(a) What do you understand by control chart for fraction defective? Explain its construction. Give the theoretical distribution on which the control limits are based. (15 marks)

(b) Each day a sample of 50 items from the production process was exami

Question

(a) What do you understand by control chart for fraction defective? Explain its construction. Give the theoretical distribution on which the control limits are based. (15 marks)

(b) Each day a sample of 50 items from the production process was examined. The number of defectives found in each sample was as follows:

| Day | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-----|---|---|---|---|---|---|---|---|---|----|----|----|
| No. of Defectives | 6 | 2 | 5 | 1 | 2 | 2 | 3 | 5 | 3 | 4 | 12 | 4 |

| Day | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
|-----|----|----|----|----|----|----|----|----|----|----|----|----|
| No. of Defectives | 4 | 1 | 3 | 5 | 4 | 1 | 4 | 3 | 5 | 4 | 2 | 3 |

Draw a suitable control chart and check for control. What control limits would you suggest for subsequent use? (15 marks)

(c) A factory has 1000 bulbs installed. Cost of individual replacement is US $3 while cost of that of group replacement is US $1 per bulb respectively. It is decided to replace all the bulbs simultaneously at fixed interval and also to replace the individual bulbs that fall in between. Determine the optimum replacement policy. Failure probability are given below:

| Week | 1 | 2 | 3 | 4 | 5 |
|------|-----|------|------|------|------|
| Failure probability(p) | 0·10 | 0·25 | 0·50 | 0·70 | 1·00 | (20 marks)

UPSC Answer Check · Accepted Answer

Explain the theoretical foundations of p-charts in part (a), then solve the numerical problems in (b) and (c) with systematic working. Allocate approximately 25-30% time to (a) as it requires conceptual elaboration, 30-35% to (b) for control chart construction and interpretation, and 35-40% to (c) as it carries the highest marks and involves multi-step replacement policy optimization. Present calculations in tabular format where possible and conclude with clear managerial recommendations.
- Part (a): Definition of control chart for fraction defective (p-chart), construction steps using sample proportion p̂ = d/n, and identification of Binomial distribution as the theoretical basis with Normal approximation for large samples
- Part (b): Calculation of center line (CL = p̄), control limits UCL/LCL = p̄ ± 3√[p̄(1-p̄)/n], plotting of 24 sample points, identification of Day 11 as out-of-control, and revised limits after removing assignable cause
- Part (c): Computation of expected failures Np, N·q·p₂, etc., individual replacement cost, group replacement cost for each policy period, and determination of optimal replacement interval at minimum average cost per week
- Correct handling of variable control limits when sample sizes differ (though here n=50 constant), and recognition that p-chart is appropriate for attribute data with varying sample sizes
- Economic interpretation: trade-off between individual replacement flexibility and group replacement economies of scale, with explicit cost comparison across weeks 1-5

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	For (a): Correctly identifies p-chart purpose, states Binomial distribution with parameters n and p as theoretical basis, and notes Normal approximation validity. For (b): Properly sets n=50, calculates p̄=4/50=0.08, and establishes correct control limit formula. For (c): Correctly interprets failure probabilities as conditional, computes survival probabilities, and sets up cost structure with $3 individual and $1 group replacement costs.	Identifies p-chart and Binomial distribution but misses Normal approximation justification; calculates p̄ with minor arithmetic errors; in (c) confuses unconditional with conditional probabilities or misstates cost components.	Wrong chart type selected (e.g., c-chart or X̄-R chart); fundamental misunderstanding of Binomial basis; in (c) completely wrong problem setup with inverted costs or probability misinterpretation.
Method choice	20%	10	For (b): Uses standard 3-sigma p-chart limits with correct standard error formula √[p̄(1-p̄)/n]; applies Western Electric rules or clear point-by-point analysis for control assessment; computes revised limits after eliminating out-of-control points. For (c): Systematically computes expected failures per week using N·P(X≥t\|X>t-1), builds cost table comparing pure individual vs. group+individual strategies for each replacement interval.	Uses correct p-chart method but applies fixed limits incorrectly or skips revision step; in (c) computes expected failures but makes errors in conditional probability chain or cost aggregation method.	Wrong methodology: uses c-chart formulas for (b) or simple average for control limits; in (c) uses simple failure probability multiplication without proper conditional structure or ignores group replacement cost entirely.
Computation accuracy	20%	10	Precise calculations: (b) p̄ = 96/(24×50) = 0.08, UCL = 0.08 + 3√(0.08×0.92/50) = 0.08 + 0.115 = 0.195 (or 9.75 defectives), LCL = 0 (since negative), correctly identifies Day 11 (12/50=0.24) as above UCL; revised p̄ = 84/(23×50) = 0.073, new UCL ≈ 0.184. (c) Accurate expected failure calculations: Week 1: 1000×0.10=100, Week 2: 900×0.25=225, etc., with correct cost minimization at optimal week.	Minor arithmetic slips in control limit calculations (e.g., rounding errors in square roots) or in failure expectations; correct method but final numerical values slightly off; cost calculations mostly correct but aggregation errors.	Major computational errors: incorrect p̄ calculation, wrong standard error formula, misplotted points; in (c) fundamental errors in probability calculations leading to completely wrong expected failures and cost comparisons.
Interpretation	20%	10	Clear interpretation: (a) Explains why Binomial applies (independent Bernoulli trials, constant p) and when Normal approximation is valid (np>5, n(1-p)>5). (b) Correctly diagnoses Day 11 as assignable cause (special cause variation), concludes process was otherwise in control; explains rationale for revised limits. (c) Interprets cost trade-offs: group replacement reduces per-unit cost but may replace working bulbs; individual replacement avoids waste but has higher unit cost; identifies global minimum with managerial recommendation.	States control/loss of control without explaining why; mentions assignable cause vaguely; in (c) identifies minimum cost period but weak explanation of why costs behave as they do across weeks.	Misinterprets control status (e.g., claims process in control when Day 11 is clearly out); no interpretation of economic trade-offs in (c); purely mechanical answers without SQC or operations management insight.
Final answer & units	20%	10	Precise final answers: (a) Complete p-chart construction summary. (b) Neatly drawn control chart (sketch acceptable), explicit statement that process is out-of-control due to Day 11, revised limits p̄=0.073, UCL≈0.184, LCL=0 for future use. (c) Optimal replacement period clearly stated (typically Week 3 or 4 depending on exact calculation), with minimum average cost per week in $, and explicit policy recommendation: 'Replace all bulbs every X weeks, replacing individual failures in between.' All monetary values carry $ units.	Correct final answers but missing units or incomplete policy statement; revised limits calculated but not explicitly recommended; optimal period identified but cost figure missing or approximate.	Missing final answers for one or more parts; no clear recommendation; units omitted throughout; contradictory conclusions (e.g., states process in control despite evidence).

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