CPA
Foundation Leval
Quantitative Analysis September Pilot 2015
Suggested solutions

Assumptions of Markov Analysis:

• The system being analyzed can be divided into distinct states.
• The process moves from one state to another in a series of discrete steps.
• The probability of transitioning from one state to another remains constant over time.
• The Markov property holds, meaning the future state depends only on the current state and not on the sequence of events that preceded it.
• Transitions between states are mutually exclusive.
• There is a well-defined starting point and ending point in the system.
• The Markov chain has a finite number of states.

i) Converting the above data into a matrix of transition probabilities

(i) The level of output that maximizes profit

i) The average number of patrons waiting in line to purchase the tickets

(i) The least squares regression function relating the hours worked and earnings. Interpret your results.

(i) A linear programming model of the production problem facing Lanex company.

To determine the daily production volume for each of the two dyes that will maximize contribution.

Contribution per unit of

Light = 13 - 9 = Sh.4 per unit

Dark = 16 - 10 = Sh.6 per unit

The objective function = Zmax = 4𝑥 + 6y

Subject to constraints:

Skilled labour hour constraints;

6𝑥 / 60 + (12 / 60)y ≤ 400

Therefore, 0.1𝑥 + 0.2 y ≤ 400

Blending chemical gramme constraint;

0.05𝑥 + 0.02y ≤ 100

Processing capacity constraint;

𝑥 + y ≤ 3000

Production capacity constraint;

𝑥 + y ≥ 2000

(ii) Using the graphical approach, determine the optimum daily production plan and consequent contribution.

Non negativity constraint
X ≥ 0
Y ≥ 0
Where;

𝑥 = Number of units of litres of product "light" to be manufactured
Y = Number of units or litres of product "dark" to be manufactured

Hence the optimal dail production plan is

X = 1500 litres of light dye
Y = 1250 litres of dark dye

Optimal contribution:

Zmax = 4(1,500) + 6(1,250) = 6,000 + 7,500

Sh 13,500 per day

(i) The best course of action on the basis of prior information

(H) because it results in the highest profit. -

(ii) The expected value of perfect information (EVPI)

The Expected Value of Perfect Information (EVPI) represents the maximum amount a decision-maker should be willing to pay for perfect information before making a decision under uncertainty. It is the difference between the expected value of making a decision without perfect information and the expected value of making a decision with perfect information.

EVPI = EMV_{with perfect information} − EMV_{without perfect information}

Where:

EMV stands for Expected Monetary Value,

with perfect information refers to the scenario where perfect information is available,

without perfect information refers to the scenario where the decision is made without perfect information.

EMV_{with perfect information}

0.30(6000,000) + 0.3(0) = Shs 1,800,000

Payoff matrix table

	State of demand
	High(H)	Moderate(M)	Low(L)
Decision alternative Open new plant Do nothing	0.3 6,000,000 0	0.3 1,500,000 0	0.4 -2,500,000 0

EMV_{without perfect information}

0.30(6000,000) + 0.30(1,500,000) + 0.40(-2500,000)

1800,000 + 450000 - 1,000,000 = Sh. 1,250,000

EVPI = 1,800,000 - 1,250,000 = 550,000

(iii) Advise Brightshine Limited whether the market research should be conducted. Show your workings using a decision tree.

Revision of prior probabilities by use of contingent table

		State of nature
		High(H)	Moderate(M)	Low(L)	Total
		0.30	0.30	0.40
Market Survay Report	G (0.7 | 0.6 | 0.2) P (0.3 | 0.4 | 0.8)	(0.7 x 0.3) = 0.21 (0.3 x 0.3) = 0.9	(0.6 x 0.3) = 0.18 (0.4 x 0.3) = 0.12	(0.2 x 0.4) = 0.08 (0.8 x 0.4) = 0.32	0.47 0.53
Total		0.3	0.3	0.4	1.0

Therefore,

P(H / G) = 0.21 / 0.47 = 0.45

P(M / G) = 0.18 / 0.47 = 0.38

P(L / G) = 0.08 / 0.47 = 0.17

Total = 1.00

P(H / P) = 0.09 / 0.53 = 0.17

P(M / P) = 0.12 / 0.53 = 0.23

P(L / P = 0.32 / 0.53 = 0.60

Total = 1.00

Therefore,

Expected value with perfect information EV_wPI = Sh 1,265,600

Expected value of sample information EVI = 1265,600 + 6000 - 1250,000 = Sh. 75,600

Hence, Brightshine Itd should conduct the market research since the cost is less than the amount that it would incur if it were to conduct a market study

(i) Distinguishing Zero-sum game and Non-zero sum game:

(i) A network diagram for the project.

(ii) The critical path and the expected project duration.

A - E - I - L

Duration → 50 days

(iii) The time schedules for activities F, G and H.

Earliest start (ES) = EF of the one before it

Earliest finish (EF) = ES + Duration of the activity

Latest Finish LF = is the smallest of the start times for the next activity.

Latest Start (LS) = LF − Duration of the activity

Float = LS - ES or LF - EF

Activity F: ES = 12, EF = 29, LS = 19, LF = 36, Float = 7

Activity G: ES = 20, EF = 32, LS = 22, LF = 34, Float = 2

Activity H: ES = 20, EF = 34, LS = 20, LF = 34, Float = 0

i) The probability that the machine is effective

(i) The centred moving average trend values.

Components of a Time Series:

• Trend: The long-term movement or direction in a time series. It represents the underlying pattern that may show an increase, decrease, or remain stable over time.
• Seasonality: The regular and predictable fluctuations in a time series that occur at specific intervals, often related to calendar time (e.g., daily, monthly, or annually).
• Cycle: The repeating up and down movements or fluctuations in a time series that are not of fixed period, typically representing economic or business cycles.
• Irregularity (or Residual): The random fluctuations or noise in a time series that cannot be attributed to the trend, seasonality, or cycle. It represents the unpredictable and unsystematic components.