Explain Single Sampling Plan with the help of an example. Also, write the importance of an Operating Characteristics Curve in a sampling plan. (10 marks)

Solve the above assignment problem.

Depot
I II III IV V
A 160 130 175

Question

Explain Single Sampling Plan with the help of an example. Also, write the importance of an Operating Characteristics Curve in a sampling plan. (10 marks)

Solve the above assignment problem.

Depot
        I    II   III  IV   V
A      160  130  175  190  200
Town B 135  120  130  160  175
C      140  110  145  170  185
D       50   50   80   80  110
E       55   35   80   80  105
(10 marks)

Use algebraic method to solve the above game.

Player B
        B₁     B₂     B₃     B₄
A₁     0·25   0·20   0·14   0·30
Player A A₂     0·27   0·16   0·12   0·14
A₃     0·35   0·08   0·15   0·19
A₄    −0·02   0·08   0·13   0·00
(10 marks)

Consider the Markov Chain with transition probability matrix:

0    1    2
0    (0    1    0)
1    (½    0    ½)
2    (0    1    0)

Show that the states are periodic and persistent non-null. (10 marks)

State the importance of the hazard function. If the hazard rate of a component is given by:

h(t) = { 0.015,  t ≤ 200
       { 0.025,  t > 200

then find an expression for the reliability function of the component. (10 marks)

UPSC Answer Check · Accepted Answer

The directive 'solve' requires demonstrating complete analytical solutions for all five numerical problems. Structure your answer by addressing each sub-question sequentially: (1) Single Sampling Plan with OC curve illustration, (2) Assignment problem using Hungarian method, (3) Game theory problem via algebraic method for mixed strategies, (4) Markov chain periodicity and persistence proof, and (5) Reliability function derivation from piecewise hazard rate. Each solution must show method, calculation, and final interpretation.
- Single Sampling Plan: Define n (sample size) and c (acceptance number) with concrete example; sketch OC curve showing P(A) vs p with points at p=0, p=AQL, p=LTPD, p=1
- Assignment Problem: Apply Hungarian method—row reduction, column reduction, minimum lines to cover zeros, optimality check, and final assignment with minimum cost
- Game Theory: Verify no saddle point exists, formulate as LPP or use algebraic method for 2×2 subgames, find mixed strategy probabilities and game value
- Markov Chain: Compute P² to show period d=2, verify irreducibility, calculate stationary distribution to confirm persistent non-null states
- Reliability: Derive R(t) = exp(-∫h(u)du) giving R(t)=exp(-0.015t) for t≤200 and R(t)=exp(-3)exp(-0.025(t-200)) for t>200 with continuity at t=200

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies parameters for each problem: (n,c) for sampling plan, balanced assignment matrix, game payoff matrix structure, transition matrix properties, and piecewise hazard function with proper initial condition R(0)=1	Identifies most parameters but misses subtle requirements like continuity condition for reliability or periodicity definition for Markov chain	Incorrect parameter identification, wrong matrix dimensions, or fundamental misunderstanding of problem setup such as treating game as pure strategy only
Method choice	20%	10	Selects optimal methods: OC curve construction, Hungarian algorithm for assignment, algebraic/LPP method for game theory, Chapman-Kolmogorov for periodicity, and integration of hazard for reliability; justifies method selection	Uses correct but inefficient methods or applies standard methods without adaptation (e.g., trial and error for assignment instead of Hungarian)	Chooses inappropriate methods such as dominance rule for game when algebraic method is demanded, or uses simulation instead of analytical solution
Computation accuracy	20%	10	All calculations error-free: correct minimum assignment cost, accurate mixed strategy probabilities (p,q) summing to 1, precise stationary distribution π=(¼,½,¼), and exact reliability expressions with proper exponential constants	Minor arithmetic errors in one or two problems such as wrong game value or incorrect reliability constant, but methodologically sound	Major computational errors: incorrect determinant calculations, wrong probability normalization, or failure to maintain R(t) continuity at t=200
Interpretation	20%	10	Interprets OC curve's producer/consumer risk trade-off, explains assignment's practical resource allocation meaning, interprets game value as fair payoff, explains periodicity's real-world implication (e.g., seasonal patterns), and relates reliability to component lifecycle	Provides basic interpretation for most problems but misses deeper insights like economic interpretation of sampling risks or physical meaning of hazard rate change at t=200	No interpretation provided or completely misinterprets results such as confusing probability of acceptance with probability of rejection
Final answer & units	20%	10	All five answers clearly stated with proper units: sampling plan parameters (n,c), assignment minimum cost in rupees/units, game value V with strategy probabilities, period d=2 with persistence classification, and R(t) as piecewise function with time units	Answers present but units missing or inconsistent, or final answers buried in working without clear highlighting	Missing final answers, incorrect units (e.g., probability >1), or answers stated without supporting derivation making verification impossible

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