1. Scheduling
Two people are camping out and wish to cook a simple supper consisting of soup and hamburgers.
In order to prepare the supper, several tasks must be performed.
T1: Go to the store to buy matches – 10 minutes
T2: Collect firewood– 8 minutes
T3: Light the fire– 6 minutes
T4: Get water from a well– 12 minutes
T5: Cook soup (requires constant stirring)– 15 minutes
T6: Make hamburger patties – 9 minutes
T7: Cook hamburgers– 7 minutes
Each task requires only one of the campers and once a task is begun it cannot be interrupted.
Some of the tasks can be done simultaneously, but some cannot be done until others are completed.
What is the shortest amount of time it will take for all of the tasks to be completed?
How should the work be divided between the two campers?

2. Chapter 3: Planning and Scheduling Lesson Plan
For All Practical Purposes
Scheduling Tasks
Assumptions and Goals
List-Processing Algorithm
Optimal Schedules
Strange Happenings
Critical-Path Schedules
Independent Tasks
Decreasing-Time-List Algorithm
Bin Packing
Bin-Packing Heuristics
Decreasing-Time Heuristics
Resolving Conflict via Coloring
Vertex Coloring and Chromatic Number
Mathematical Literacy in Today’s World, 8th ed. 2
© 2009, W.H. Freeman and Company

3. Chapter 3: Planning and Scheduling Scheduling Tasks
Accurate planning and scheduling is required of both people and machines.
Examples: Around-the-clock scheduling of doctors and nurses, scheduling equipment (X-rays, MRI scans) for maximum efficiency
Processors
A person, machine, robot, operating room, or runway (for airplanes) whose time must be scheduled.
Machine-Scheduling Problem
The problem of deciding how the tasks should be scheduled so the entire job is completed as early as possible. 3

4. Chapter 3: Planning and Scheduling Scheduling Tasks
Assumptions
When a processor starts a task, it will be completed without interruption.
Any processor can work on any of the tasks.
No processor stays idle voluntarily.
Order-requirement digraph is used to show the task order and has the task times highlighted within each vertex.
Priority lists arrange the tasks in order, independent of the order requirements. Used to “break ties” if more than one task is ready.
Goals to Consider
Minimize the completion time of the job.
Minimize the total time the processors are idle.
Find the minimum number of processors necessary to finish the job by a specified time. 4

5. Chapter 3: Planning and Scheduling Scheduling Tasks
Order-Requirement Digraph
For the next example, tasks are labeled as T1–T8.
The length of time for each task is indicated in a colored circle.
At the beginning (time = 0), tasks T1, T7, T8 are ready to be scheduled on machines.
Order Requirement Digraph — Example
Ready task – A task is ready if its predecessors have been completed by that time as given by the order-requirement digraph.
Processors
For this example, tasks will be scheduled on two machines according to the order-requirement digraph and priority list.
Priority List
A list that ranks the tasks in some criterion of importance.
For the problem, the priority list is T1, T2, T3, T4, T5, T6, T7, T8. 5

6. Chapter 3: Planning and Scheduling Scheduling Tasks
List Processing Algorithm
At a given time, assign to the lowest numbered free machine (or processor) the first task on the priority list that is ready at that time (and not already been assigned).
Summary of Steps
Look at timeline for available machine. If more than one is free, use the lower-number machine—the top machine.
Use order-requirement digraph to choose the ready tasks.
When there is a tie (two or more tasks ready), use priority list to break ties and assign the ready tasks in order.
Order-Requirement Digraph
Schedule Tasks on Machines
Priority List: T1, T2, T3, T4, T5,T6, T7, T8 6

7. Chapter 3: Planning and Scheduling Scheduling Tasks
When is the schedule optimal?
From Chapter 2, remember the longest path in an order requirement is the critical path.
The earliest time for the job made up of all the tasks is the length of the longest path (critical path) in the order-requirement digraph.
Referring to the previous example:
The critical path is T1, T2, T3, T4 = = 27.
If the job is completed in the same time as the length of the critical path, this is the earliest completion time possible and it is an optimal schedule.
If the schedule completion time = the length of critical path,
then the schedule is optimal (earliest completion time). 7

8. Chapter 3: Planning and Scheduling Scheduling Tasks
Strange Happenings
Four factors affect the final schedule:
Times of the tasks
Number of processors
Order-requirement diagram
Ordering of the tasks on the priority list
While keeping the priority list constant, you would expect the following strategies would make the completion time earlier:
Reduce task times.
Use more processors.
“Loosen” the constraints of the order-requirement digraph.
However, with a constant priority list, applying one of these strategies may not always make the completion time shorter — strange happenings. 8

9. Example Schedule the tasks in the digraph on two processors with priority list T1, T4 , T5 , T6 , T 7, T8, T9, T10, T2, T3. What is the completion time?

