Paper II — Q3
(a) With respect to a given Linear Programming Problem (LPP), explain the following concepts : 15 marks (i) Extreme Point…
With respect to a given Linear Programming Problem (LPP), explain the following concepts : 15 marks Extreme Point Solutions
Duality Theorem
Complementary Slackness Principle
Define a Transition Probability Matrix (TPM). When is it said to be Regular and Ergodic ? Check whether the following TPM is Regular or Ergodic. Hence or otherwise obtain the limlimitsₙ → ∞ Pⁿ, where P = 0.88 & 0.12 0.15 & 0.85 . 15 marks
The reliability function R(t) of a cutting assembly is given by : R(t) = (1-t/(t₀))², & 0 ≤ t ≤ t₀ 0 & , t ≥ t₀ Determine the failure rate.
Does the failure rate increase or decrease with time ?
Determine the mean time to failure. 8+4+8=20 marks
हिंदी में प्रश्न पढ़ें
एक दी गई रैखिक प्रोग्रामन समस्या (एल पी पी) के संदर्भ में, निम्नलिखित संकल्पनाओं की व्याख्या कीजिए : 15 अंक चरम बिन्दु समाधान
द्वैत प्रमेय
पूरक शिथिलता सिद्धांत
संक्रमण प्रायिकता मैट्रिक्स (टी पी एम) को परिभाषित कीजिए। इसे कब नियमित और अभ्यतिग्राय (एर्गोडिक) कहते हैं ? परीक्षण कीजिए कि क्या निम्नलिखित संक्रमण प्रायिकता मैट्रिक्स (टी पी एम) नियमित है या अभ्यतिग्राय (एर्गोडिक) है। इस प्रकार या अन्य प्रकार से limlimitsₙ → ∞ Pⁿ को प्राप्त कीजिए, जहाँ P = 0.88 & 0.12 0.15 & 0.85 है। 15 अंक
एक कतन समुच्चय (कटिंग असेंबली) का विश्वसनीयता फलन R(t) दिया गया है : R(t) = (1-t/(t₀))², & 0 ≤ t ≤ t₀ 0 & , t ≥ t₀ विफलता दर निर्धारित कीजिए।
क्या विफलता दर समय के साथ बढ़ती या घटती है ?
विफलता का औसत समय निर्धारित कीजिए। 8+4+8=20 अंक
Model answer
Drafted with AI by UPSC Answer Check against this question's marking rubric, to the expected length. UPSC does not publish answers for Mains — this is one way to score well, not an official key. How we build rubrics and answers.
(a) Concepts in Linear Programming
(i) Extreme Point Solutions
In an LPP, the feasible region defined by the constraints is a convex polyhedron. A feasible solution is an extreme point (vertex) if it cannot be expressed as a strict convex combination of two other distinct feasible solutions, i.e. it lies at a corner where at least n linearly independent constraints intersect (n being the number of variables). The Fundamental Theorem of LPP states that if an optimal solution exists, at least one optimal solution occurs at an extreme point, because a linear objective function attains its maximum/minimum over a convex polytope at a vertex. This is why the simplex method searches only among basic feasible solutions (extreme points) rather than the infinite feasible set.
(ii) Duality Theorem
Every primal LPP has an associated dual. Weak duality states that for any feasible primal x and dual y, the primal objective ≤ dual objective (for maximisation primal). Strong duality states that if either problem has a finite optimal solution, so does the other, and their optimal objective values are equal. The dual variables are shadow prices: they measure the marginal change in the optimal objective per unit relaxation of a constraint, giving the economic worth of scarce resources.
(iii) Complementary Slackness Principle
At optimality, for each primal constraint, the product of its slack and the corresponding dual variable is zero; similarly for each dual constraint, the product of its surplus and the corresponding primal variable is zero. Thus a resource with positive slack (not fully used) has zero shadow price, and a resource with positive shadow price is fully exhausted. It is a powerful optimality-verification tool.
(b) Transition Probability Matrix: Regularity, Ergodicity and lim Pⁿ
A TPM P = [p_ij] gives the one-step transition probabilities of a Markov chain, satisfying p_ij ≥ 0 and each row summing to 1 (row-stochastic). A chain is Regular if some power Pⁿ has all entries strictly positive. It is Ergodic if it is irreducible (all states communicate) and aperiodic; a regular chain is ergodic.
For P = [[0.88,0.12],[0.15,0.85]], all entries are positive, so P itself has all positive entries; hence P is regular and ergodic. The limiting distribution π satisfies πP = π, π₁+π₂=1:
0.88π₁+0.15π₂ = π₁ → 0.15π₂ = 0.12π₁ → π₁/π₂ = 5/4.
With π₁+π₂=1: π = (5/9, 4/9). Hence lim Pⁿ has both rows equal to (5/9, 4/9). Explicitly, the eigenvalues of P are 1 and 0.88 + 0.85 − 1 = 0.73, and Pⁿ = (1/0.27)[[0.15, 0.12], [0.15, 0.12]] + ((0.73)ⁿ/0.27)[[0.12, −0.12], [−0.15, 0.15]]; since (0.73)ⁿ → 0, Pⁿ → [[5/9, 4/9], [5/9, 4/9]].
