Decision Theory
Unit: Quantitative Analysis
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August 2025
2 Questions
Question 6b
Jumbo Ltd. has to make a decision on whether to invest in project A, B or C. The economic conditions could be
favourable, moderate or unfavourable.
The table below shows the potential payoffs in thousands of shillings (Sh.“000”) depending on the market
conditions and their associated probabilities:
| Economic conditions | |||
| Project | Favourable | Moderate | Unfavourable |
| A | 7,300 | 5,600 | 4,100 |
| B | 15,100 | 6,700 | 0 |
| C | 9,500 | 6,000 | 2,500 |
| Probability | 0.20 | 0.30 | 0.50 |
Required:
Advise the management of Jumbo Ltd. of the most optimal decision using:
(i) Maximax criterion.
(ii) Maximin criterion.
(iii) Laplace criterion.
(iv) Expected monetary value (EMV) criterion.
Question 2a
Outline FOUR applications of set-theory in business.
April 2025
2 Questions
Question 1a
Discuss THREE emerging issues in quantitative analysis that have significantly impacted decision making in
modern industries.
Question 5b
Pegra Ltd. is considering launching a new electric product. However, demand for the proposed product is uncertain
and the company can either launch the new product immediately or conduct market research before making a
decision which could return either a favourable or unfavourable outcome. If the research outcome is favourable,
Pegra Ltd. can proceed with launch. If the research outcome is unfavourable, the company has the option to abandon
the launch.
Probabilities and payoffs:
| Option 1: | Launch immediately:
|
| Option 2: | Conduct market research at a cost of Sh.50,000
|
If research outcome is favourable:
- Probability of high demand will be 80% with a projected profit of Sh.500,000
- Probability of low demand will be 20% with a projected loss of Sh.200,000
If research outcome is unfavourable:
- The company can choose to abandon the launch incurring only the Sh.50,000 research cost.
Required:
(i) Construct a decision tree based on the given probabilities and outcomes.
(ii) Compute the expected monetary value (EMV) for each option.
(iii) Recommend the best investment decision for the company.
December 2024
1 Questions
Question 4b
Faida Ltd. is in the process of reviewing the selling price for product “Excel”.
The selling prices under consideration are Sh.50, Sh.55 and Sh.60.
The following additional information is provided about the forecasted demand for product “Excel” under three
different market conditions:
| Market condition (Demand in Units) | |||
| Selling price | Good | Moderate | Bad |
| Sh.50 | 20,000 | 18,000 | 14,000 |
| Sh.55 | 18,000 | 16,500 | 12,000 |
| Sh.60 | 16,000 | 14,000 | 8,500 |
The fixed costs are estimated at Sh.300,000 and variable cost per unit is Sh.30.
Required:
(i) Construct the pay-off trade for product “Excel”.
(ii) Advise Faida Ltd. on the best selling price for product “Excel” based on maximax criterion.
(iii) Advise Faida Ltd. on the best selling price for product “Excel” based on maximin criterion.
(iv) Determine the best selling price for product “Excel” based on expected monetary value (EMV) criterion
given that probability for good, moderate and bad market criterion is 0.30, 0.50 and 0.20 respectively.
(v) Determine the expected value of perfect information.
August 2024
1 Questions
Question 5b
A company is contemplating an investment in either of the two machines; Machine A or Machine B to enhance its
production capabilities. The decision requires assessing the costs and potential returns of each machine under
varying market conditions.
The following information relates to the two machines:
Machine A
- Cost: Sh.100,000,000.
- Potential returns: Sh.250,000,000, if market demand is high and Sh.60,000,000 if market demand is low.
- Probability of high market demand is 0.70.
- Probability of low market demand is 0.30.
Machine B
- Cost: Sh.150,000,000.
- Potential returns: Sh.320,000,000 if market demand is high and Sh.80,000,000 if market demand is low.
- Probability of high market demand is 0.50.
- Probability of low market demand is 0.50.
Additionally, the company can choose to conduct a survey at a cost of Sh.20,000,000 to obtain more information
before making the investment decision. The survey can indicate either favourable or unfavourable results with
probabilities of 0.80 and 0.20 respectively, influencing the probabilities of market conditions.
If survey is favourable for both Machine A and Machine B:
- Probability of high market demand is 0.90.
- Probability of low market demand is 0.10.
If survey is unfavourable for both Machine A and Machine B:
- Probability of high market demand is 0.20.
- Probability of low market demand is 0.80.
The potential returns of Machine A and Machine B remain unchanged regardless of the survey outcome.
Required:
(i) Draw the decision tree for the above scenario.
(ii) Advise the management of the company on the best decision using the Expected Monetary Value (EMV)
approach.
(iii) Determine the Expected Monetary Value from the decision made in (b) (ii) above.
April 2024
2 Questions
Question 2c
Highlight FOUR considerations in determining whether to employ a census or a sample in collecting primary data.
Question 6b
A property developer has Sh.1,000,000 to invest in a new property. The developer has a choice of three different
properties going at the same price in different towns; one in Nairobi, another in Nakuru and the other in Nanyuki.
With the three properties, his ultimate profit depends on whether the economy is strong, weak or mixed.
He estimates that the pay-off matrix representing his monthly profits in thousands of shillings is as follows:
| States of nature | ||||
| Weak | Mixed | Strong | ||
| Nairobi | 85 | 30 | 75 | |
| Strategies | Nakuru | 45 | 45 | 110 |
| Nanyuki | 60 | 95 | 85 | |
Required:
Determine the town that the developer should invest in using:
(i) Maxi-max criterion.
(ii) Maxi-min criterion.
(iii) Mini-max regret criterion.
(iv) Suppose the investor reads in the newspaper that there is 50% chance of a weak economy, 20% chance of
a mixed economy and a 30% chance of a strong economy.
Advise on the best strategy, given the new information.
(v) Compute the expected value of perfect information.
December 2023
3 Questions
Question 4b
Consider the following data for 120 students of a College concerning the languages they are studying namely;
French, German and Russian.
- 15 students study Russian and German.