11. Example Using the same digraph, schedule the tasks on three processors with priority list T 1, T 2, T3 , T 4, T 5, T6, T 7, T8, T9, T10. What is the completion time?

13. Chapter 3: Planning and Scheduling Critical Path Schedules
A systematic method of choosing/creating a priority list, L, that yields optimal, or nearly optional, schedules.
Critical-Path Scheduling Algorithm (to create priority list L)
Find a task that heads a critical (longest) path in the order-requirement digraph. If a tie, choose the lowest-number task.
Place the task found in step 1 next on the list L (the first time through the process this task will head the list).
Remove the task found in step 1 and the edges attached to it from the current order-requirement digraph, obtaining a new (modified) order-requirement digraph.
If there are no vertices left in the new order-requirement digraph, the procedure is complete; if there are vertices left, go to step 1.
This procedure will terminate when all the tasks in the original order-
requirement digraph have been placed on the list L (example on next slide). 13

14. Chapter 3: Planning and Scheduling Critical Path Schedules
Example of Critical-Path Scheduling Algorithm
For this order-requirement digraph, there are two critical paths of length 64: T1, T2, T3 and T1, T4, T3. Place T1 on the list L.
With T1 “gone,” there is a new critical path of length 60: T5, T6, T4, T3. Place T5 next on the list L.
With T1 and T5 “gone,” the next longest path would be 56: T6, T4, T3. Place T6 next on the list L. (Continue the algorithm until list is complete.)
The new priority list would be: L = T1, T5, T6, T2, T4, T3, T8, T9, T7, T10.
Using the list-processing algorithm with the original order-requirement digraph and the new priority list L, the following schedule is obtained: 14

15. Example Determine the earliest completion time for the order-requirement digraph. Use critical-path scheduling to construct a priority list for the tasks.

16. Chapter 3: Planning and Scheduling Independent Tasks
Tasks that are independent of each other if they can be performed in any order (no edges in the order-requirement digraph).
Label the tasks by their task times rather than their task number.
The list-processing algorithm can be used to schedule the tasks onto machines using the given list but may not be efficient.
Decreasing-Time List Algorithm
First put the tasks in decreasing order by time, then apply the list processing algorithm. This will help pack the longest tasks first, so they do not “stick out” on the right end.
Example: Apply decreasing-time list algorithm on independent tasks 10, 9, 7, 7, 5, 4 (already in order)
This result is an optimal schedule! 16

17. When a path is not given you can add up the TOTAL amount of time of tasks and divide that number by the number of processors.
Example from previous page: 10, 9, 7, 7, 5, 4
= 42
42/2= 21
The minimum time required to complete the given tasks on two processors is 21 minutes.

18. Example 2: What is the minimum time required to complete nine independent tasks on three processors when the sum of all the times of the nine tasks is 72 minutes?
Example 3: What is the minimum time required to complete 12 independent tasks on two processors when the sum of all the times of the 12 tasks is 84 minutes?
Example 4: What is the minimum time required to perform eight independent tasks with a total task time of 48 minutes on four machines?

19. Chapter 3: Planning and Scheduling Bin Packing
The problem of finding the minimum number of bins into which the weight can be packed.
Find the minimum number of bins (containers) of capacity, W.
Where weights w1, w2, …,wn are packed into the bins.
Each weight of object is less than or equal to W (wi W).
Instead of fixing the number of machines and finding a minimum completion time, we want to find the minimum number of machines (bins), each with a fixed capacity W .
Heuristic Algorithms
Methods that can be carried out quickly but cannot guarantee to produce optimal results.
Once again, there is no fast or optimal algorithm, so we will look at several methods and compare them. 19

