(a) What do you understand by Statistical Quality Control (SQC)? Discuss briefly its need and utility in Industry. Discuss the causes of variation in quality. (10 marks)

(b) Consider an item with failure rate $Z(t) = \frac{t}{t+1}$. Write down the s

Question

(a) What do you understand by Statistical Quality Control (SQC)? Discuss briefly its need and utility in Industry. Discuss the causes of variation in quality. (10 marks)

(b) Consider an item with failure rate $Z(t) = \frac{t}{t+1}$. Write down the survivor function $R(t)$ and hence evaluate Mean Time To Failure (MTTF). Also obtain the conditional survival function and Mean Residual Life (MRL). (10 marks)

(c) Solve the following linear programming problem by using graphical approach:

Minimize $4x_1 + 5x_2 + 6x_3$

Subject to $x_1 + x_2 \geq 11$
$x_1 - x_2 \leq 5$
$x_3 - x_1 - x_2 = 0$
$7x_1 + 12x_2 \geq 35$
$x_1 \geq 0, x_2 \geq 0, x_3 \geq 0$ (10 marks)

(d) In a two-person zero-sum game, write the payoff matrix in general notation. Consider the two-person zero-sum game where each player tosses an unbiased coin simultaneously. Player B pays ₹7 to A if {H, H} occurs or {T, T} occurs otherwise player A pays ₹3 to B. Write down A's payoff matrix. Explain the Max Min criterion for player A and hence define the saddle point. (10 marks)

(e) Let Xₜ be the state of a flea at time t

Find the transition Matrix P. Also obtain Pᵣ[X₂ = 3 | X₀ = 1]. (10 marks)

UPSC Answer Check · Accepted Answer

This multi-part question requires solving five distinct problems: (a) discuss SQC concepts with industrial applications, (b) derive reliability functions from given failure rate, (c) solve LP graphically, (d) construct and analyze game matrix, and (e) compute Markov chain probabilities. Allocate approximately 15-18 minutes per part, with extra attention to (c) and (e) where computational errors are common. Begin each part clearly labeled, show all derivation steps, and conclude with boxed final answers.
- Part (a): Define SQC as statistical methods for maintaining quality standards; explain need (mass production complexity, cost reduction, customer satisfaction) and utility (process control, acceptance sampling); classify variations into chance causes (random, inherent) and assignable causes (identifiable, correctable)
- Part (b): Derive R(t) = exp(-∫Z(t)dt) = (t+1)e^(-t); compute MTTF = ∫R(t)dt = 2; obtain conditional survival R(x|t) = R(t+x)/R(t) and MRL = ∫R(x|t)dx
- Part (c): Use constraint x₃ = x₁ + x₂ to reduce to 2-variable problem; minimize 10x₁ + 11x₂; identify feasible region vertices from intersection of x₁+x₂≥11, x₁-x₂≤5, 7x₁+12x₂≥35; optimal solution at (3,8) with value 118
- Part (d): General payoff matrix [aᵢⱼ] where i=1,...,m strategies for A, j=1,...,n for B; specific matrix with entries +7 for (H,H) and (T,T), -3 otherwise; apply Max Min: maximize minimum row payoff; saddle point exists if Max Min = Min Max
- Part (e): Construct transition matrix P from flea movement probabilities (typically given in diagram); compute P² and extract P[X₂=3|X₀=1] = (P²)₁₃ using Chapman-Kolmogorov equations

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies all problem structures: for (a) distinguishes chance vs assignable causes with industrial examples (e.g., ISI certification); for (b) sets up proper integral limits and initial condition R(0)=1; for (c) correctly substitutes x₃ before graphing; for (d) constructs proper 2×2 payoff matrix with A's gains; for (e) reads transition diagram correctly	Identifies most structures but has minor setup errors: incomplete cause classification in (a), wrong initial condition in (b), attempts 3D graph in (c), or mislabels payoffs in (d)	Fundamental setup errors: treats SQC as only inspection, integrates wrong function in (b), ignores x₃ constraint in (c), constructs payoff for B instead of A in (d), or misreads transition diagram in (e)
Method choice	20%	10	Selects optimal methods throughout: uses hazard-survivor relationship in (b), reduces dimensionality before graphical solution in (c), applies von Neumann minimax theorem in (d), uses matrix multiplication or Chapman-Kolmogorov in (e); justifies method choices briefly	Uses acceptable methods but with inefficiencies: numerical integration for MTTF, attempts 3D graphical method, or uses step-by-step probability tree instead of matrix power in (e)	Inappropriate methods: treats failure rate as constant in (b), uses simplex for graphical problem in (c), confuses zero-sum with non-cooperative game in (d), or attempts steady-state for finite-time probability in (e)
Computation accuracy	20%	10	All calculations precise: correct integration yielding R(t)=(t+1)e^(-t), MTTF=2, MRL=(t+2)/(t+1); LP vertices calculated accurately; game value computed correctly; matrix powers and multiplication error-free	Minor computational slips: one integration constant error, single vertex miscalculation, arithmetic error in matrix multiplication, or sign error in payoff—correctable with shown work	Major computational failures: incorrect integration by parts, wrong feasible region identification, fundamental errors in probability calculations, or incorrect matrix operations without verification
Interpretation	20%	10	Provides meaningful interpretations: explains increasing failure rate in (b) indicates wear-out phase; interprets LP solution as resource allocation; explains saddle point as stable strategy pair; connects Markov result to flea movement dynamics; uses Indian industrial context (e.g., SQC in textile/steel sectors)	States numerical results with minimal interpretation; mentions 'optimal' or 'saddle point' without explaining significance; generic statements about quality control without sectoral examples	No interpretation provided; leaves answers as bare numbers; or provides incorrect interpretation (e.g., calling MTTF a probability, confusing saddle point with Nash equilibrium in non-zero-sum sense)
Final answer & units	20%	10	All five parts with clearly boxed/concluded answers: (a) structured summary of SQC components; (b) R(t), MTTF=2, MRL with proper functional forms; (c) x₁=3, x₂=8, x₃=11, Z=118; (d) payoff matrix, Max Min=Min Max=-3, no saddle point (or value if exists); (e) P matrix and P[X₂=3\|X₀=1] value; consistent units throughout	Most answers present but format inconsistent: missing units on MTTF, unboxed LP solution, incomplete game analysis, or missing one final probability value	Missing multiple final answers; answers scattered without conclusion; wrong units (e.g., probability >1, negative MTTF); or fundamental misunderstanding of what constitutes final answer for each part type

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