(a) It is planned to raise a research team to a strength of 50 chemists, which is to be maintained. The wastage of recruits depends on their length of service which is as follows:

Year : 1 2 3 4 5 6 7 8 9 10

Total percentage who have left by the en

Question

(a) It is planned to raise a research team to a strength of 50 chemists, which is to be maintained. The wastage of recruits depends on their length of service which is as follows:

Year : 1 2 3 4 5 6 7 8 9 10

Total percentage who have left by the end of year : 5 36 55 63 68 73 79 87 97 100

What is the required number of recruitments per year necessary to maintain the required strength? There are 8 senior posts for which the length of service is the main criterion. What is the average length of service after which the next entrant expects promotion to one of these posts? (20 marks)

(b) Explain the structure of a queuing system. Explain M/M/1 queuing system and obtain steady-state solution. Also calculate busy period distribution. (15 marks)

(c) A company that operates for 50 weeks in a year is concerned about its stocks of copper cable. This costs ₹ 240 a metre and there is a demand for 8000 metres a week. Each replenishment costs ₹ 1,050 for administration and ₹ 1,650 for delivery, while holding costs are estimated at 25 percent of value held a year. Assuming that no shortages are allowed, what is the optimal inventory policy for the company? How would this analysis differ if the company wants to maximize its profits rather than minimize cost? What is the gross profit if the company sells the cable for ₹ 360 a metre? (15 marks)

UPSC Answer Check · Accepted Answer

This is a multi-part numerical problem requiring you to solve three distinct operations research scenarios. Allocate approximately 40% of time to part (a) given its 20 marks, and 30% each to parts (b) and (c). Begin with clear problem identification for each sub-part, show all working steps with proper formulae, and conclude with precise numerical answers with units. For part (b), balance theoretical explanation with mathematical derivation.
- Part (a): Calculate annual wastage rates from cumulative percentages, determine survival probabilities, compute required annual recruitment using renewal equation, and find average service length for promotion using weighted probability distribution
- Part (a): Correctly interpret 'total percentage who have left' as cumulative distribution and derive conditional probabilities of leaving in each specific year
- Part (b): Define queuing system components (arrival process, service mechanism, queue discipline) and derive steady-state probabilities for M/M/1 using balance equations with ρ = λ/μ < 1
- Part (b): Obtain explicit formulas for P₀ = 1-ρ, Pₙ = ρⁿ(1-ρ), and derive busy period distribution using Takács formula or generating function approach
- Part (c): Apply EOQ model with D = 400,000 metres/year, C₀ = ₹2,700, Cₕ = ₹60/metre/year, calculate optimal Q*, cycle time, and total minimum cost
- Part (c): Distinguish cost minimization from profit maximization by incorporating revenue function, show that optimal quantity remains unchanged under constant price, and compute gross profit as (360-240) × 400,000

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	For (a): correctly constructs life table with lₓ values and computes annual wastage qₓ; for (b): properly defines state space and draws state transition diagram; for (c): identifies all cost parameters correctly with consistent time units (weeks converted to year where needed)	Sets up most problems correctly but has minor errors in parameter identification (e.g., uses weekly demand without annualizing holding cost) or incomplete state definitions	Major setup errors: misinterprets cumulative percentages as annual rates, confuses arrival rate with service rate in queuing, or uses wrong cost components in inventory model
Method choice	20%	10	For (a): applies renewal theory/stationary population model correctly; for (b): uses birth-death process balance equations and generating functions for busy period; for (c): uses EOQ formula with proper square root derivation and compares marginal analysis for profit maximization	Uses correct broad approach but with shortcuts (e.g., states EOQ formula without derivation, gives steady-state probabilities without showing balance equations)	Wrong methodology entirely: uses simple averages instead of renewal equation, treats M/M/1 as deterministic, or applies wrong inventory model
Computation accuracy	20%	10	All calculations precise: annual recruitment ≈ 50/μ where μ is mean service time computed correctly from life table; steady-state probabilities sum to 1; EOQ = √(2×400000×2700/60) = 6000 metres with reorder level and annual orders calculated	Correct formulas with minor arithmetic slips (e.g., calculation errors in weighted average for promotion timing, or off-by-one errors in busy period)	Significant computational errors: wrong square roots, probabilities not summing to 1, unit conversion errors (₹ per week vs ₹ per year), or order of magnitude mistakes
Interpretation	20%	10	For (a): interprets result as sustainable recruitment policy and explains promotion criterion as expected value E[X\|X≥threshold]; for (b): explains physical meaning of ρ and stability condition; for (c): explains why EOQ doesn't change for profit maximization under constant price and computes total profit = ₹48,000,000	States numerical answers with basic interpretation but lacks insight into policy implications or economic reasoning	No interpretation provided, or nonsensical conclusions (e.g., negative recruitment, unstable queue considered acceptable, inventory policy ignoring business context)
Final answer & units	20%	10	All answers boxed with precise units: (a) recruitment per year (chemists/year), average service years; (b) explicit Pₙ formula, mean queue length L = ρ/(1-ρ), busy period density; (c) Q* = 6000 metres, cycle time = 0.75 weeks, reorder point, total cost = ₹3,60,000/year, gross profit = ₹4.8 crore/year	Correct numerical values but inconsistent or missing units, or incomplete set of required outputs	Missing final answers, wrong units (e.g., metres instead of chemists, dimensionless probabilities without formulas), or answers without context

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