- 58 students study German or French but not Russian.
- 28 students study French only.
- 90 students study French or German.
- 20 students study French and Russian.
- 44 students study at least two of the three languages.
- 20 students do not study any of the three languages.
Required:
(i) Present the above information in a Venn diagram.
(ii) Number of students who study all the three languages.
(iii) Proportion of students who study French.
(iv) Number of students who study at least one language.
Question 2a
List SEVEN steps of the decision making process
Question 5a
Explain the terms “minimax criterion” and “maximax criterion” as used in decision analysis.
August 2023
4 Questions
Question 3a
State FOUR characteristics of the normal distribution.
Question 4b
The following pay-off matrix was developed by a company showing profits (in shillings) obtained from launching
four different products P_1, P_2, P_3 and P_4 under four different states of nature:
| State of nature | ||||
| Product | \(S_1\) | \(S_2\) | \(S_3\) | \(S_4\) |
| \(P_1\) | 5,000 | 9,000 | 7,000 | 8,000 |
| \(P_2\) | 7,000 | 4,000 | 6,000 | 12,000 |
| \(P_3\) | 10,000 | 8,000 | 9,000 | 7,000 |
| \(P_4\) | 14,000 | 5,000 | 7,000 | 6,000 |
The probabilities for \(S_1\), \(S_2\), \(S_3\) and \(S_4\) are given as 0.30, 0.40, 0.20 and 0.10 respectively.
Required:
(i) Advise on the best course of action using the Mini-Max Regret Criterion.
(ii) Advise on the best course of action using the Expected Opportunity Loss Criterion.
(iii) An expert has offered to provide perfect information at a cost of Sh.2,500.
Advise the management of the company on whether or not to acquire the perfect information.
Question 1a
Explain the meaning of the following terms as used in decision theory:
(i) Decision alternative.
(ii) State of nature.
(iii) Conditional payoff.
(iv) Opportunity cost
Question 2b
Explain the following terms as used in set theory:
(i) Disjoint set.
(ii) Complement of a set.
(iii) Union of a set
April 2023
2 Questions
Question 5a
Outline FOUR properties of a set.
Question 5c
Red Rose Limited has developed a new washing detergent. The company is considering whether or not to market test
the detergent prior to its launch. Market testing will cost Sh.50 million which may increase the likelihood of success
of the product if it passes the test. The probability of passing the test is 0.8. If the product fails the test, it is regarded
as worthless.
Previous data collected was as follows:
| Probability of | No testing | Market testing |
| High success | 0.20 | 0.40 |
| Moderate success | 0.35 | 0.40 |
| Low success | 0.45 | 0.20 |
Launching the product will cost Sh.150 million and the estimated profits are as follows:
| High success | Sh.900 million |
| Moderate success | Sh.450 million |
| Low success | Sh.225 million |
Red Rose Limited has the option of selling the product design for Sh.250 million.
Required:
(i) A decision tree showing the pay-off and the expected monetary value (EMV) of each decision.
(ii) Advise the management of the company on the best course of action and the related expected monetary
value (EMV).
December 2022
5 Questions
Question 7b
Define the following terms as used in decision making:
(i) Value of perfect information.
(ii) Regret
Question 1b
Explain the meaning of the following terms as used in set theory:
(i) Venn diagram.
(ii) Complement of a set.
(iii) Union of a set.
Question 5a
Highlight FOUR advantages of decision tree analysis as a tool for decision making
Question 5c
The management of a wall paint manufacturing company is faced with the problem of choosing one of three products
to add to its existing product line. The potential demand for each product may turn out to be good, moderate or poor
with probabilities estimated as 0.75, 0.15 and 0.10 respectively.
The estimated profit or loss under the three states of demand with respect to each product is outlined below:
| Type of demand | |||
| Product | Good | Moderate | Poor |
| Silk | 70,000 | 55,000 | -10,000 |
| Matt | 100,000 | 40,000 | -6,000 |
| Gloss | 120,000 | 50,000 | -40,000 |
Required:
(i) Advise the management on the choice of product based on the expected monetary valve (EMV) criterion.
(ii) Compute the expected opportunity loss for each decision.
Which decision would you recommend based on the expected opportunity loss?
(iii) Compute the expected value of perfect information.
Question 7a
Explain THREE decision making environments.
August 2022
3 Questions
Question 5b
A survey of 850 residents of a certain town aimed at finding out the brand of vehicles owned, produced the following
results:
- 316 residents owned Nissan vehicles.
- 160 residents owned Honda vehicles.
- 360 residents owned Toyota vehicles.
- 120 residents owned both Nissan and Toyota vehicles.
- 56 residents owned both Nissan and Honda vehicles.
- 80 residents owned both Honda and Toyota vehicles.
- 240 residents did not own any of the three brands of vehicles.
Required:
(i) Present the above information in the form of a Venn diagram.
(ii) The number of residents who own all the three brands of vehicles.
(iii) The number of residents who own just one of the three brands of vehicles.
Question 3c
The following payoff matrix shows the potential profits in millions of shillings which are expected to arise from
launching four products S1, S2, S3 and S4 for each level of demand; low, moderate or high.
| Payoff matrix | |||
| Demand | |||
| Product | Low | Moderate | High |
| \(S_1\) | 15 | 22 | 29 |
| \(S_2\) | 22 | 24 | 28 |
| \(S_3\) | 32 | 23 | 34 |
| \(S_4\) | 35 | 22 | 33 |
Required:
Assuming that the products are mutually exclusive, evaluate the optimal decision using:
(i) Maximax criterion.
(ii) Maximin criterion.
(iii) Minimax regret criterion.
Question 3a
(i) Explain the term “Markov analysis”.
(ii) Highlight four assumptions of Markov analysis.
April 2022
1 Questions
Question 7b
An investment manager in an investment fund has a choice between:
1. A diversified portfolio promising Sh.15 million with a probability of 0.7 and Sh.8 million with a
probability of 0.3.
2. A risky investment consisting of two contracts with independent outcomes one promising Sh.7 million with a probability of 0.7 and the other Sh.3.5 million with a probability of 0.3.
