ZAMBIAN OPEN UNIVERSITY

SCHOOL OF BUSINESS STUDIES

JUNE 2024 EXAMINATION

BBA 313- OPERATIONS RESEARCH

INSTRUCTIONS

a) The answer paper should be typed using Microsoft Word

b) PDF, Picture, scanned and handwritten answer paper will not be marked
c) The answer paper should be uploaded into Moodle LMS only.
d) The answer paper will automatically be tested for plagiarism when
uploaded. Answer papers with a high plagiarism score will not be marked.
e) Answer papers submitted using email or in person will not be marked
f) There are two sections in this paper. Section A is COMPULSORY and answer
any FOUR (4) questions in Section B.
YOU ARE REQUIRED TO UPLOAD THE ANSWER PAPER BY MIDNIGHT (00.00HRS)
THURSDAY 27 TH JUNE, 2024.
THE MOODLE LMS WILL AUTOMATICLALY CLOSE AFTER NIGHT AND STUDENTS
WILL NOT BE ABLE TO UPLOAD ANSWER PAPERS AFTER THE INDICATED CUT OFF
TIME
________________________________________________________________

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SECTION A (COMPULSORY)

QUESTION ONE

(a) Mr. Sakala is interested in developing and marketing a new drug. The cost of extensive
research to develop the drug would be K100,000. The manager of research programme said
that there is 60% chance that the drug will be developed successfully. The market potential
is assessed as follows with present value of profit:
Market conditions Probability Present value of profits (K)
Large market potential 0.1 500,000
Moderate market potential 0.6 220,000
Low market potential 0.3 80,000

The present value figures do not include the cost of research. While Mr. Sakala was
considering this proposal, another similar proposal came up which also required the
investment of K100,000. The present value of profit for the second proposal was K120,000.
The return on the investment in the second proposal is almost certain.
i. Draw a decision tree for Mr. Sakala indicating all choices and events (4 marks)
ii. What decision Mr. Sakala should take regarding the investment of K100,000?
(2 marks)
iii. If Mr. Sakala is a risk averter, should he change the decision given by you?
(2 marks)
(b) A tax consulting firm has 5 counters in its office to receive people who have problems
concerning their income, wealth and sales taxes. On the average 75 persons arrive in an 8
–hour day. Each tax adviser spends 25 minutes on an average on an arrival. If the arrivals
are Poissonly distributed and service times are according to exponential distribution, find
i. The probability of there being no customer in the system (4 Marks)
ii. The average number of customers in the system (3 Marks)
iii. Average number of customers waiting to be served (2 Marks)
iv. Average time a customer spends in the system (2 Marks)
v. Average waiting time for a customer (1 Marks)
[TOTAL: 20 MARKS]
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SECTION B

(CHOOSE FOUR (4) QUESTIONS)

QUESTION TWO

(a) Use the simplex method technique to

Maximize Ζ = 3x1 + 5 x2 + 4 x3

Subject to 2 x1 + 3 x2 ≤ 8

2 x2 + 5 x3 ≤ 10

3 x1 + 2 x2 + 4 x3 ≤ 15

x1 , x2 , x3 ≥ 0 (9 Marks)

(b) Two grades of paper X and Y are produced on a paper machine. Because of raw materials
restrictions, not more than 400 tonnes of grade X and 300 tonnes of grade Y can be
produced in a week. There are 160 production hours in a week. It requires 0.2 and 0.4 hour
to produce one tonne of products X and Y respectively with corresponding profits of K20
and K50 per tonne
i. Formulate the problem as a linear programming problem (4 Marks)
ii. Find the optimum product mix by using the simplex method (7 Marks)
[TOTAL: 20 MARKS]

QUESTION THREE

(a) Write the dual of the following linear programming problem:

Maximize Ζ = 3 x1 + 5 x2 + 4 x3 ,

Subject to 2 x1 + 3 x2 ≤ 8,

2 x2 + 5 x3 ≤ 10

3 x1 + 2 x2 + 4 x3 ≤ 15

x1 , x2 , x3 ≥ 0 (5 Marks)
(b) Find the optimal solution to the given problem in (a) above, using the simplex method.
(9 Marks)

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 (c) From the final optimal table in (b) above, find the solution to the dual problem. (3 Marks)
(d) Give the economic interpretation to the solution in (c) above. (3 Marks)

