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EXAMINER : Mr M Bamber
DEPARTMENT : Math.Sciences TEL.NO 50345 / 44756
MAY 2010 EXAMINATIONS
DECISION AND DISCRETE MATHEMATICS
TIME ALLOWED: Two Hours
INSTRUCTIONS TO CANDIDATES
This paper is divided into two sections. All the questions are based on material studied in
workshops and tutorials for this module.
There are 5 questions in Section A – you should answer ALL the questions in Section A.
There are 3 longer questions in Section B. You should answer TWO questions from Section B.
Use INSERT 1 provided with this paper to submit your answer to Question A2.
Use INSERT 2 provided with this paper to submit your answer to Question A5.
If you choose to answer Question B3, use INSERT 3 to answer parts (a) and (d).
MATH016 Page 1 of 11 Continued
PAPER CODE NO.
MATH 016
SECTION A
ANSWER ALL THE QUESTIONS FROM THIS SECTION (40 Marks)
A
The organiser of a sponsored walk wishes to allocate each of six volunteers, Alan, Geoff, Laura, Nicola, Philip and Sam to one of the checkpoints along the route. Two volunteers are needed at checkpoint 1 (the start) and one volunteer at each of checkpoint 2, 3, 4 and 5 (the finish). Each volunteer will be assigned to just one checkpoint. The table shows the checkpoints each volunteer is prepared to supervise.
Name Checkpoints Alan 1 or 3 Geoff 1 or 5
Laura 2, 1 or 4 Nicola 5
Philip 2 or 5 Sam 2
Initially Alan, Geoff, Laura and Nicola are assigned to the first checkpoint in their individual list.
(a) Draw a bipartite graph to model this situation and indicate the initial matching in a distinctive way. (2)
(b) Starting from this initial matching, use the maximum matching algorithm to find an improved matching. Clearly list any alternating paths you use. (3)
(c) Explain why it is not possible to find a complete matching. (2) [7 marks]
A
(a) State two differences between Kruskal’s algorithm and Prim’s algorithm for finding a minimum spanning tree. (2)
(b) Listing the arcs in the order that you consider them, find a minimum spanning tree for the network in the diagram above, using
(i) Prim’s algorithm,
(ii) Kruskal’s algorithm. (6) [8 marks]
A4.
(a) The list of numbers above is to be sorted into descending order. Perform a Quick Sort to obtain the sorted list, giving the state of the list after each pass and indicating the pivot elements clearly. (4) The numbers in the list represent the lengths, in mm, of some pieces of wood. The wood is sold in one metre lengths.
(b) Use the first-fit decreasing bin-packing algorithm to determine how these pieces could be cut from the minimum number of one metre lengths. (You should ignore wastage due to cutting.) (3)
(c) Determine whether your solution to part (b) is optimal. Give a reason for your answer.
[9 marks]
A
Use INSERT 2 to answer the whole of this question
The diagram above describes an algorithm in the form of a flow chart, where a is a positive integer.
List P , which is referred to in the flow chart, comprises the prime numbers 2, 3, 5, 7, 11, 13, 17, ...
(a) Starting with a = 90, implement this algorithm. Show your working in the table on INSERT 2. (6)
(b) Explain the significance of the output list. [Answer on INSERT 2.] (2)
(c) Write down the final value of c for any initial value of a. [Answer on INSERT 2.] (1) [9 marks]
START
Input a
Is an integer?
c
Output b
Is a = b?
END
Increase to next integer in List
b
P
Let a = c
YES
YES
NO
NO
Let b = First number in List P
Let c = ab
List P : 2, 3, 5, 7, 11, 13, …
B2.
The tableau below is the initial tableau for a maximising linear programming problem.
Basic variable x y z r s t Value r 7 10 10 1 0 0 3600 s 6 9 12 0 1 0 3600 t 2 3 4 0 0 1 2400 P – 35 – 55 – 60 0 0 0 0
(a) Write down the three inequalities and the objective function represented in the initial tableau above. (4) (b) Explain the purpose of the slack variables r, s and t. (2)
(c) Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. State the row operations that you use. (11)
(d) State the values of the objective function and each variable. (3) [20 marks]
B3.
Answer parts (a) and (d) of this question on INSERT 3
A building project is modelled by the activity network shown in the diagram above. The activities involved in the project are represented by the arcs. The numbers in brackets on each arc gives the time, in days, taken to complete the activity.
(a) Complete the boxes on the copy of the network above on INSERT 3 by calculating the earliest and latest event times. (5) (b) Hence write down the critical activities and the length of the critical path. (3) (c) Obtain the total float for each non-critical activity. (2)
(d) On the grid on INSERT 3, draw a cascade (Gantt) chart showing the information found in parts (b) and (c). (3)
Given that each activity requires one worker, and that any worker can perform any task,
(e) Determine the minimum number of workers required to complete the project on time and draw up a schedule which shows the tasks that each worker will perform. (4)
Due to unforeseen circumstances, activity C takes 30 days rather than 20 days.
(f) Determine how this affects the length of the critical path and state the critical activities now. (3) [20 marks]
A (30) 7
B (52)
C (20)
D (18)
E (10)
F (18)
H (12)
I (10)
G (15)
INSERT 2 (for answering Question A5)
Please detach this INSERT and attach it securely to your script.
a b c Integer? Output List
a = b?
(b) Explain the significance of the output list.
……………………………………………………………………………………..................................................................
(c) Write down the final value of c for any initial value of a.
……………………………………………………………………………………..................................................................
INSERT 3 (for answering parts (a) and (d) of Question B3)
Please detach this INSERT and attach it securely to your script.
B3 (a)
(d)
END
A (30) 7
B (52)
C (20)
D (18)
E (10)
F (18)
H (12)
I (10)
G (15)
KEY: Earliest Event Time
Latest Event Time
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
