑗∈{1,⋯,𝑛}𝐶1𝑗. Then, eliminate that job and consider the second person. Find the job that satisfies𝑗2 ∈ arg min𝑗∈{1,⋯,𝑛}/{𝑗1}𝐶2𝑗 for the second person. Continue the process until the last person isassigned.
(b) Algorithm 2: Among all persons and jobs, find the first person job pair (𝑖1, 𝑗1), 𝑖1 ∈ {1, ⋯ , 𝑛}and 𝑗1 ∈ {1, ⋯ , 𝑛}, for which 𝐶𝑖1𝑗1 is minimal. Assign person 𝑖1 to job 𝑗1. Eliminate cost valuesinvolving either person 𝑖1 or job 𝑗1 , and then find the second person job pair (𝑖2, 𝑗2), 𝑖2 ∈{1, ⋯ , 𝑛}/{𝑖1} and 𝑗2 ∈ {1, ⋯ , 𝑛}/{𝑗1}, with the smallest cost 𝐶𝑖2𝑗2 in the same fashion. Continuethe process until the last person is assigned
