Question

A greedy solution for the Job Sequencing with Deadlines problem is to sort the jobs by their profits in nonincreasing order, then consider the jobs one at a time. Give a counterexample to show that this doesn’t necessarily yield

Answer 1

Job sequencing with deadlines-

The sequencing of jobs on a single processor with deadline constraints is named as job sequencing with deadlines.

Problem-

You are given a set of jobs where each job has a defined deadline and some profit associated with it. The profit of that job is given only when that job is completed within its deadline.

The problem is-

“How can the total profit be maximized if only one job can be completed at a time?”

Constraints-

• Only one processor is available for processing all jobs.
• Processor takes one unit of time to complete a job.
• All the jobs have to be completed within their respective deadlines to obtain the profits associated with them.

Solution-

For the given problem,

• A feasible solution would be a subset of jobs where each job of the subset gets completed within its deadline.
• Value of the feasible solution would be the sum of profits of all the jobs contained in the subset.
• An optimal solution of the problem would be a feasible solution which gives the maximum profit.

Job Sequencing with deadlines Greedy Algorithm-

We adopt a greedy algorithm to determine how the next job is selected for an optimal solution. The greedy algorithm described below always gives an optimal solution to the job sequencing problem.

Step-01:

Sort all the given jobs in the decreasing order of their profits.

Step-02:

Check the value of maximum deadline. Then, draw a Gantt chart where maximum time on Gantt chart is the value of maximum deadline.

Step-03:

Pick up the jobs one by one and then put them on the Gantt chart as far as possible from 0 (starting point) such that the job gets completed before its deadline.

The following practice problem illustrates these steps in detail.

PRACTICE PROBLEM BASED ON JOB SEQUENCING WITH DEADLINES

Problem-

Given the following jobs, their deadlines and associated profits as shown-

Jobs	J1	J2	J3	J4	J5	J6
Deadlines	5	3	3	2	4	2
Profits	200	180	190	300	120	100

Answer the following questions-

1. Write the optimal schedule that gives maximum profit.
2. Are all the jobs completed in the optimal schedule?
3. What is the maximum earned profit?

Solution-

Step-01:

Sort all the given jobs in the decreasing order of their profits-

Jobs	J4	J1	J3	J2	J5	J6
Deadlines	2	5	3	3	4	2
Profits	300	200	190	180	120	100

Step-02:

Value of maximum deadline = 5

So, draw a Gantt chart with maximum time on Gantt chart = 5 units as shown-

Step-03:

Now, we will take each job one by one in the order they appear in step-01 and place them on Gantt chart as far as possible from 0 (starting point)

• First, we take job J4.
• Since, its deadline is 2, so we place it in the first empty cell before deadline 2 as-

• Now, we take job J1.
• Since, its deadline is 5, so we place it in the first empty cell before deadline 5 as-

• Now, we take job J3.
• Since, its deadline is 3, so we place it in the first empty cell before deadline 3 as-

• Now, we take job J2.
• Since, its deadline is 3, so we place it in the first empty cell before deadline 3.
• Since, the second and third cells are already filled, so we place job J2 in the first cell as-

• Now, we take job J5.
• Since, its deadline is 4, so we place it in the first empty cell before deadline 4 as-

Now,

• The only job left is job J6 whose deadline is 2.
• All the slots before deadline 2 are already occupied.
• Thus, job J6 can not be completed.

Now, the given questions may be answered as-

Part-01:

The optimal schedule is-

J2 , J4 , J3 , J5 , J1

This is the required order in which the jobs must be completed in order to obtain the maximum profit.

Part-02:

• All the jobs are not completed in the optimal schedule.
• This is because job J6 could not be completed within its deadline.

Part-03:

Maximum earned profit

= Sum of profits of all jobs in optimal schedule

= Profit of job J2 + Profit of job J4 + Profit of job J3 + Profit of job J5 + Profit of job J1

= 180 + 300 + 190 + 120 + 200

= 990 units