Quantitative Techniques

Quantitative Techniques — 2024

KNEC Diploma in Supply Chain Management, Module II. Original theory examination questions with explained model answers for revision.

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November 2024

Paper codes: 2903/204
Time: 3 hours
Instructions: Answer any FIVE questions. All questions carry equal marks. Answer the questions in English.

Question 1(a) — 10 marks

The following are the average revenue (AR) and average cost (AC) functions of a product manufactured and sold by Baraka Limited.

AR = 40 − 0.5q
AC = 15 + 2q

Where q is the quantity produced and sold.

(i) Derive the profit function.

(ii) Determine the: I. quantity that maximizes profit; II. maximum profit.

Model answer

Question 1(b) — 10 marks

Given the following matrices:

A =

3	2
4	-1
6	0

B =

4	1
3	2

C =

4

5

Determine: (i) Aᵀ; (ii) B⁻¹; (iii) B × C; (iv) Cᵀ × B.

Model answer

(i) Transpose A by swapping rows and columns: Aᵀ =

3	4	6
2	-1	0

(ii) det B = 4×2 − 1×3 = 5. Swap diagonal entries and change the signs of off-diagonal entries, then divide by 5: B⁻¹ = (1/5)

2	-1
-3	4

(iii) BC: first row 4×4 + 1×5 = 21; second row 3×4 + 2×5 = 22.

21

22

(iv) CᵀB: [4,5]B = [4×4 + 5×3, 4×1 + 5×2] =

31

14

Question 2(a) — 10 marks

Explain five limitations of quantitative techniques in business decision making.

Model answer

1. Results depend on reliable data; incomplete, biased or outdated figures can produce misleading recommendations.
2. Models simplify reality and depend on assumptions that may not hold in changing business conditions.
3. Qualitative factors such as morale, ethics and customer relationships are difficult to express fully as numbers.
4. Building and maintaining models may require costly data systems, specialist staff and time.
5. Managers may misinterpret precise-looking results or apply them beyond the model’s scope; judgement and sensitivity checks remain necessary.

Question 2(b) — 10 marks

Odok Limited hired three contractors A, B and C to construct a road. The contractors have been assigned 60%, 25% and 15% of the road respectively. The probabilities that contractors A, B and C will fail to complete the construction within the scheduled time is 0.4, 0.3 and 0.2 respectively.

(i) Present the information above using a tree diagram.

(ii) Determine the probability that the road construction will: I. not be completed as scheduled; II. be completed as scheduled.

Model answer

The usual weighted-tree interpretation selects a road section at random, with contractor probabilities equal to the assigned proportions. Multiply probabilities along each branch, then add mutually exclusive paths.

Interpretation: these are the probabilities for a randomly selected section (or expected proportions of road), as implied by the supplied weights. If “the road” literally means every contractor finishing on time, the weighted sum is not that event. Under an additional independence assumption, P(all finish) = 0.6×0.7×0.8 = 0.336 and P(at least one late) = 0.664. Independence is not stated, so the whole-road probability is otherwise undetermined.

Question 3(a) — 8 marks

Outline the steps that are followed in the test of hypothesis.

Model answer

1. State the null hypothesis H₀ and the alternative hypothesis H₁ in terms of the population parameter.
2. Choose the significance level α before inspecting the result.
3. Select a suitable test and verify its assumptions and the sampling design.
4. Specify whether the test is one-sided or two-sided from the alternative hypothesis.
5. Identify the sampling distribution and the critical region, or plan to use a p-value.
6. Calculate the test statistic from the sample data.
7. Reject H₀ if the statistic enters the critical region or p ≤ α; otherwise fail to reject H₀.
8. Interpret the conclusion in the business context, stating what the evidence supports without claiming that failure to reject proves H₀ true.

Question 3(b) — 12 marks

A survey conducted in a certain country in Kenya found that 55% of the families own mobile phones. Ten families were randomly selected from the county. Determine the probability that:

(i) none of the 10 families have mobile phones;

(ii) at least one of the 10 families have mobile phones;

(iii) all the 10 families have mobile phones;

(iv) fewer than 4 families have mobile phones;

(v) more than 8 families do not have mobile phones.

Model answer

Let X be the number owning phones. With independent selections and constant ownership probability, X ~ Binomial(10, 0.55), so P(X=k) = C(10,k)(0.55)ᵏ(0.45)¹⁰⁻ᵏ.

Question 4(a) — 10 marks

Explain five requirements of a linear programming problem.

Model answer

1. An explicit objective must be specified, such as maximizing profit or minimizing cost, expressed as a linear function of the decision variables.
2. Clearly defined decision variables represent the quantities the business can choose.
3. Resource limits or business requirements must be expressible as linear equality or inequality constraints.
4. Relationships must be proportional and additive: coefficients remain constant, with no products or powers of decision variables; continuous divisibility is assumed unless integer restrictions are added.
5. Coefficients are treated as known with certainty and feasible sign restrictions, normally non-negativity, must be specified so the solution has business meaning.