20. Chapter 3: Planning and Scheduling Bin Packing
Bin Packing — Six Methods
Next Fit (NF) Next Fit Decreasing (NFD)
First Fit (FF) First Fit Decreasing (FFD)
Worse Fit (WF) Worse Fit Decreasing (WFD)
Next Fit (NF) – A new bin is opened if the weight to be packed next will not fit in the bin that is currently being filled; the current bin is then closed.
First Fit (FF) – The next weight to be packed is placed in the lowest-numbered bin already opened into which it will fit. If it does not fit in any open bins, a new bin is opened.
Worse Fit (WF) – The next weight to be packed is placed into the open bin with the largest amount of room remaining. If it does not fit in any bins, open a new bin.
Decreasing-Time Heuristics (NFD, FFD, WFD) – Create the priority list by listing the tasks in order of decreasing size before the bin-packing method. 20

21. Chapter 3: Planning and Scheduling Bin Packing
Next Fit (NF)
Advantages – Does not require knowledge of all the weights in advance; only need to know the remaining space in the bin.
Disadvantages – The bin packed early on may have had room for small items that come later in the list.
Best method for assembly-line packing.
Example Using Next Fit Bin Packing
Pack list 6, 6, 5, 5, 5, 4, 4, 4, 4, 2, 2, 2, 2, 3, 3, 7 7, 5, 5, 8, 8, 4, 4, 5 21

22. Example
Pack boxes of sizes 5, 7, 2, 6, 5, 1, 3, 4, 2, 3, 6, 3 into bins of capacity 10. How many bins are
required
a) using the next-fit algorithm?
b) using the first-fit algorithm?
c) using the worst-fit algorithm?

23. a) There are six bins required.

24. b) There are five bins required.

25. c) There are five bins required.

26. Example Pack boxes of sizes 5, 7, 2, 6, 5, 1, 3, 4, 2, 3, 6, 3 into bins of capacity 10. How many bins are required a) using the next-fit decreasing algorithm? b) using the first-fit decreasing algorithm? c) using the worst-fit decreasing algorithm?

27. Solution First, rearrange the boxes in decreasing order of size
Solution First, rearrange the boxes in decreasing order of size. 7, 6, 6, 5, 5, 4, 3, 3, 3, 2, 2, 1 a) There are six bins required.

28. b) There are five bins required.

29. c) There are five bins required.

30. Chapter 3: Planning and Scheduling Resolving Conflict
Goals of Three Scheduling Problems
Optimization issues – Try to maximize profit, minimize cost.
Example: Scheduling machine time for earliest completion time
Equity – Try to make things fair for all participants.
Example: Schedule baseball games (same number home and away games)
Conflict Resolution – Try to prevent conflicts from happening.
Example: Scheduling college final examinations for end of term
Resolving Conflict via Coloring
Vertex Coloring – The vertex coloring problem for a graph requires assigning each vertex of the graph a color (label), such that two vertices joined by an edge are assigned different colors.
Chromatic Number – The chromatic number is the minimum number of colors needed to label the vertices of a graph so that no two vertices of the graph joined by an edge get the same color. 30

31. Vertex Coloring Color the vertices of the following graphs in such a way that adjacent vertices have different colors. What is the minimum number of colors required for each graph?

32. Example Discuss the coloring of the following graphs
Example Discuss the coloring of the following graphs. Can you make any generalizations regarding the chromatic number of such figures?

33. In general, any pie chart with an even number of sectors can be colored with 2 colors and with an odd number of sectors (greater than one) requires 3.

34. Example What is the chromatic number for each of the following?

35. SOLUTION:

36. Chapter 3: Planning and Scheduling Resolving Conflict
Example: Scheduling Exams Using Vertex Coloring
There are eight (8) finals to schedule with only two (2) air-conditioned rooms. (8 courses: French-F, Math-M, History-H, Philosophy-P, English-E, Italian-I, Spanish-S, Chemistry-C).
In the graph, courses are represented by vertices and two course are joined by an edge for every student enrolled in both courses.
c) 4 colors — 4 time slots but need 3 rooms for time slot 2.
d) 4 colors with 4 time slots — each color appears twice so only need 2 air condition rooms.
a) Represents conflict info about the courses.
b) 8 colors represent 8 time slots — not optimal.
The chromatic number is 4 as in d (we scheduled 8 exams in 4 time slots without a conflict). 36

37. Conflict Resolution A man wants to set up an aquarium in his home to contain five different types of fish—A, B, C, D, and E. However, some of these fish will eat other ones and thus cannot share a tank. The chart below displays the incompatibilities. (An X indicates that the two species cannot share a tank.) What is the minimum number of tanks required, and which fish go into which tank?

38. Conflict Solution: