bsc (hons) business information systems bsc (hons ... · ... direct cost (rs ’000) ... compute...

BSc (Hons) Business Information Systems

BSc (Hons) Software Engineering

Cohorts BIS / 07 / Full Time & BSE / 07 / Full Time

Examinations for 2009 - 2010 / Semester 1

MODULE: Operations Research MODULE CODE: MATH 3113 Duration: 2 Hours 30 Minutes

Instructions to Candidates:

1. Answer ALL the questions.

2. Questions may be answered in any order but your answers must show

the question number clearly.

3. Always start a new question on a fresh page.

4. All questions carry equal marks.

5. Total marks 100.

This question paper contains 4 questions and 8 pages.

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ANSWER ALL QUESTIONS

Question 1: (25 Marks)

(a) Suppose the entire kola industry produces only two kolas. Given that a person

last purchased kola A, there is 85% chance that the person next purchase will

be kola A. Given that a person last purchased kola B, there is an 75% chance

that the person next purchase will be kola B. If a person is currently a kola

B purchaser, what is the probability that the person will purchase kola A two

purchases from now?

(5 marks)

(b) The XYZ company considers an account as uncollectible if the account is more

than three months overdue. Thus, at the beginning of each month, each account

may be categorized into one of the following states:

State 1 New account

State 2 Payment on account is one month overdue

State 3 Payment on account is two months overdue

State 4 Payment on account is three months overdue

State 5 Account has been paid

State 6 Account is written off as bad debt

Based on past data, it has been found that the following Markov chain describes

how the status of an account changes from one month to the next month:

New 1 month 2 months 3 months Paid Bad debt

New 0 1/2 0 0 1/2 0

1 month 0 0 2/5 0 3/5 0

2 months 0 0 0 3/5 2/5 0

3 months 0 0 0 0 4/5 1/5

Paid 0 0 0 0 1 0

Bad debt 0 0 0 0 0 1

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(i) Classify the states of the Markov chain.

(2 marks)

(ii) Write down a matrix Q which represents transitions between transient

states.

(1 mark)

(iii) Write down a matrix R which represents transitions from transient states

to absorbing states.

(1 mark)

(iv) Use the Gauss-Jordan method to find (I −Q)−1 and hence compute (I −Q)−1R.

(12 marks)

(v) Write down the probability that a new account will eventually be col-

lected.

(1 mark)

(vi) Write down the probability that a one-month overdue account will even-

tually become a bad debt.

(1 mark)

(vii) If the company’s sales average Rs 2 500 000 per month, how much money

per year will be uncollected?

(2 marks)

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The demand for the product of the XYZ company during each of the next four months

is given in Table 1.

Month Demand (number of units)

1 1

2 3

3 4

4 2

Table 1: Demand for the next four months.

At the beginning of each month, the company must determine how many units should

be produced during the current month. During a month in which any units are

produced, a setup cost of Rs 30 is incurred. In addition, there is a variable cost of

Rs 10 for every unit produced. At the end of each month, a holding cost of Rs 5

per unit on hand is incurred. Capacity limitations allow a maximum of 5 units to

be produced during each month. The size of the company’s warehouse restricts the

ending inventory for each month to at most 4 units. Assume that zero units are on

hand at the beginning of the first month and define

• ft(i) to be the minimum cost of meeting demands for months t, t + 1, . . . , 4 if

i units are on hand at the beginning of month t;

• c(x) to be the cost of producing x units during a month;

• xt(i) to be a production level during month t that minimizes the total cost

during months t, t + 1, . . . , 4 if i units are on hand at the beginning of month

t.

(a) State the value of c(0) and for x > 0, write down an expression for c(x).

(2 marks)

(b) For i = 0, 1, 2, 3 and 4, compute the values of f4(i) and x4(i).

(5 marks)

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(c) For each possible production level x during month 3, find an expression for the

total cost during months 3 and 4. Hence, deduce that

f3(i) = minx∈I3

{5 (i + x− 2) + c(x) + f4(i + x− 2)} (1)

and define the set I3.

(5 marks)

(d) Find the values of f3(0) and x3(0).

(3 marks)

(e) Find expressions similar to (1) for f2(i) and f1(i).

(4 marks)

(f) Use the information provided in Table 2 to compute f1(0) and x1(0).

i 0 1 2 3 4

f2(i) 160 150 140 120 105

x2(i) 5 4 3 0 0

Table 2: Values of f2(i) and x2(i).

(3 marks)

(g) Hence, use the additional information given in Table 3 to determine a production

schedule that will meet all demands on time and will minimize the sum of

production and holding costs during the four months. (You must state the total

cost associated with the optimal production schedule.)

i 1 2 3 4

f3(i) 100 70 65 60

x3(i) 5 0 0 0

Table 3: Values of f3(i) and x3(i).

(3 marks)

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The following table gives the activities in a construction project and other relevant

information:

Activity Immediate Time (months) Direct Cost (Rs ’000)

Predecessor Normal Crash Normal Crash

A − 4 3 60 90

B − 6 4 150 250

C − 2 1 38 60

D A 5 3 150 250

E C 2 2 100 100

F A 7 5 115 175

G D, B, E 4 2 100 240

Indirect costs vary as follows:

Months 15 14 13 12 11 10 9 8 7 6

Cost (Rs ’000) 600 500 400 250 175 100 75 50 35 25

(a) Draw a network diagram for this project.

(3 marks)

(b) Determine the critical path of the project.

(6 marks)

(c) Determine the normal project completion time and the associated total cost.

(2 marks)

(d) Compute the cost slope for each activity in the project.

(3 marks)

(e) Determine the project duration which will result in minimum total project cost.

(11 marks)

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(a) In the figure below is shown the road system (without the curves) in a partic-

ular recreational park. The location O is the entrance into the park and the

other letters designate the locations of ranger stations. The numbers give the

distances of the winding roads in kilometres.

O

A

B D

C E

T

2

4

5

7

4

2

1

3

4

1 7

5

Telephone lines must be installed under the roads to establish telephone com-

munication among all the stations (including the park entrance). Because the

installation is both expensive and disruptive to the natural environment, lines

will be installed under just enough roads to provide some connection between

every pair of stations.

(i) Determine where the lines should be laid so that a minimum total number

of kilometres of line is installed.

(4 marks)

(ii) Hence, state the minimum total number of kilometres of line installed.

(1 mark)

(b) The following payoff table shows profit for a decision analysis problem with two

decision alternatives and three states of nature.

State of Nature

Decision Alternative S1 S2 S3

D1 250 100 25

D2 100 100 75

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(i) Construct a decision tree for this problem.

(3 marks)

(ii) If the decision maker knows nothing about the probabilities of the three

states of nature, what is the recommended decision using the optimistic,

pessimistic and minimax regret approaches?

(6 marks)

(iii) Suppose that the decision maker has obtained the probability assessments:

P (S1) = 0.65, P (S2) = 0.15, and P (S3) = 0.20.

1. Use the expected value approach to determine the optimal decision.

(4 marks)

2. What is the optimal decision strategy if perfect information were avail-

able?

(3 marks)

3. What is the expected value for the decision strategy developed in part

(2)?

(2 marks)

4. What is the expected value of perfect information?

(2 marks)

***END OF PAPER***

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