(a) Distinguish between process control and product control. Explain the various sources of variation encountered in a process…

(a) Process and product control. Process control is the in-process use of statistical methods, especially control charts and process capability studies, to monitor quality characteristics while production is taking place. Its object is to keep the process in statistical control, detect departures early, and reduce variation before defective items are made. Product control is the examination of finished goods or lots after production, usually by acceptance sampling, to decide whether a lot meets specification. The property process control has that product control lacks is direct, real-time feedback to the production system; product control only accepts or rejects output and does not by itself correct the process. Thus process control is preventive and process-improving, while product control is detective and lot-disposition oriented. Variation in a process control study arises from chance causes and assignable causes. These sources are often summarised as man, machine, material, method, measurement and environment. Chance causes are numerous, small, random and inherent, such as minor fluctuations in temperature, humidity, operator attention and measurement noise; they cannot be traced to a single source and are reduced by redesigning the process, improving machines, tightening tolerances, standardising methods, better training and environmental control. Assignable or special causes are identifiable and avoidable, such as worn tooling, wrong material batch, machine misalignment, faulty measurement instrument, untrained operator or method change; they are detected through control-chart signals, Pareto diagrams, stratification and cause-and-effect analysis, and eliminated by immediate corrective action such as replacing material, recalibrating or repairing equipment, revising the method, retraining personnel and improving measurement systems.

(b) Simulation. Assign two-digit random numbers to cumulative probabilities. Selling price: 00–19 gives ₹3, 20–69 gives ₹4, 70–99 gives ₹5. Variable cost: 00–29 gives ₹1, 30–89 gives ₹2, 90–99 gives ₹3. Sales volume: 00–29 gives 2000, 30–59 gives 3000, 60–99 gives 5000. For each trial the first number gives price, second variable cost, third volume, and profit is (price − variable cost) × volume − 4000. Trial 1: 81,32,60 gives price 5, cost 2, volume 5000, profit 11000. Trial 2: 04,46,31 gives price 3, cost 2, volume 3000, profit −1000. Trial 3: 67,25,24 gives price 4, cost 1, volume 2000, profit 2000. Trial 4: 10,40,02 gives price 3, cost 2, volume 2000, profit −2000. Trial 5: 39,68,08 gives price 4, cost 2, volume 2000, profit 0. Trial 6: 59,66,90 gives price 4, cost 2, volume 5000, profit 6000. Trial 7: 12,64,79 gives price 3, cost 2, volume 5000, profit 1000. Trial 8: 31,86,68 gives price 4, cost 2, volume 5000, profit 6000. Trial 9: 82,89,25 gives price 5, cost 2, volume 2000, profit 2000. Trial 10: 11,98,16 gives price 3, cost 3, volume 2000, profit −4000. The total profit is 21000, so the simulated average profit is ₹2100.

(c) Conditional Poisson. Let N(t) be a Poisson process with rate λ, so N(t) has mean λt and independent increments. For s<t, and 0≤k≤n, P(N(s)=k | N(t)=n) = P(N(s)=k, N(t)−N(s)=n−k)/P(N(t)=n) = [e^−λs(λs)^k/k! × e^−λ(t−s)(λ(t−s))ⁿ−k/(n−k)!] / [e^−λt(λt)ⁿ/n!] = n!/[k!(n−k)!] (s/t)^k (1−s/t)ⁿ−k. Thus the conditional distribution is Binomial(n, s/t). The comment is that once the total number of events in [0,t] is fixed, the number occurring by time s depends only on the fraction of the interval, s/t, and not on the rate λ. Equivalently, the n event times are distributed as order statistics of n independent uniform random variables on [0,t], so the count before s is binomial.

(d) Game theory. The theory of games assumes two rational players, each with a finite set of pure strategies, who choose simultaneously. They have complete knowledge of the payoff matrix, no cooperation or collusion, and conflicting interests; in the two-person zero-sum case one player's gain is exactly the other's loss. The maximin principle is used by player A: A examines the minimum payoff in each row and chooses the row with the largest such minimum. This gives A's security level, the best guaranteed payoff. The minimax principle is used by player B: B examines the maximum payoff to A in each column and chooses the column with the smallest such maximum. This gives B's security level. If maximin equals minimax, the game has a saddle point and pure strategies are optimal. If not, the players use mixed strategies. For larger games the same idea is implemented through linear programming, but the 2×2 algebraic form shows the logic clearly. For a 2×2 game with payoff matrix (a11, a12; a21, a22), let A play row 1 with probability p and row 2 with 1−p. B's expected losses in columns 1 and 2 are p a11+(1−p)a21 and p a12+(1−p)a22. To make B indifferent, set them equal; this gives p=(a22−a21)/(a11−a12−a21+a22), and the value is v=p a11+(1−p)a21. Let B play column 1 with probability q. To make A indifferent between rows, q a11+(1−q)a12 = q a21+(1−q)a22, giving q=(a22−a12)/(a11−a12−a21+a22), and the same value follows. Thus the algebraic method solves the simultaneous indifference equations to obtain the optimal mixed probabilities and the value of the game.

(e) Censoring and exponential estimation. Censoring is important in life-testing experiments because complete failure data may require prohibitively long times and high cost, may be unethical in medical or safety tests, and may be unnecessary when only a few failures are needed to estimate reliability. It also allows study of heavy-tailed lifetimes where waiting for all failures is impractical. Type-II censoring is often preferred when a fixed number of failures is required for a reliable estimate; the censored observations contribute survival information, not just failure times. In type-II censoring, n items are put on test and the experiment stops at the r-th failure. The observed data are x_(1),...,x_(r) and n−r items censored at x_(r). For an exponential lifetime with mean θ, f(x)=θ⁻¹e^-x/θ, S(x)=e^-x/θ. The likelihood is L=θ^-r exp−[Σᵢ₌₁^r xᵢ+(n−r)x_(r)]/θ=θ^-re^−T/θ, where T is total time on test. The log-likelihood is −r log θ−T/θ. Differentiating gives −r/θ+T/θ²=0, so θhat=T/r. Since T=Σᵢ₌₁^r (n−i+1)(X_(i)−X_(i−1)) and each spacing has mean θ/(n−i+1), E(T)=rθ; hence θhat is unbiased. The spacings are independent with variance θ²/(n−i+1)², so Var(T)=rθ² and Var(θhat)=θ²/r. Thus type-II censoring makes life testing feasible while retaining an unbiased estimator of θ with variance θ²/r.