(c)(i) Failure Rate
R(t) = (1 − t/t₀)², 0 ≤ t ≤ t₀; 0 for t ≥ t₀.
λ(t) = −R′(t)/R(t). R′(t) = −(2/t₀)(1 − t/t₀). So λ(t) = [(2/t₀)(1 − t/t₀)] / (1 − t/t₀)² = 2/(t₀ − t), for 0 ≤ t < t₀.
(c)(ii) Does the Failure Rate Increase or Decrease?
λ′(t) = 2/(t₀ − t)² > 0, so the failure rate increases with time — an Increasing Failure Rate (IFR) distribution, characteristic of a wear-out phase.
(c)(iii) Mean Time to Failure
MTTF = ∫₀^t₀ R(t)dt = ∫₀^t₀(1 − t/t₀)²dt. Let u = 1 − t/t₀, dt = −t₀du: = t₀∫₀¹u²du = t₀/3.
How this answer will be evaluated
What earns full marks: All parts fully answered with correct derivations, clear definitions, and proper interpretation.
Evaluation rubric
Each sub-part is marked on its own, against the marks and word limit printed on the paper.
- (a) (a) Explain extreme point solutions, the duality theorem and the complementary slackness principle for an LPP (15 marks) 15 marks
explain— definition/context → points in order → small example → short close
Must cover
- Feasible region is a convex polyhedron; an extreme point is not a strict convex combination of two other feasible points and corresponds to a basic feasible solution
- Fundamental theorem: if an optimum exists, it is attained at an extreme point, which justifies the simplex search
- Duality: weak duality inequality and strong duality (equal optimal values; one finite optimum implies the other)
- Complementary slackness: slackᵢ × yᵢ = 0 and surplusⱼ × xⱼ = 0 at optimality, with shadow-price interpretation
Loses marks
- Confusing extreme points with any boundary point
- Stating strong duality as primal ≤ dual only
Earns more
- Writes the primal–dual pair in symmetric form
- Mentions the unbounded/infeasible duality relation
Extra mark
- Small numerical illustration
- (b) (b) Define TPM, regular and ergodic chains; check P=[[0.88,0.12],[0.15,0.85]] and find lim Pⁿ (15 marks) 15 marks
define— precise definition → the distinguishing feature → one example
Must cover
- TPM: non-negative entries, rows summing to 1
- Regular: some Pᵐ has all entries positive; ergodic: irreducible, aperiodic (positive recurrent)
- P has all positive entries, so it is regular and ergodic
- Solves πP=π: π=(5/9, 4/9), so lim Pⁿ has both rows (5/9, 4/9)
Loses marks
- Column-stochastic confusion
- Arithmetic error in the stationary distribution
Earns more
- Explicit spectral form with eigenvalue 0.73
- Interpretation as long-run proportions
Extra mark
- Notes the limit is independent of the initial state
- (c)(i) (c)(i) Failure rate for R(t)=(1−t/t₀)², 0≤t≤t₀ (8 marks) 8 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- Uses h(t)=f(t)/R(t)=−R′(t)/R(t)
- f(t)=(2/t₀)(1−t/t₀)
- h(t)=2/(t₀−t)
- States the range 0 ≤ t < t₀
Loses marks
- Using h=R′/R without the minus sign
- Algebraic errors
Earns more
- Checks f integrates to 1
- Notes h(t) → ∞ as t → t₀
Extra mark
- Cumulative hazard H(t)=−2 ln(1−t/t₀)
- (c)(ii) (c)(ii) Does the failure rate increase or decrease with time? (4 marks) 4 marks
comment— context → arguments both sides → judgment → close
Must cover
- h′(t)=2/(t₀−t)² > 0
- Failure rate increases with time
- IFR distribution
- Interpretation: wear-out/ageing of the cutting assembly
Loses marks
- Claiming a constant or decreasing rate
- No justification
Earns more
- Links to the wear-out phase of the bathtub curve
- Clear reasoning
Extra mark
- Mentions preventive replacement implications
- (c)(iii) (c)(iii) Mean time to failure (8 marks) 8 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- MTTF=∫₀^{t₀} R(t) dt
- Substitution u=1−t/t₀
- MTTF=t₀∫₀¹u² du
- MTTF=t₀/3
Loses marks
- Integrating over 0 to ∞ without using R=0 beyond t₀ correctly
- Answer t₀/2 or other
Earns more
- Alternative via ∫t f(t) dt giving the same result
- Clear limits of integration
Extra mark
- Computes the median life t₀(1−1/√2)
What "Explain" is asking you to do
Make the working of something clear — what sets it off, what follows from what, and what it produces. Explain is the Commission's mechanism word: it dominates the technical papers and the “explain why” stems, where the marks sit in the causal chain and not in the label.
Structure that answers it
State what it is → the initiating condition → the chain of cause, step by step → an instance where it plays out → what the chain produces
Where marks are lost
Describing what something looks like instead of why it works that way. Naming the stages without linking them reads as description too.
Practice this exact question
Write your answer and it is marked point by point against the model answer above — what you covered, what you missed, what you got wrong.
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