SCHOOL OF MANAGEMENT AND COMMERCE
MAT 2203 Management Mathematics I
Date:16th July 1013
INSTRUCTION: Answer Question 1 and any other 2 Questions
Time: 2 Hours
Question 1 (30 Marks)
(a) Differentiate between the following
(i) Feasible and optimal solution
(ii) Recurrent and transient states of Markov chains
(b) Explain two differences between transportation and assignment models
(2 Marks)
(2 Marks)
(2 Marks)
CONTACTOR
(c) A project consists of four major jobs for which four contractors have submitted tenders.
The tender amounts quoted in millions Kenya shillings are given in the matrix below.
JOBS
I
II
III
IV
A
16 10 14 11
B
14 11 15 15
C
15 15 13 12
D
13 12 14 15
Find the assignment which minimizes the total cost of the project. Each contactor has to be
assigned at least one job.
(6 Marks)
(d) On January 1st this year, Company A had 60% of its local market share while the other
two companies B and C had 30% and 10% respectively of the market share. Based upon a
study by a marketing research firm, the following facts were compiled:
Company A retains 90% of its customers while gaining 5% of B’s customers and 10% of C
customers.
Company B retains 85% of its customers while gaining 5% of A’s customers and 7% of C’s
customers.
Company C retains 83% of its customers and gains 5% of A’s customers and 10% of B’s
customers.
(i)
Develop the state transition matrix
(3 Marks)
(ii)
What will each firm’s share be in two time period
(5 Marks)
Page 1 of 5
(c)The management of an oil refinery wants to decide on the optimal mix of two possible
blending processes 1 and 2. The table below gives the inputs and outputs in production
Input (units)
Output (units)
Process Crude
Crude B Gasoline X Gasoline Y
A
1
5
3
5
8
2
4
5
4
4
(e) The maximum amounts available of crudes A and B are 200 units and 150 units
respectively. At least 100 units of Gasoline X and 80 units of Y are required. The profit per
production run from process 1 and 2 are $300 and $400 respectively.
(i) Formulate the above as a linear programming problem
(5Marks)
(ii) Use a graphical method to solve the model in (e) (i) above
(5 Marks)
QUESTION 2 (20 Marks)
(a) The director of data processing for a consulting firm wants to assign four programming
tasks to four of her programmers. She has estimated the total number of days each
programmer would take if assigned each of the programs. The table below summarizes
these estimates.
Estimated days per programming
task
Programming task
Programmer
1
2
3
4
Peres
Patrick
Paul
Phoebe
14
16
18
20
13
15
14
13
17
16
20
15
14
15
17
18
If the objective is to assign one programmer per task in such a way as to minimize the total
number of days required to complete the task:
(i) Formulate the objective function and constraints
(5 marks)
(ii)Solve using the Hungarian method for the appropriate assignments
(5 Marks)
(b) A Company has 3 plants and 4 warehouses. The supply and demand in units and the
corresponding transportation costs are given in the table below
Page 2 of 5
Warehouses
I
II
III
11
13
17
16
18
14
21
24
13
200 225 275
plants
A
B
C
Demand
IV
14
10
10
250
supply
250
300
400
Required:
(i) Determine an initial basic feasible solution for the above model using the North
West corner cell method
(7 Marks)
(ii) Test for the optimality using the stepping stone method
(3 Marks)
Question 3 (20 marks)
(a) An NGO has commissioned a research to establish the level of addiction to common
drugs by street children undergoing day care rehabilitation. It was found out by their survey
that the mobility of the street children , in percent in the three classifications of their
addition states;
Severe (S), Mild(M) and Recovered (R) states were in the following
percentages.
S
S
M
R
M
R
50 30 20
10 70 20


10 40 50
If the present population in states S, M and R are in proportions of 0.7, 0.2, and 0.1
respectively determine the proportions of the children in the three states
(i)
after two years
(4 Marks)
(ii)
in the long run
(6 Marks)
(b) A technological matrix for a three-industry input-output model is given by:
0.3 0.3 0.2
B   0.1 0.2 0.3
0.2 0.1 0.4
Page 3 of 5
If the non-industry demand for the output of these industries is d1  Kshs 50 Million,
d 2  Kshs 30 Million, d 3  Kshs 60 Million, determine the equilibrium output levels for
three industries
( 10 Marks)
Question 4 (20 Marks)
(a) Suppose a certain activity has a total float of 12 weeks and the head event slack is 3
weeks. Compute the free float for this activity.
(2 Marks)
(b) Explain the difference between the term Critical path analysis and the PERT analysis for
a project network.
(3 Marks)
(c) A student entrepreneur has opened a snack bar in a University student centre. She wishes
to launch a new product to increase variety of foods she sells. The following table gives
various activities that are scheduled to be undertaken by the entrepreneur before launching a
new product in a market.
Description
Required
Predecessor
(None)
(None)
A
A
A
C
D
B, E
H
F, G, I
Duration
(Months)
5
1
2
3
2
3
4
2
1
1
(i)
Activity
A
Product design
B
Market research
C
Production analysis
D
Product model
E
Sales brochure
F
Cost analysis
G
Product testing
H
Sales training
I
Pricing
J
Project report
Draw the project network
(ii)
Determine the critical path
(8 marks)
(iii)
Hence, find the project completion time
(2 marks)
Page 4 of 5
(5 marks)
Question 5(20 Marks)
The following is a Linear Programming, (LP) problem
Maximize Z=10X1+15X2+20X3
Subject to
2X1+4X2+6X3  24
3X1+9X2+6X3  30
X1, X2 and X3  0
Required:
(i)
Develop the dual of the problem above.
(5 Marks)
(ii)
Write the standard form for the above primal Linear Programming model
(5 Marks)
(iii)
Solve for the optimum values of solution from the primal Linear Programming
problem using the simplex method.
(10 Marks)
Page 5 of 5