Required:
(i) Construct a decision tree depicting the above information using the expected monetary value (EMV) criterion.
(ii) Advise on the best decision using the EMV criterion.
September 2021
4 Questions
Question 5c
ABC Limited is a construction company. An engineer identified the following activities to complete a project and
their estimated expected time in weeks.
| Activity | Preceding Activity | Expected estimated time (in weeks) |
| A | - | 5 |
| B | - | 7 |
| C | - | 3 |
| D | A | 7 |
| E | A | 6 |
| F | B | 8 |
| G | C | 10 |
| H | E,F | 3 |
| I | E,F | 4 |
| J | D,I | 2 |
| K | G,H,J | 4 |
| L | D,I | 7 |
Required:
(i) Draw a network diagram to represent the above project.
(ii) Obtain the critical path and project duration.
Question 2a
Explain any four assumptions underlying the game theory.
Question 4c
A medium-sized company manufactures product "Ndovu". The current variable production cost per unit of product
"Ndovu" is Sh.800 while the selling price is Sh.1,150 per unit. However, the demand for product "Ndovu" is
uncertain and has the following probability distribution:
| Number of units of product "Ndovu" | Probability demanded |
| 10 | 0.36 |
| 20 | 0.42 |
| 30 | 0.22 |
Required:
Advise the company on the optimal number of units of product "Ndovu" to produce using the:
(i) Expected monetary value (EMV) criterion.
(ii) Hurwicz criterion. Assume a coefficient of optimism of 0.8.
Question 5a
Summarise four practical limitations of the "queuing model"
May 2021
6 Questions
Question 2c
An investor intends to purchase shares in one of three companies, A, B and C. The three companies have varying
degrees of sensitivity to the state of the economy. There are three states of the economy classified as weak,
moderate or strong. The investor has constructed the following pay off table for the profits under the three states
of the economy, in millions of shillings.
| State of the economy | |||
| Company | Weak | Moderate | Strong |
| A | -4.0 | 3.5 | 6.0 |
| B | -2.0 | 2.5 | 4.5 |
| C | -2.4 | 2.8 | 3.5 |
The probabilities for the three states of the economy are 0.2, 0.4 and 0.4 for weak, moderate and strong respectively.
Required:
Advise the investor on the best course of action based on the:
(i) Maxmin criterion.
(ii) Maxmax criterion.
(iii) Minimax regret criterion.
(iv) Expected value of perfect information.
Question 4b
The following is a pay-off matrix of a zero sum two person game:
| Player B strategy | ||||||
| I | II | III | IV | V | ||
| I | -2 | 0 | 0 | 5 | 3 | |
| Player A strategy | II | 4 | 2 | 1 | 3 | 2 |
| III | -4 | -3 | 0 | -2 | 6 | |
| IV | 5 | 3 | -4 | 2 | -6 | |
Required:
The optimal plan for both players.
Question 3a
A baker makes and sells cakes to students through their cafeteria system. The distribution of cakes produced and
cakes sold for the last 250 weeks is as follows:
| Number of weeks | ||
| Number of cakes | Cakes Produced | Cakes Sold |
| 150 | 20 | 35 |
| 250 | 50 | 50 |
| 350 | 80 | 80 |
| 450 | 80 | 65 |
| 500 | 20 | 20 |
Each cake costs Sh.80 to make and is sold for Sh.120 if sold during the week of production, otherwise it is sold
during the second week at Sh.60. If not sold during the second week, the cake's value drops to zero and the baker
suffers the total loss of production. Weekly demand is satisfied from the week's production and any demand
remaining unsatisfied is satisfied from the stock of the previous week. A stock out costs the baker Sh.20 per cake.
The following random numbers are applicable:
| Cakes produced | 33 | 86 | 50 | 41 | 31 | 78 | 30 | 22 | 26 | 88 |
| Cakes sold | 79 | 03 | 40 | 13 | 58 | 61 | 72 | 49 | 82 | 86 |
Required:
Simulate the baker's average weekly profit over an 8-week period.
Question 4a
Enumerate eight applications of quantitative analysis in business.
Question 5b
X Ltd. is considering undertaking a project which fequires the following resources:
| Activity | Preceding | Duration in days | Total cost | Number of | |
| activity | Normal | Crash time | normal Sh. | persons per day | |
| A | - | 7 | 5 | 7,500 | 5 |
| B | - | 6 | 3 | 6,000 | 4 |
| C | - | 2 | 2 | 2,500 | 6 |
| D | A | 5 | 4 | 6,000 | 5 |
| E | B | 5 | 4 | 7,000 | 5 |
| F | E | 6 | 2 | 8,000 | 6 |
| G | E | 7 | 6 | 6,000 | 4 |
| H | C | 6 | 5 | 7,200 | 6 |
| I | H | 8 | 5 | 9,800 | 9 |
| J | D | 4 | 4 | 3,500 | 3 |
| K | J | 6 | 5 | 3,600 | 2 |
| L | F | 3 | 2 | 7,000 | 12 |
| M | G,I | 8 | 4 | 9,200 | 6 |
| N | K,L,M | 4 | 2 | 7,700 | 15 |
The cost of crashing an activity per day is Sh.1,000.
Required:
Using network analysis, determine:
(i) The project's normal duration, normal cost and critical path.
(ii) The minimum time in which the project could be completed and the cost of the project.
Question 4c
A beauty therapist has observed that the mean arrival rate in her beauty parlour is 6 customers per hour and the mean service rate is 8 customers per hour. The beauty parlour operates a 12 hour day.
A more efficient machine for use by the therapist is available for purchase. If the machine is purchased by the therapist, it would increase the average service rate at the parlour to 12 customers per hour. The cost of each hour lost due to a customer waiting for service is Sh.875.
Required:
(i) The average waiting cost per day.
(ii) Evaluate the effect of purchasing the more efficient machine on the average daily waiting cost.
November 2020
4 Questions
Question 5b
In the context of critical path analysis (CPA) method:
(i). Discuss two strengths and two weaknesses of CPA method.
