Barr the bear has started a business to sell fish to poe and his fellow penguins. the penguin customers submit many fish orders, but barr can only process one order at a time. suppose that barr currently has orders from n penguin customers (label them as 1, 2, . . , n). customer i’s order takes ti time to complete. barr is going to process each penguin’s order one by one, and the scheduling of orders can be described as a permutation of the customers. let ci denote the completion time of order i. for instance, if customer j’s order is the first to be completed, then we would have cj = tj (assume barr begins processing orders at time 0); if customer k’s order is the second to be completed after that, then ck = cj + tk = tj + tk, and so on. each customer is of different importance to barr’s business, and we denote this relative weight by wi (for customer i). barr wishes to minimize the weighted sum of the completion times of n orders, pn i=1 wi · ci . intuitively, the more important a customer is, the sooner barr wishes to complete the customer’s order. design an o(n log n) algorithm to solve this problem to barr. you are given a set of n orders with a processing time ti and a weight wi for each customer i (assume ti , wi are positive integers). you want to decide an ordering of the customer orders so as to minimize the weighted sum of the completion times. prove correctness of your algorithm (i. e., optimality) and analyze its runtime.

Algorithm below designed using Java

Explanation:

import java.util.SortedMap;

import java.util.TreeMap;

import java.util.Vector;

public class SchedulingProblem {

static Vector<Integer> processTime = new Vector<>();

static Vector<Integer> processWeight = new Vector<>();

static SortedMap<Integer, Double> timeWeightRatio = new TreeMap<Integer, Double>() {

};

static int number = 2;

public static void main(String[] args) {

processTime.add(1);

processWeight.add(10);

processTime.add(3);

processWeight.add(2);

for (int i = 0; i < number; i++) { // O(N)

timeWeightRatio.put(i, processWeight.get(i) / (processTime.get(i) * 1.0)); // O(log(n))

}

// O(nlog(n))

int sum = 0, tmpTime = 0;

while (timeWeightRatio.size() > 0) { // O(n)

tmpTime += processTime.get(timeWeightRatio.firstKey()); // O(n)

sum += processWeight.get(timeWeightRatio.firstKey()) * tmpTime; // O(n)

System.out.print(processWeight.get(timeWeightRatio.firstKey()) + " * " + tmpTime + ((timeWeightRatio.size() > 1) ? " + " : " = " + sum));

timeWeightRatio.remove(timeWeightRatio.firstKey()); // O(log(n))

}

// >>>> O(nlog(n))

}

}

the benefits of raising bc’s minimum wage to $15 through a

series of pre-announced staged increases far outweigh the likely

costs. claims that such an increase will lead to massive job losses

in low-wage sectors of the economy are not credible.

• an increase to $15 would result in some job losses. however,

the likely impact is less than a 1 per cent reduction in the overall

provincial employment rate, and a 7.6 per cent reduction in the

employment rate for those currently making between $10.25

and $15 per hour.

• the large job loss impacts predicted by some opponents of

minimum wages misrepresent the existing economic research. in

reading and using that research, it is important to recognize that

estimates showing job loss effects of minimum wage increases

apply only to teenagers. estimated effects for young adult and

adult workers range from insignificant to non-existent.

• focusing on teenage workers plays up an inaccurate stereotype

about who earns very low wages in bc and leads to exaggerated

claims about job losses. only 26 per cent of those currently making

between $10.25 and $12 are teenagers, and only 5 per cent

of those making between $12 and $15 are teenagers.