Quantitative Techniques

Quantitative Techniques — 2023

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

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July 2023

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

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 1(b) — 10 marks

The following information relates to ten machines at Okwot Enterprises.

Age of machine (years)	Annual maintenance cost Ksh 000
12	60
8	52
10	55
4	30
7	40
2	20
11	58
10	54
10	50
15	70

(i) Calculate the Pearson’s correlation co-efficient.

(ii) Interpret the results in (i) above.

Model answer

Let x be age and y be annual maintenance cost in Ksh thousands. n = 10.

r = [nΣxy − ΣxΣy]/√{[nΣx² − (Σx)²][nΣy² − (Σy)²]} = 5,019.0000/5,112.6726 = 0.981678.

There is a strong positive linear association: older machines tend to have higher annual maintenance costs. Correlation alone does not establish that age is the only cause of the cost difference.

Question 2(a) — 10 marks

Explain five roles of quantitative techniques in business decision making.

Model answer

1. Forecasting demand and sales from data helps determine future production, purchasing and staffing requirements.
2. Optimisation allocates scarce labour, materials and machine time to the activities that best meet a stated objective.
3. Inventory models balance ordering, holding and shortage costs to guide replenishment decisions.
4. Project scheduling identifies activity dependencies, completion times and critical tasks so managers can control delivery.
5. Probability and statistical analysis measure uncertainty, compare alternatives and test claims, supporting more consistent risk-informed decisions.

Question 2(b) — 10 marks

The total cost of one pencil and one eraser in a shop is Ksh 70. The total cost of three similar pencils exceeds the total cost of two similar erasers by Ksh 110.

(i) Formulate simultaneous equations from the information above.

(ii) Using matrix method, solve the simultaneous equations in (i) above to determine the cost of each item.

(iii) Determine the total cost of five pencils and eight erasers.

Model answer

Let x = pencil price and y = eraser price. Then x + y = 70 and 3x − 2y = 110. In matrix form AX = b, with A =

1	1
3	-2

and b =

70

110

det A = −2 − 3 = −5; A⁻¹ = (−1/5)

-2	-1
-3	1

X = A⁻¹b = (−1/5)[−140−110, −210+110]ᵀ = [50,20]ᵀ.

Check: 50 + 20 = 70; 3×50 − 2×20 = 110.

Question 3(a) — 10 marks

Explain the importance of Time Series analysis in decision making.

Model answer

1. It reveals the long-term trend in sales, costs or demand, helping managers plan growth and capacity.
2. It identifies seasonal patterns so purchasing, staffing and cash availability can be adjusted before predictable peaks and troughs.
3. It supports forecasts based on observed historical movement, providing inputs for budgets and operating plans.
4. It separates regular patterns from unusual movements, helping managers investigate shocks or changes in performance.
5. It provides a benchmark for comparing actual results with expected patterns and assessing whether corrective action is needed.

Question 3(b) — 10 marks

The following is the Marginal Revenue (MR) function of a firm.

MR = 20 − 0.5q

Where q is the level of output sold by the firm. Determine the:

(i) total revenue function;

(ii) level of output that will maximize the total revenue;

(iii) maximum total revenue;

(iv) price at maximum total revenue.

Model answer

Question 4(a) — 8 marks

Explain four problems that may be encountered when constructing index numbers.

Model answer

1. Selecting a representative base year is difficult when available years contain unusual inflation, shortages or other shocks.
2. Choosing representative commodities and obtaining comparable prices can be difficult because consumption differs among households and locations.
3. Selecting suitable weights is challenging, and old weights become misleading as expenditure patterns change.
4. Quality changes, new products, disappearing products and different units of sale reduce comparability over time and may require adjustments.

Question 4(b) — 12 marks

Figure 1 is a network diagram of a project to be undertaken by a firm. The time duration is given in days.

(i) Prepare a table showing the activity, preceding activity and the duration.

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

Model answer

The event numbers are added to the redrawing for reference; activity labels and durations retain the supplied network. An activity leaving a merge event waits for all activities entering that event.

Forward pass: take the maximum finish of all predecessors. Backward pass: take the minimum start of successors. The zero-float path is A → B → D → J → K → L, taking 5 + 10 + 10 + 14 + 12 + 10 = 61 days.

Question 5(a) — 8 marks

Explain four poor stock management practices that may make a firm to accumulate high levels of stock.

Model answer

1. Ordering quantities beyond realistic demand or buying merely to obtain discounts creates excess inventory and ties up cash.
2. Poor or outdated demand forecasting causes purchases to continue after demand has fallen or product preferences have changed.
3. Inaccurate stock records and weak coordination allow duplicate orders for stock already held or on order.
4. Failure to review slow-moving and obsolete stock, enforce reorder limits or dispose of unwanted items allows old stock to accumulate.

Question 5(b) — 12 marks

The following are prices and quantities of four commodities in a market for the years 2015 and 2018.

Commodity	2015 price p₀ Ksh	2015 quantity q₀	2018 price p₁ Ksh	2018 quantity q₁
A	40	100	50	75
B	60	500	80	600
C	80	350	120	400
D	100	120	80	100

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

Model answer

Question 6(a) — 12 marks

Distinguish between each of the following terms as used in the test of hypothesis: (i) Type I error and Type II error; (ii) critical region and acceptance region; (iii) Z-test and T-test.

Model answer

Question 6(b) — 8 marks

John intends to invest Ksh 3,000,000 for a period of five years in either option A or option B.

Option A: deposit the amount in an account paying interest at the rate of 10% per annum compounded semi-annually.

Option B: invest the amount in a fund paying interest at the rate of 8% per annum compounded quarterly.

(i) Determine the total amount that will be available to John under each option at the end of five years.

(ii) Calculate the interest that will be earned under each option.

(iii) Based on the results in (ii) above, advise John on the option to invest in.

Model answer

Compound amount F = P(1 + r/m)ᵐᵗ. Interest = F − P.

On the rates given and assuming equal risk, charges and access conditions, choose A: it earns Ksh 428,841.69 more over five years. This is the comparison requested by the examination data.

Question 7(a) — 10 marks

In the “Jua Kali” sector, the percentage of male entrepreneurs is 70%. A random sample of ten entrepreneurs was selected from the sector. Determine the probability of selecting:

(i) one male entrepreneur;

(ii) no male entrepreneur;

(iii) at least one male entrepreneur;

(iv) at most eight male entrepreneurs.

Model answer

Let X ~ Binomial(10,0.7), assuming independent selections with constant probability. P(X=k)=C(10,k)0.7ᵏ0.3¹⁰⁻ᵏ.

Question 7(b) — 10 marks

The following are the sales of a company for ten consecutive years.

Year	Sales Ksh million
2000	40
2001	38
2002	42
2003	50
2004	48
2005	52
2006	65
2007	65
2008	70
2009	72

(i) Using three year moving averages, determine the trend values.

(ii) Comment on the trend in the sales.

Model answer

Average each consecutive group of three years and place the result against the middle year.

The smoothed sales trend rises from Ksh 40.00 million in 2001 to Ksh 69.00 million in 2008, with faster growth in the later years. There is no centred three-year value for 2000 or 2009 because an adjoining year is missing at each end.