(ii). Explain three practical applications of CPA method.
Question 3a
Dolce Ltd. is in the process of launching a new product into the market. Three variables are uncertain, selling price, variable cost and sales volume.
The following information is provided
| Selling price (Sh) 600 700 800 Variable cost (Sh.) 300 400 500 Sales volume (units) 40,000 50,000 60,000 | Probability 0.30 0.50 0.20 Probability 0.40 0.50 0.10 Probability 0.30 0.50 0.20 |
The following random numbers have been provided:
44, 84, 82, 50, 85, 40, 96, 88, 16, 16, 97, 92, 44, 82, 39, 33, 83, 42, 16, 07, 77 , 66, 50, 20, 50, 95, 83, 39, 58, 44, 77, 11, 08, 38, 89, 45, 09, 99, 81, 97, 50, 83,
Required:
The average contribution of Dolce Ltd using Monte Carlo simulation with 10 simulations.
Question 2a
A baker must decide whether to bake brown bread or white bread for a new market. Demand at the market can either be small or large with probability estimated to be 0.3 and 0.7 for brown bread and white bread respectively.
Additional information:
1. If brown bread is baked and demand proves to be high, the baker may choose not to expand (pay off = Sh.350,000) or to expand (pay off = Sh.420,000).
2. If brown bread is baked and demand is low, there is no reason to expand and the payoff is Sh.310,000.
3. If white bread is baked and demand proves to be low, the choice is to do nothing (Sh.90,000) or to stimulate demand through local advertising. The response to advertising may be either modest or sizeable, with their probabilities estimated to be 0.4 and 0.6 respectively. If it is modest, the pay off is estimated to be Sh.50.000 the pay off grows to Sh.340,000 if the response is sizeable.
4. If white bread is baked and the demand turns out to be high, the payoff is Sh. 1,400,000.
Required:
(i). A decision tree showing the payoff and expected monetary value of cach alternative decision.
(ii). Advise the management of the bakery on the best product to introduce into the market
Question 1b
A game between two players, A and B lias the following pay off matrix:
Required:
Required:
(i). The optimum strategy for each player.
(ii). The saddle point
(iii). The value of the game.
November 2019
7 Questions
Question 5a
A businessman has three alternatives open to him, each of which can be followed by any of the five payoff
conditional possible events (in millions of shillings) as given below.
| Pay off conditional on events | |||||
| Alternative | E1 | E2 | E3 | E4 | E5 |
| A | 6 | 2 | -2 | -12 | 4 |
| B | -6 | -3 | 10 | 16 | 0 |
| C | 12 | 8 | 4 | 0 | 6 |
Required:
Advise the businessman on the best alternative under:
(i) Maximin criterion.
(ii) Maximax criterion.
(iii) The Hurwitz criterion, assuming a degree of optimism of 0.6.
(iv) Laplace criterion.
Question 4d
Simulation models have various applications in business.
Required:
Discuss how simulation models can be applied in:
(i) Predicting business outcome.
(ii) Managing business risks.
Question 4c
Kikwetu Company Ltd. is the sole producer of 3 cosmetic products; Meta, Nzuri and Safi which currently have a
market share of 40%, 40% and 20% respectively. Each week, some brand switching takes place. Of those who
bought Meta the previous week, 60% buy it again while 20% switch to Nzuri and 20% to Safi. Of those who bought
Nzuri the previous week, 50% buy it again while 40% switch to Meta and 10% to Safi. Of those who bought Safi,
80% remain loyal while 10% switch to Meta and 10% to Nzuri.
Required:
(i) Construct a probability transition matrix of the switching probabilities.
(ii) Construct a vector to represent the initial market share in percentages.
(iii) Calculate a new market share a week after the current market share.
Question 4a
(i) Explain the meaning of a "transition matrix'
(ii) Outline two features of a transition matrix.
(ii) Outline two features of a transition matrix.
Question 2d
A bank teller can open new accounts at an average rate of 3 accounts per hour. Customers requiring to open an account
arrive at an average rate of 2 customers per hour.
The management of the bank has established a single channel single phase queuing system.
Required:
(i) The average number of customers in the system.
(ii) The average time spent by a customer in the system.
(iii) The average number of customers in the queue.
(iv) The utilisation factor of the service utility.
Question 2c
Explain two advantages and two disadvantages of decision trees as used in decision theory.
Question 1a
Business analytics is today emerging as a critical component of driving and sustaining business growth, particularly in
the face of rising competition and other market dynamics.
Required:
In the context of the above statement, describe what "business analytics" entails.
May 2019
5 Questions
Question 3a
Explain the following terms as used in network planning and analysis
(i) Free float.
(ii) Total float.
(iii) Project crashing.
Question 2c
A manufacturing company intends to introduce a new product into the market. Three products have been proposed
namely; P₁, P₂ and Pt. The company can only introduce one of the three products. The following are the estimates of
the probabilities and annual profits of the three products at three given states of demand, namely; high, moderate and
low.
| Annual profit (Sh."000") | ||||
| State of demand | Probability | P1 | P2 | P3 |
| High | 0.35 | 15,000 | 34,000 | 22,000 |
| Moderate | 0.40 | 25,000 | 30,000 | 15,000 |
| Low | 0.25 | (5,000) | (3,000) | 8,000 |
Required:
(i) A decision tree showing the payoff and expected monetary value of each alternative action.
(ii) Advise the management of the company on the best product to introduce into the market.
Question 3c
Two competing companies, A and B, that deal in the sale of computers, have an equal share of the market. Both
companies intend to increase their market share through adoption of three different media of advertisement. namely:
newspaper, radio and television. The payoff table for the two companies, showing the gain or loss of customers from
adoption of the different media of advertisement is as shown below:
| Company A | Company B | ||
| Newspaper | Radio | Television | |
| Newspaper | 40 | 50 | -17 |
| Radio | 10 | 25 | -10 |
| Television | 100 | 30 | 60 |
Required:
(i) The optimal strategies for companies A and B.
(ii) The value of the game.
Question 5a
Outline four merits of using the project evaluation and review technique (PERT) to plan and analyse a project in an
organisation.
