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Management Mathematics 1 Question Paper
Management Mathematics 1
Course:Bachelor Of Commerce
Institution: Strathmore University question papers
Exam Year:2011
STRATHMORE UNIVERSITY
FACULTY OF COMMERCE
Bachelor of Commerce
END OF SEMESTER EXAMINATION
MAT 2203: MANAGEMENT MATHEMATICS 1
DATE: 5th March 2009 TIME: 2 Hours
INSTRUCTIONS:
THE QUESTION PAPER CONSISTS OF FIVE QUESTIONS.
ANSWER QUESTION ONE AND ANY OTHER TWO QUESTIONS.
QUESTION ONE (30 marks)
a) (i) A firm manufactures a medical product containing three ingredients X, Y and
Z. Each unit processed must contain at least 100g of X, 30g of Y and 75g of Z.
The product is made by mixing the inputs A and B which come in containers costing
respectively KSh. 40 and KSh. 60. The following information is also provided: 1
container of A contains 50g of X, 20g ox Y and 15g of Z, and 1 container of B
contains 20g of X, 10g of Y and 50g of Z.
Required:
Formulate a linear programming problem (4 marks)
(ii) A Markov chain has three states, A, B and C. The probability of going from state
A to state B in one trial is 0.1, the probability of going from state A to state C in one
trail is 0.3, the probability of going from state B to state A in one trial is 0.2, the
probability of going from state B to state C in one trial is 0.5, and the probability of
going from state C to C in one trial is 1.
Required:
Draw the transition diagram and write down the transition matrix.
(3 marks)
b) (i) Define the terms (1) feasible region and (2) a corner or extreme point of a
feasible region.
(3 marks)
(ii) Solve the following system of linear inequalities graphically and find the corner
points and optimal solution:
Maximise Z = 6x + 8y
Subject to 4x + 3y = 60
4x + 4y = 40
x = 0, y = 0.
(8 marks)
2
c) (i) Consider
0.4 0.6
0.3 0.7
P and S = (0.2 0.8) as probability matrices.
Find (i) P2, P8, P32, (ii) the value P in the long run (iii) the value of S in the long run
(8 marks)
d) Form the dual of the following primal problem:
Minimise Z = 20x1 + 50x2
Subject to 5x1 + 20x2 = 60
10 x1 + 8x2 = 40
4x1 + 10x2 =30
x1, x2 = 0
(4 marks)
QUESTION TWO (20 marks)
a) Paul Nganga, a private financial consultant, has been informed of a land
investment opportunity that is suitable for one of his clients. The land investment
opportunity allows investors to purchase shares in one or both of two developments.
No investor is allowed to purchase more than KS. 6000 worth of shares in
development 1 and not more than KSh. 7000 worth of shares in development 2. It is
expected that a shilling invested in development 1 will return a profit of KSh. 0.25
per year, and a shilling invested in less risky development 2 will return a profit of
KSh. 0.10
Upon consultation with his client, Paul was allowed to invest up to KSh. 10,000
(total) in the development, with provision that at least 30% of the total amount
investment be placed in development 2. Find how much Paul is required to invest in
each development in order to maximize the client’s annual profit. Use graphical
(10 marks)
b) An economy is based on three sectors, coffee, tea and transportation. Production of
a shilling’s worth of coffee requires an input of KSh. 0.20 from coffee sector, and
KSh. 0.40 from the transportation sector. Production of a shilling’s worth of tea
requires an input of KSh. 0.10 from the coffee sector and KSh. 0.20 from the
transport sector. Production of a shilling’s worth of transportation requires an input of
KSh. 0.40 from the coffee sector, KSh 0.20 from tea sector, and KSh. 0.20 from the
transportation sector.
Find (i) the technology matrix M
(ii) (I – M)-1
(iii) Find the output from each sector that is required to satisfy a final demand
of KSh.40 billion for coffee, KSh.30 billion for tea and KSh.60 billion for
transportation.
(10 marks)
3
QUESTION THREE (20 marks)
a) The manager of A dairy Company is trying to determine the correct blend of two types
of feed. Both feeds contain various percentages of four essential ingredients. The table
below shows the quantities involved:
Ingredients Feed 1(%) Feed 2 (%) Minimum
Requirements 9KG0
1 40 20 4
2 10 30 2
3 20 40 3
4 30 10 6
Cost per unit 50 40
.
(i) Formulate a LP problem
(ii) Solve the LP problem and find the least cost blend.
(8 marks)
(b) Consider the following linear programming problem,
Maximise Z = 5x + 3y
Subject to 6x + 2y = 36
5x + 5y = 40
2x + 4y = 28
x = 0, y = 0
(i) Transform the set of constraints into standard form
(ii) Find the optimal solution by using the simplex method
(iii) Find the shadow prices
(12 marks)
QUESTION FOUR (20 marks)
a) CFCLife Insurance Company found out that on the average, over a period
of 15 years, 20% of the drivers in a particular community who were
involved in an accident one year were also involved in an accident the
following year. They also found out that only 12% of the drivers who
were not involved in an accident one year were involved in an accident
the following year and 5% of the drivers who were involved in accident
one year were not involved in an accident the following year. Use these
percentages as approximate empirical probabilities for the following:
(i) Draw a transition diagram,
(ii) Find the transition matrix P and state marix
(iii) What is the probability that a driver chosen at random from the
community will be involved in an accident after two transitions, after
three transitions?
(iv) Find the transition and state matrices for a long run
(14 marks)
4
b) Consider the following transport problem involving 3 sources (factories) and 3
destinations (depots). Develop a linear programme (LP) model for this problem
Destination (depots)
Source
(Factory)
1 2 3 Factory
capacity
A 4 6 8 400
2 3 5 2 200
3 3 9 6 600
Depot
requirements
400 450 350 1200
(6 marks)
QUESTION FIVE (20 marks)
(a) (i) Define the term network, nodes, and arcs
(ii) The figure given below is a network diagram, which illustrates the route
structure for a small commuter airline, which services five cities.
Construct an adjacency matrix and square the adjacency matrix. Explain the one-stop
service, which exists between the cities.
(6 marks)
b) Consider the following LP problem
Maximize Z = 10x + 8y
Subject to 5x + 10y = 80
4x + 4y = 60
x, y = 0
(i) Solve the LP problem by using the simplex method
(ii) What will be the effect on the solution if the right-hand side constants of
the constraints are changed from 80 and 60 to 50 and 30 respectively?
(14 marks)
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