[TOTAL: 20 MARKS]

QUESTION FOUR

(a) Use the big – M simplex method to

Maximize Ζ = 3x1 + 2 x2 + 3 x3

Subject to x1 − x2 + x3 ≥ 4

x1 + x2 + 2 x3 ≤ 8

x1 + x3 ≥ 2

x1 , x2 , x3 ≥ 0 (10 Marks)

(b) The research department of Kwasa Kwasa Ltd has recommended to the marketing
department to launch a shampoo of three different types. The marketing manager has to
decide one of the types of shampoo to be launched under the following estimated pay offs
for various levels of sales:

Estimated levels of sales

15, 000 10,000 5, 000

1. Egg shampoo 3000 3000 1000

2. Clinic shampoo 1000 3000 1000
3. Deluxe shampoo -1000 250 1000

What will be the marketing manager’s decision if

i. Maximum criterion is applied? (2 Marks)
ii. Minimax criterion is applied? (2 Marks)
iii. Maximax criterion is applied? (2 Marks)

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 iv. Laplace criterion is used? (2 Marks)
v. Hurwicz criterion is used? Use α = 0.75 (2 Marks)
[TOTAL: 20 MARKS]

QUESTION FIVE

(a) Use the two – phase simplex method

Minimize Ζ = 3 x1 + 2 x2 + x3

Subject to x1 + 4 x2 + 3 x3 ≥ 50

2 x1 + x2 + x3 ≥ 30

−3 x1 − 2 x2 − x3 ≤ −40

x1 , x2 , x3 ≥ 0 (9 Marks)
(b) Five men are available to do five different jobs. From past records the time in hours that
each man takes for each job is known and is given below

Machine

Men I II III IV V

A 3 10 3 8 2

B 7 9 8 7 2

C 5 7 6 4 2

D 5 3 8 4 2

E 6 4 10 4 2

Find the assignment of men to job that will minimize the total time taken. (7 Marks)
(c) Explain the significance of the following variables with reference to linear programming
problem:
i. Entering variable (2 Marks)
ii. Leaving variable (2 Marks)

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 [TOTAL: 20 MARKS]

QUESTION SIX

(a) Auto vehicles arrive at a petrol pump, having one petrol unit, in Poisson fashion with an
average of 10 units per hour. The service time is distributed exponentially with a mean of
3 minutes. Find the following:
i. Average number of units in the system (2 Marks)
ii. Average waiting time (2 Marks)
iii. Average length of queue (2 Marks)
iv. Probability that a customer arriving at the pump will have to wait (2 Marks)
v. The utilization factor for the pump unit (1 Mark)
vi. Probability that the number of customers in the system is 2. (2 Marks)
(b) Use the simplex method to
Maximize Ζ = 2 x1 + x2 − 3 x3 + 5 x4

Subject to x1 + 7 x2 + 3 x3 + 7 x4 ≤ 46

3 x1 − x2 + x3 + 2 x4 ≤ 8

2 x1 + 3 x2 − x3 + x4 ≤ 10

x1 , x2 , x3 , x4 ≥ 0 (9 Marks)

[TOTAL: 20 MARKS]

QUESTION SEVEN

(a) A firm makes products A and B and has a total production capacity of 9 tonnes per day, A
and B requiring the same production capacity. The firm has a permanent contract to supply
at least 2 tonnes of A and 3 tonnes of B per day to another company. Each tonne of A
requires 20 machine- hour production time and each tonne of B requires 50 machine-hour
production time, the daily maximum possible production time available is 360 hours. Profit
per unit of product A is K80 and that of B is K120. Formulate a linear programming model
for the problem. (4 marks)

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(b) Duration and requirement of workforce for each activity is tabulated below for a network

Activity Duration Number of men

1–2 3 5
2–4 2 3
2–3 3 7
3–4 0 0
3–5 3 2
4–5 7 2
3–6 2 1
5–6 6 6
4–6 5 5

(i) Draw the network and comment on the scheduling of activities to smoothen the
development of the workforce (4 marks)
(ii) Indicate the maximum crew size in each case (5 marks)
(iii) Re-schedule the activities for smooth development (4 marks)
(iv) Find the project schedule if only 4 men are available (3 marks)

[TOTAL: 20 MARKS]

END OF EXAMINATION