Question 4(b) — 10 marks

The following are the prices and quantities of four commodities in a market for the years 2020 and 2024.

Commodity	2020 price Ksh p₀	2020 quantity q₀	2024 price Ksh p₁	2024 quantity q₁
X	20	200	24	180
Y	35	400	40	320
W	40	500	42	450
Z	75	600	80	500

Determine the: (i) Laspayre’s quantity index; (ii) Paasche’s quantity index; (iii) Fisher’s ideal quantity index.

Model answer

The quantity indices value the change in quantities using base-year and current-year prices respectively. The source labels its quantity columns “Ksh”; the numerical entries are treated as quantities as required by the question.

Question 5(a) — 10 marks

Explain five ways in which the concept of time value of money may be applied in business.

Model answer

1. Investment appraisal discounts future project cash flows to present value so receipts at different dates can be compared with the initial outlay.
2. Loan evaluation uses discounting and compounding to relate borrowed principal, repayment instalments and interest over time.
3. Savings and sinking-fund planning calculate the regular contributions needed to accumulate a target future amount.
4. Lease-versus-buy analysis compares the present values of payments, ownership costs and residual values occurring at different dates.
5. Valuation of bonds, shares or a business discounts expected future cash receipts at a rate reflecting time and risk.

Question 5(b) — 10 marks

The following relates to material KT used by Pere Enterprises in its manufacturing process.

Item	Value
Normal usage	200 units per day
Minimum usage	120 units per day
Maximum usage	260 units per day
EOQ	10,000 units
Lead time	15–20 days

Determine the: (i) re-order level; (ii) average re-order period; (iii) minimum stock level; (iv) maximum stock level; (v) average stock level.

Model answer

The usual safety-stock-plus-half-order convention gives 6,700 units. If “average stock” is instead defined as the midpoint of the maximum and minimum levels, it is (13,400 + 1,700)/2 = 7,550 units; the two definitions should not be mixed.

Question 6(a) — 12 marks

The following information relates to a project to be undertaken by a company.

Activity	A	B	C	D	E	F	G	H	I	J	K
Preceding activity	—	—	—	A	B	B	C	D	E	H,I	F,G
Duration (weeks)	5	4	6	8	12	10	4	6	8	10	7

(i) Draw a network diagram of the project.

(ii) Determine the: I. critical path; II. project duration.

Model answer

Forward pass: ES = latest predecessor EF; EF = ES + duration. Backward pass: LF = earliest successor LS; LS = LF − duration. Float = LS − ES. Project duration = 34 weeks. Red identifies critical activities and their controlling links.

Critical path: B → E → I → J. Duration = 4 + 12 + 8 + 10 = 34 weeks. Both terminal branches J and K must finish, so the later finish controls completion.

Question 6(b) — 8 marks

The following are the average prices of a certain commodity in a market.

Year	Price Ksh
2013	15
2014	18
2015	20
2016	20
2017	19
2018	25
2019	26
2020	28
2021	28
2022	30

(i) Using the least squares method, determine the trend line equation.

(ii) Based on the equation in (i) above, estimate the price of the commodity in the year 2023.

Model answer

Centre time at 2017.5: x = year − 2017.5. Since Σx = 0, a = Σy/n and b = Σxy/Σx².

Trend: ŷ = 22.9000 + 1.630303(year − 2017.5). Estimated 2023 price = Ksh 31.87.

Question 7(a) — 9 marks

The annual requirement of material TD at Ware Manufacturing Company is 1,200 units. The ordering cost is Ksh 2,500 per order, while the carrying cost is Ksh 600 per unit per year. Determine the:

(i) Economic Order Quantity (EOQ);

(ii) number of orders per year;

(iii) average inventory;

(iv) total relevant cost with the inventory.

Model answer

This assumes steady demand, no shortages and no safety stock. Unit purchase price is not given and purchase expenditure is not relevant to the basic EOQ choice when unit price is constant.

Question 7(b) — 11 marks

Three assessors P, Q and R ranked 10 employees at Pendo Limited according to their accuracy in a certain task as follows:

Employee	Assessor P	Assessor Q	Assessor R
A	1	9	2
B	4	7	4
C	6	2	5
D	2	5	1
E	7	1	8
F	10	4	9
G	3	6	3
H	8	10	7
I	5	8	10
J	9	3	6

(i) Determine the Spearman’s Rank coefficient of correlation between: I. assessor P and assessor Q; II. assessor P and assessor R; III. assessor Q and assessor R.

(ii) Based on the results in (i) above, determine the pair of assessors whose judgement of accuracy were: I. most consistent; II. most inconsistent.

Model answer

No tied ranks occur. For each pair rₛ = 1 − 6Σd²/[n(n²−1)], with n = 10 and denominator 990.

P and R are most consistent because they have the highest positive correlation. P and Q are most inconsistent because they have the lowest (most negative) correlation.