Question 5b
A food processing company intends to install a computerised order processing system. The activities to be carried out
during the installation of the system and their time estimates are gíven below:
| Time estimates (days) | |||
| Activity | Optimistic time | Most likely time | Pessimistic time |
| A | 7 | 17 | 27 |
| B | 5 | 11 | 23 |
| C | 3 | 8 | 19 |
| D | 23 | 31 | 45 |
| E | 9 | 21 | 39 |
| F | 9 | 11 | 25 |
| G | 2 | 5 | 14 |
| H | 9 | 10 | 17 |
The above time estimates were analysed using a computer and the results of the analysis were as follows:
| Activity | Earliest start time (days) | Latest start time (days) | Earliest finish time (days) | Latest finish time (days) |
| A | 0 | 0 | 17 | 17 |
| B | 17 | 17 | 29 | 29 |
| C | 19 | 43 | 38 | 52 |
| D | 19 | 29 | 61 | 61 |
| E | 38 | 52 | 60 | 74 |
| F | 61 | 61 | 74 | 74 |
| G | 61 | 79 | 67 | 85 |
| H | 74 | 74 | 85 | 85 |
Required:
(i) The expected completion time and variance of each of the activities.
(ii) The total float of each activity.
(iii) The expected completion time and variance of the project.
(iv) The 95% confidence interval for the project's completion time.
November 2018
3 Questions
Question 5b
The manager of a certain project has identified the following information relating to the project:
| Activity | Immediate predecessor (s) | Duration (weeks) | Probability |
| A | - | 3 4 5 | 0.25 0.50 0.25 |
| B | - | 4 5 6 7 8 | 0.15 0.30 0.20 0.20 0.15 |
| C | A | 1 3 5 | 0.20 0.65 0.15 |
| D | B,C | 4 5 | 0.80 0.20 |
| E | D | 3 4 5 6 | 0.15 0.25 0.25 0.35 |
| F | D | 5 7 | 0.20 0.80 |
| G | E,F | 2 3 | 0.50 0.50 |
Required:
(i) A network diagram for the project.
(ii) The expected duration of the project.
(iii) Simulate the durations of the project on the basis of two runs.
Question 2b
X Limited, Y Limited and Z Limited deal in the production of detergents. On 1 January 2017, the three companies introduced a similar new detergent in the market. Prior to introduction of the new detergent, the three companies
had an equal share of the market. A survey conducted on the market shares of the three companies as at 31 December 2017 revealed the following:
- X Limited had retained 90 per cent of its customers but had lost 3 per cent and 7 per cent of its customers to Y Limited and Z Limited respectively.
- Y Limited had retained 75 per cent of its customers but had lost 10 per cent and 15 per cent of its customers to X Limited and Z limited respectively.
- Z Limited had retained 80 per cent of its customers but had lost 5 per cent and 15 per cent of customers to X Limited and Y Limited respectively. its
- There were no significant changes in the buying habits of the customers during the year.
Required:
(i) The market shares ofthe three companies as at 31 December 2018.
(ii) The long run market shares of the three companies.
Question 1a
Explain the following terms as used in decision theory:
(i) Opportunity loss.
(ii) Expected value of perfect information.
May 2018
5 Questions
Question 2a
(i) Distinguish between a "single server queuing model" and a "multiple server queuing model".
(ii) Highlight two assumptions of the queuing theory.
Question 2b
Outline three advantages and three disadvantages of the simulation model as used in quantitative analysis.
Question 3c
A manufacturing company is considering production of one of the three different types of pens, P₁, P₂ and P3. The fixed
and variable costs of producing the pens are as given below:
| Type of pen | Fixed cost (Sh.) | Variable cost (Sh.) |
| P1 | 2,000,000 | 100 |
| P2 | 3,200,000 | 80 |
| P3 | 6,000,000 | 60 |
The demand of the pens unde. three different states of demand is provided below:
| State of demand | Number of pens |
| Low | 250,000 |
| Moderate | 1000,000 |
| High | 1,500,000 |
The unit se lling price of the pens is Sh.200.
Required:
(i) The payoff table of the company.
(ii) The type of pen to produce using the maximin criterion.
(iii) The type of pen to produce using the maximax criterion.
(iv) The type of pen to produce using the minimax regret criterion.
Question 4a
Define the following terms as used in game theory:
(i) Mixed strategy.
(ii) Value of the game.
Question 5b
The following information relates to a certain construction project:
| Activity | Preceding activity | Time estimates (weeks) | ||
| Most optimistic | Most likely | Most pessimistic | ||
| A | - | 2 | 4 | 12 |
| B | - | 10 | 12 | 26 |
| C | A | 8 | 9 | 10 |
| D | A | 10 | 15 | 20 |
| E | A | 7 | 7.5 | 11 |
| F | B,C | 9 | 9 | 9 |
| G | D | 3 | 3.5 | 7 |
| H | E,F,G | 5 | 5 | 5 |
Required:
(i) The network diagram for the project.
(ii) The critical path.
(iii) The probability of completing the project within a 30 - week duration.
December 2017
6 Questions
Question 5a
Explain the following terms as used in game theory:
(i) Saddle point.
(ii) Dominance.
(i) Saddle point.
(ii) Dominance.
Question 4b
Two airlines, K and Q are interested in increasing their market shares. Airline K has three available strategies,
advertising its special fare, advertising its unique features or advertising its safety record. On the other hand, Airline Q
also has three available strategies; do nothing, advertise its special fare or advertise its special features.
The matrix below shows the gains and losses associated with the different available strategies in millions of shillings.
Positive values favour Airline K and negative values favour Airline Q.
| Airline Q | ||||
| Q1 | Q2 | Q3 | ||
| K1 | -30 | 0 | -90 | |
| Airline K | K2 | -40 | -15 | -20 |
| K3 | 80 | 20 | -50 | |
Where: K1 = Advertise special fare.
K2 = Advertise unique features.
K3 = Advertise safety record.
Q1 = Do nothing.
Q2 = Advertise special fare.
