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MAP602S- MATHEMATICAL PROGRAMMING - JAN 2020 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BAMS
LEVEL: 6
COURSE CODE: MAP602S
COURSE NAME:
Mathematical Programming
SESSION: January 2020
DURATION: 3 hours
PAPER: Theory
MARKS: 100
SUPPLEMENTARY/SECOND OPPORTUNITY QUESTION PAPER
EXAMINERS
MR. B.E OBABUEKI
MRS. S. MWEWA
MODERATOR:
DR. A.S EEGUNJOBI
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
2. Graph papers to be supplied by Examinations Department
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Excluding this front page)
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Question 1 (8 marks)
Your school is planning to make toques and mitts to sell at the winter festival as a fundraiser.
The school’s sewing classes divide into two groups — one group can make toques, the other
group knows how to make mitts. The sewing teachers are also willing to help. Considering the
number of people available and time constraints due to classes, only 150 toques and 120 pairs
of mitts can be made each week. Enough material is delivered to the school every Monday
morning to make at least a total of 200 items per week. Because the material is being donated
by community members, each toque sold makes a profit of $2 and each pair of mitts sold
makes a profit of $5. Model this statement into a linear programme. (DO NOT SOLVE but
variables must be unambiguously declared, and constraints must be identifiably named.) (8)
Question 2 (16 marks)
Consider the following linear programming model:
Maximize P=20a+16b
Subject to
Sa+8b<40
8a+3b<24
9a+6b <36
a;b=0
2.1. Use graphical method to show that the solution of the model is a=4, b=, P=.
(It is expected that your values from the graph will be in decimal equivalence of these
values.)
(10)
2.2
Hence determine the value of the slack variable for each of the three constraints(6.)
Question 3 (10 marks)
The model in question 2 above is the dual of the primal model
Minimize P =40x+24y+36z
Subject to
5x+8y+9z220
8x+3y+6z216
x3;y3;z20
Use the solution of the dual to transit to and solve for the solution of this primal model. (10)
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Question 4 (10 marks)
Consider the following primal model:
Maximize
Subject to
z=2x,+3x,
3x, —4x, 22
& F2e, 26
x, 20; x, unrestricted in sign.
Determine the dual of this primal model.
(10)
Question 5 (19 marks)
Solve the following linear programme using ONLY the big-M method:
(19)
Maximize P=6x+9y
Subject to
8x+6y<48
4x—-6y=12
4x+3y212
x; y20
Question 6 (17 marks)
Consider the linear programme:
Maximize Q=10x+7y
Subject to
6x+5y $360 (units of sodium)
2x+y<S96 (units of iron)
x<40
(units of fibre)
y<50
(units of sugar)
x;y20
Sodium comes in packets of 20 units per packet, iron comes in packets of 12 units per packet,
fibre and sugar come in packets of 10 units per packet. Use graphical method to determine
the shadow price of a packet of sodium.
Use a scale of 2cm to 10 units on each axis.
(17)
Question 7 (20 marks)
The following table shows the cost of transporting one unit of a product from three
warehouses (W1, W2, W3 and W4) to three destinations (D1, D2, D3 and D4) as well as the
supply and demand capacities of the warehouses and the destinations. No product may be
transported from W1 to D3 and from W2 to D4.
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D1
D2
D3
D4
Supply
W1
5
9
-
4
28
W2
6
10
3
-
32
Ww3
4
2
5
7
60
Demand
48
29
40
33
6.1 Use the Least-cost method to determine the initial feasible distribution of product to
minimize total cost.
(8)
6.2 Use the North-west method to determine the initial feasible distribution of product to
minimize total cost.
(8)
6.3 Which of the two methods above is preferable and why?
(4)
END OF PAPER |
TOTAL MARLS 100
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