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MAP602S - MATHEMATICAL PROGRAMMING - 1ST OPP - NOVEMBER 2023 |
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nAmlBIA untVERSITY
OF SCIEnCE AnDTECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoolof NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: •264 612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION : BACHELORof SCIENCEIN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM
LEVEL:6
COURSE:MATHEMATICAL PROGRAMMING
COURSECODE: MAP602S
DATE: NOVEMBER 2023
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY EXAMINATION: QUESTION PAPER
Mr. Benson E. Obabueki
Professor Adetayo S. Eegunjobi
INSTRUCTIONS
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
6. Show all your working/calculation steps.
PERMISSIBLE MATERIALS:
1. Non-Programmable Calculator.
2. Metric graph paper to be supplied by the examination department.
ATTACHEMENTS
1. None
This paper consists of 3 pages excluding this front page.
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Question 1 (8 marks)
Impact Printing makes two kinds of computer paper using premium or ordinary quality stock. They
have a contract to supply at least 5000 cases of paper. There is only enough stock to make 4000
cases of premium paper, but ample stock for ordinary paper. Both kinds are made with the same
machine and 1200 hours of machine time are available. Premium paper takes 18 minutes per case
and ordinary paper takes 12 minutes per case. The profit on each is $4/case and $3/case,
respectively.
Model the above statement into a linear programme. You must clearly define your variables
unambiguously and name your constraints. DO NOT SOLVE.
(8)
Question 2(13 marks)
Using a scale of 1cm to 1 unit, solve the following linear program graphically:
Minimize T=10x+15y
Subject to 12x+4y~48
4x+8y~32
2x+5y 10
(13)
O~x<6
y~O
Question 3(26 marks)
Consider the primal linear program:
Minimize T = 32a + 32b
Subject to 4a + 8b 12
8a+4b 14
a,b 0
3.1 Write down the dual of the linear program.
(5)
3.2 Solve the dual of the linear program completely using the simplex method.
(12)
3.3 Use the solution of the dual to determine the solution of the primal program. (9)
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Question 4 (18 marks)
The following linear program is to be solved using the 2-phase method:
Maximize Z =3x + 4y
Subject to 2x + y 5: 600
x+ y~150
5x+4y5: 1000
x+2y ~225
x,y 0
4.1 Determine the objective function for phase one.
(3)
4.2 Write down the initial tableau for phase one.
(2)
4.3 One of the tableaux in the process of phase one is as follows:
(5)
X
y
s1 s2 s3 s4 A2 A4
H
30 2
0
0
1
0
-1
0
10 0
-2
0
1
2
-1
0
30
0
0
1
2
0
-2
0
12 0
0
0
-1
0
1
0
10 0
-2
0
1
0
-3
2
Use this tableau to determine the optimal tableau for phase one.
975
75
550
225
· 75
4.4 Express the original function Z =3x+4y in terms of the non-basic variables of the final
phase one tableau.
(3)
4.5 Determine the optimal solution of the given linear program.
(6)
Question 5 (16 marks)
A brewing company has three plants that produce Soul-Ale. The products are moved from the
plants to three warehouses. The costs of moving a crate of 50 bottles from each plant to the
different warehouses, the capacities of the plants as well as the demands from the warehouses,
are given in the following table:
Plant 1
Plant 2
Plant 3
Demand
Warehouse 1
9
6
9
21
Warehouse 2
15
8
3
14
Warehouse 3
12
13
11
25
Supply
10
23
27
Use the North-west corner method to distribute the products in such a way that the total cost of
transportation is minimal.
(16)
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Question 6 (17 marks)
Consider a situation where four jobs (Jl, J2, J3, and J4) need to be executed by four workers (Wl,
W2, W3, and W4), one job per worker. The table below shows the cost of assigning a certain worker
to a certain job.
Jl J2 J3 J4
Wl 82 83 69 92
W2 77 37 49 92
W3 11 69 5 86
W4 8 9 98 23
Minimize the total cost of the assignment using the Hungarian method.
(17)
End of paper
Total marks: 100
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