Q3 = Advertise special features.
Required:
(i) The optimal strategies for each airline.
(ii) The value of the game.
Question 4a
In a certain busy business facility, the mean arrival rate of clients is 800 clients per hour. The mean service rate is 820
clients per hour. The facility operates between 6.00 a.m. and 6.00 p.m. every day. The management of the facility are
concerned about the average number of customers in the queuing system and wish to improve the facility in order to serve an average of 847 clients per hour. The cost of improving the facility amounts to Sh.18,500 per day. Each hour lost
costs the facility Sh. 125.
Required:
(i) The average waiting cost per day.
(ii) Advise the management on whether they should improve the facility.
(iii) Compare the probabilities that the total number of clients in the queue and those being served is greater than 17
in the existing and in the improved facilities.
Question 3d
The management of New Era Computer Systems Limited is planning to launch a new product branded Zimsang. The
fixed cost of Zimsang is Sh.80,000. However, the selling price, variable costs and annual sales volume of Zimsang are
uncertain. The data below relate to product Zimsang:
| Unit selling Price(sh.) | Probability | Variable cost (Sh.) | Probability | Sales yolume (units) | Probability |
| 60 | 0.25 | 20 | 0.25 | 40,000 | 0.30 |
| 80 | 0.45 | 40 | 0.55 | 60,000 | 0.35 |
| 100 | 0.30 | 60 | 0.20 | 100,000 | 0.35 |
Required:
Simulate the average profit of product Zimsang on the basis of 10 trials. Use the following random numbers:
81 32 60 04 16 31 67 25 24 10 40 02 39
68 08 59 66 90 12 64 79 31 86 68 82 89
25 11 98 16
Question 5b
Suggest two areas in which advanced information technology could be used to solve quantitative analysis problems.
Question 5c
The data below relate to normal duration and cost along with crash duration and cost for each activity of a certain
project.
| Activity | Normal duration (Days) | Crash duration (Days) | Normal cost (Sh.) | Crash cost (Sh.) |
| 1 - 2 | 6 | 4 | 2,800,000 | 3,800,000 |
| 1 - 3 | 8 | 5 | 4,000,000 | 5,600,000 |
| 2 - 3 | 4 | 2 | 2,200,000 | 3,000,000 |
| 2 - 4 | 3 | 2 | 1,600,000 | 2,800,000 |
| 3 - 4 | Dummy | - | - | - |
| 3 - 5 | 6 | 3 | 1,800,000 | 3,200,000 |
| 4 - 6 | 10 | 6 | 5,000,000 | 7,000,000 |
| 5 - 6 | 3 | 2 | 1,000,000 | 1,600,000 |
The indirect cost of the project is Sh.600.000 per day.
Required:
(i) The normal duration and the corresponding total cost.
(ii) The minimum duration and the corresponding total cost.
(iii) The optimum duration and the corresponding total cost.
May 2017
7 Questions
Question 5b
Two companies, A and B. are competing for business whereby one company's gain is the other company's loss.
The pay-off matrix is given as follows:
| Company B's strategies | ||||
| \(B_1\) | \(B_2\) | \(B_3\) | ||
| Company A's strategies | \(A_1\) | 7 | 4 | 1 |
| \(A_2\) | 4 | 2 | 0 | |
| \(A_3\) | 3 | -1 | -2 | |
| \(A_4\) | 2 | 5 | -3 | |
Required:
The optimal strategy for each company.
Question 1b
Explain the following terms as used in Markov analysis
(i) Transition probability.
(ii) Absorbing state.
(i) Transition probability.
(ii) Absorbing state.
Question 2c
ABC Limited has recently developed a new product named "Exe". The demand for "Exe" is expected to be low.
medium or high with probabilities of 0.25, 0.45 and 0.30 respectively.
The product is to be manufactured at small or large scale production with the following annual profit estimates:
| Small scale production | Large scale production | ||||
| Profit (Sh. million) | Probability | Profit (Sh. million) | Probability | ||
| Demand | Low | 40 | 0.25 | 5 | 0.25 |
| Medium | 140 | 0.45 | 90 | 0.45 | |
| High | 180 | 0.30 | 280 | 0.30 | |
Required:
Advise ABC Limited on the best course of action based on the following approaches:
(i). Expected profit.
(ii). Minimising risk.
Question 4a
Outline five assumptions of game theory.
Question 4c
Pure Grain Society is considering the planting of wheat on a piece of land it recently acquired.
You are provided with the following random numbers:
The data below relate to the estimated selling prices, vield and cost of planting the wheat:
| Selling price per tonne (Sh."000") | Probability | Yield per acre (tonne) | Probability | Cost per acre (Sh."000") | Probability |
| 240 | 0.18 | 70 | 0.09 | 12,000 | 0.14 |
| 250 | 0.29 | 75 | 0.16 | 14,000 | 0.22 |
| 260 | 0.31 | 80 | 0.24 | 16,000 | 0.36 |
| 270 | 0.14 | 85 | 0.38 | 18,000 | 0.26 |
| 280 | 0.08 | 90 | 0.13 | 20,000 | 0.02 |
You are provided with the following random numbers:
03 91 38 55 17 46 32 43 69 72 24 22 61 96 30 32
Required:
Using eight trials, simulate the average profit of Pure Grain Society.
Question 5c
Ujenzi Limited has been awarded a contract to build an office block. The tasks of the building project have been
analysed as follows:
| Activity | Preceding activity | Duration (months) | Total Cost (Sh.million) |
| A | - | 8 | 100 |
| B | - | 2 | 75 |
| C | A | 3 | 135 |
| D | A | 7 | 70 |
| E | B | 5 | 160 |
| F | C,D | 9 | 255 |
| G | D | 2 | 30 |
| H | D,E | 4 | 90 |
| I | G,H | 3 | 55 |
The overhead costs of the project amount to Sh.5 million per month.
Required:
(i) A network diagram for the project.
(ii) The minimum cost of the project.
(iii) Ujenzi Limited has been offered a bonus of Sh.25 million if they complete the project within a period of 20 months or less.
(ii) The minimum cost of the project.
(iii) Ujenzi Limited has been offered a bonus of Sh.25 million if they complete the project within a period of 20 months or less.
The table below shows activities that would require to be crashed and their respective
total costs:
| Activity | Duration (months) | Total cost (Sh.million) |
| A | 6 | 125 |
| B | 1 | 90 |
| C | 5 | 85 |
| D | 3 | 200 |
| E | 7 | 275 |
| F | 2 | 95 |
Determine whether or not Ujenzi Limited should accept the bonus offer.
Question 5a
Enumerate five characteristics of a simple queuing system.
November 2016
5 Questions
Question 5a
Outline five limitations of game theory.
Question 5b
The data below relate to activities of a certain project that is to be undertaken by Ujuzi Consultancy Company:
| Time (weeks) | ||||
| Activity | Preceding activity | Optimistic | Most probable | Pessimistic |
| A | - | 1.5 | 2.0 | 2.5 |
| B | A | 2.0 | 2.5 | 6.0 |
| C | - | 1.0 | 2.0 | 3.0 |
| D | C | 1.5 | 2.0 | 2.5 |
| E | B,D | 0.5 | 1.0 | 1.5 |
| F | E | 1.0 | 2.0 | 3.0 |
| G | B,D | 3.0 | 3.5 | 7.0 |
| H | G | 3.0 | 4.0 | 5.0 |
| I | F,H | 1.5 | 2.0 | 2.5 |
Required:
(i). A network diagram of the project.
(ii). The expected completion time of the project.
(iii). The probability that the project will be completed between 13 weeks and 17 weeks
Question 4d
A barber shop has a total of 10 available seats for customers. The inter-arrival times for customers are exponentially
distributed with an average of 20 customers arriving each hour. Any prospective customer who finds all the seats
occupied does not wait for service but instead leaves. The barber takes an average of 12 minutes to cut each
customer's hair. Hair cut time duration is exponentially distributed.
Required:
(i) The average number of hair cuts that will be completed by the barber each hour.
(ii) The average time each customer will spend at the barber shop.
Question 4b
Summarise four decision criteria used in decision making under uncertainty.
Question 2a
Highlight four applications of Markov analysis in business.
May 2016
6 Questions
Question 3a
Explain the following terms as used in game theory:
(i) Pay-off.
(ii) Value of a game.
(i) Pay-off.
(ii) Value of a game.
Question 3b
Highlight eight steps followed in the simulation process.
Question 3d
A cashier at a departmental store can serve on average 24 customers per hour. The arrival rate of customers averages 20 customers per hour. The departmental store applies a single channel queuing system.
Required:
(i) The probability that the cashier is idle.
(ii) The average number of customers in the queuing system.
(iii) The average time a customer spends in the queue waiting to be served.
Required:
(i) The probability that the cashier is idle.
(ii) The average number of customers in the queuing system.
(iii) The average time a customer spends in the queue waiting to be served.
Question 1a
Explain four differences between the project evaluation and review technique (PERT) and the critical path analysis (CPA).
Question 5b
The table below is a summary of activities required to complete the project, the duration of the activities and their preceding activities.
Required:
(i) A network diagram of the project.
(ii) The float times of activities B and D.
(iii) The critical path of the project.
(iv) A 95 per cent confidence interval of the expected completion time of the project.
(v) The expected profit from the project.
A certain project is expected to be completed within 18 weeks. The expected net revenue if the project is complete on time is Sh.1,120,000 but a penalty of Sh.484,000 will be imposed if the project is not completed on time. The cost of the project is Sh.459,000. The standard deviation of the project's duration is 2.08 weeks.
The table below is a summary of activities required to complete the project, the duration of the activities and their preceding activities.
|
Activity
A B C D E F G H |
Duration (weeks) 5 2 4 2 5 7 6 3 |
Preceding activity
- - - B B, C C A, D G, D, E, F |
Required:
(i) A network diagram of the project.
(ii) The float times of activities B and D.
(iii) The critical path of the project.
(iv) A 95 per cent confidence interval of the expected completion time of the project.
(v) The expected profit from the project.
Question 4a
Viwanda Limited deals in the production of a product named "Nguvu". The production cost of the product is Sh.500 per unit (excluding packaging cost). The product is sold at Sh.1,000 per unit. The company is considering the purchase of one out of three different packaging systems. The cost data for the three packaging systems are as follows:
All the three packaging systems have a useful life of one year after which they would be sold at their estimated scrap values. The probability distribution for the demand for product "Nguvu" is as provided below:
Required:
Recommend the packaging system that should be purchased by Viwanda Limited
|
Packaging system
A B C |
Purchase cost
Sh. "000" 100 200 400 |
Variable cost per
unit of product Sh. "000" 1.50 1.00 0.50 |
Scrap value
Sh. "000" 10 20 40 |
All the three packaging systems have a useful life of one year after which they would be sold at their estimated scrap values. The probability distribution for the demand for product "Nguvu" is as provided below:
|
Demand (units)
100 200 400 |
Probability
0.3 0.6 0.1 |
Required:
Recommend the packaging system that should be purchased by Viwanda Limited
November 2015
5 Questions
Question 5a
A simulation model attempts to describe a business system using a number of equations. These equations are characterised by four types of variables.
Required:
With reference to the above statement, explain the four types of variables in a simulation equation.
Required:
With reference to the above statement, explain the four types of variables in a simulation equation.
Question 4d
Jane Cherop was employed by Golden Houses Limited as a sales agent last year. During the year, she was able to sell up to a maximum of 6 houses in a month. Due to good performance in the past year, the company has offered Jane Cherop one of the following three salary plans for the next year:
Plan A: A 25 per cent salary increament to Sh.50,000 per month.
Plan B: A fixed monthly salary of Sh.20,000 per month plus a commission of Sh.12.000 per house sold.
Plan C: No monthly salary but a commission of Sh.20.000 per house sold.
Required:
(i) The optimal salary plan for Jane Cherop based on the maximin criterion.
(ii) The optimal salary plan for Jane Cherop based on the minimax regret criterion.
(iii) Assume that during the past year, the distribution of the houses sold by Jane Cherop for the twelve months was as follows:
Advise Jane Cherop on the optimal salary plan based on the expected value criterion.
Plan A: A 25 per cent salary increament to Sh.50,000 per month.
Plan B: A fixed monthly salary of Sh.20,000 per month plus a commission of Sh.12.000 per house sold.
Plan C: No monthly salary but a commission of Sh.20.000 per house sold.
Required:
(i) The optimal salary plan for Jane Cherop based on the maximin criterion.
(ii) The optimal salary plan for Jane Cherop based on the minimax regret criterion.
(iii) Assume that during the past year, the distribution of the houses sold by Jane Cherop for the twelve months was as follows:
|
Number of house sold Number of months |
0 1 |
1 2 |
2 1 |
3 2 |
4 1 |
5 3 |
6 2 |
Advise Jane Cherop on the optimal salary plan based on the expected value criterion.
Question 4c
Quick Works Limited deals in the provision of typing services. On average, a typist at the company receives 22 letters per day for typing. The typist works for 8 hours a day and it takes an average of 20 minutes to type a letter. The company has determined that the cost of a letter waiting to be typed is Sh.8 per hour and the typing equipment operating cost plus the salary of the typist amount to Sh.400 per day. In an attempt to improve on the letter typing service, the company is planning to lease one of the two models of automated typewriters to be used together with the existing typing equipment. The additional cost per day and the increase in typist's efficiency of the two models is as given below:
Required:
Advise the company on the action that it should take in order to minimise the total daily cost.
|
Model
I II |
Additional cost per day (Sh.)
370 390 |
Increase in typist's efficiency (%)
50 75 |
Required:
Advise the company on the action that it should take in order to minimise the total daily cost.
Question 4a
Explain the following terms as used in game theory:
(i) Pure strategy.
(ii) Saddle point.
(i) Pure strategy.
(ii) Saddle point.
Question 3a
Outline four applications of the programme evaluation and review technique (PERT) in the planning and management of projects.
Pilot September 2015
5 Questions
Question 1a
Highlight any four assumptions of Markov Analysis.
Question 2a
The City Theatre has four auditoriums namely C1, C2, C3 and C4. Each auditorium performs a different play at any given time. The performances start at different times to avoid long queues that would occur if all the auditoriums were to start performance at the same time. The theatre has a single ticket booth and a cashier who can maintain an average service rate of 280 patrons per hour. Arrivals are poisson distributed at an average of 210 patrons per hour. The services are assumed to follow an exponential distribution.
Required:
(i) The average number of patrons waiting in line to purchase the tickets.
(ii) The average time spent waiting in line to get to the ticket, window.
(iii) The average time a patron spends in the system.
(iv) The percentage of time the cashier is busy.
(v) The probability that there are more than two people in the system.
Required:
(i) The average number of patrons waiting in line to purchase the tickets.
(ii) The average time spent waiting in line to get to the ticket, window.
(iii) The average time a patron spends in the system.
(iv) The percentage of time the cashier is busy.
(v) The probability that there are more than two people in the system.
Question 3b
Brightshine Limited based in Nairobi manufactures a detergent. The firm is considering opening a new plant in Nakuru. The opening of a new plant will, however, depend on the demand for the detergent in Nakuru.
Information concerning the demand for the detergent is shown below:
H - High demand and leads to a profit of Sh.6,000,000 per year.
M - Moderate demand and leads to a profit of Sh.1,500,000 per year.
L - Low demand and leads to a loss of Sh.2,500,000 per year.
The chances of having high, moderate and low demand are assessed at 30%, 30% and 40% respectively by the firm's management.
A market research group could be employed to provide information on which market demand would be realised. Past experience with work in the same market with this group shows its information cannot be relied upon to be absolutely accurate.
The market research group classifies its results as either being good prospects (G) or poor prospects (P). The table below gives the extent of reliability of this market research group:
The market research group would charge a fee of Sh.60,000 if it was hired.
Required:
(i) The best course of action on the basis of prior information.
(ii) The expected value of perfect information.
(iii) Advise Brightshine Limited whether the market research should be conducted. Show your workings using a decision tree
Information concerning the demand for the detergent is shown below:
H - High demand and leads to a profit of Sh.6,000,000 per year.
M - Moderate demand and leads to a profit of Sh.1,500,000 per year.
L - Low demand and leads to a loss of Sh.2,500,000 per year.
The chances of having high, moderate and low demand are assessed at 30%, 30% and 40% respectively by the firm's management.
A market research group could be employed to provide information on which market demand would be realised. Past experience with work in the same market with this group shows its information cannot be relied upon to be absolutely accurate.
The market research group classifies its results as either being good prospects (G) or poor prospects (P). The table below gives the extent of reliability of this market research group:
| Market survey | Actual state of nature | ||
| Result G P | H 0.7 0.3 | M 0.6 0.4 | L 0.2 0.8 |
The market research group would charge a fee of Sh.60,000 if it was hired.
Required:
(i) The best course of action on the basis of prior information.
(ii) The expected value of perfect information.
(iii) Advise Brightshine Limited whether the market research should be conducted. Show your workings using a decision tree
Question 4a
Distinguish between the following sets of terms:
(i) Zero-sum game and non-zero sum game.
(ii) Pure strategy game and Mixed strategy game.
(i) Zero-sum game and non-zero sum game.
(ii) Pure strategy game and Mixed strategy game.
Question 4b
An engineering firm is tendering for a contract to supply a steel fabrication. The tasks have been analysed as follows:
Required:
(i) A network diagram for the project.
(ii) The critical path and the expected project duration.
(iii) The time schedules for activities F, G and H.
| Activity A B C D E F G H I J K L M | Predecessory activity - - A A A A,B C C,D E G,H H H,I H,I,F | Time(Days) 10 12 10 9 13 17 12 14 13 12 10 14 13 |
Required:
(i) A network diagram for the project.
(ii) The critical path and the expected project duration.
(iii) The time schedules for activities F, G and H.