Statistics 2022 Paper II 50 marks Solve

Q4

(a) Use penalty method to solve the following linear programming problem : Maximize Z = x₁ + 2x₂ + 3x₃ - x₄ subject to the constraints x₁ + 2x₂ + 3x₃ = 15 2x₁ + x₂ + 5x₃ = 20 x₁ + 2x₂ + x₃ + x₄ = 10 x₁, x₂, x₃, x₄ ≥ 0 (20 marks) (b) An airline that operates seven days a week has the time-table shown below. Crew must have a minimum layover of 5 hours between flights. Obtain the pairing of flights that minimizes layover time away from home. For any given pairing, crew will be based at the city that results in the smaller layover : Delhi-Jaipur Flight No. | Departure | Arrival 1 | 7:00 AM | 8:00 AM 2 | 8:00 AM | 9:00 AM 3 | 1:30 PM | 2:30 PM 4 | 6:30 PM | 7:30 PM Jaipur-Delhi Flight No. | Departure | Arrival 101 | 8:00 AM | 9:15 AM 102 | 8:30 AM | 9:45 AM 103 | 12 Noon | 1:15 PM 104 | 5:30 PM | 6:45 PM For each pair, also mention the city where the crew should be based. (15 marks) (c) What are sequential sampling plans? Suggest a sequential sampling plan for which p₁ = 0·01, α = 0·05, p₂ = 0·06 and β = 0·10. (15 marks)

हिंदी में प्रश्न पढ़ें

(a) निम्नलिखित रैखिक प्रोग्रामन समस्या का हल दण्ड विधि का प्रयोग करके निकालिए : अधिकतमीकरण Z = x₁ + 2x₂ + 3x₃ - x₄ निम्न प्रतिबंधों के अंतर्गत x₁ + 2x₂ + 3x₃ = 15 2x₁ + x₂ + 5x₃ = 20 x₁ + 2x₂ + x₃ + x₄ = 10 x₁, x₂, x₃, x₄ ≥ 0 (20 अंक) (b) एक एयरलाइन जो सप्ताह में सातों दिन परिचालन करती है, उसकी समय-सारणी नीचे दर्शाई गई है। चालक-दल को उड़ानों के बीच कम-से-कम 5 घंटे का विश्रामकाल लेना चाहिए। उन उड़ानों की जोड़ी प्राप्त कीजिए जिनमें घर से दूर विश्रामकाल का समय न्यूनतम हो। किसी भी दी गई जोड़ी के लिए चालक-दल उस शहर पर आधारित होगा जहाँ विश्रामकाल कम होगा : दिल्ली-जयपुर उड़ान सं० | प्रस्थान | आगमन 1 | 7:00 AM | 8:00 AM 2 | 8:00 AM | 9:00 AM 3 | 1:30 PM | 2:30 PM 4 | 6:30 PM | 7:30 PM जयपुर-दिल्ली उड़ान सं० | प्रस्थान | आगमन 101 | 8:00 AM | 9:15 AM 102 | 8:30 AM | 9:45 AM 103 | 12 मध्याह्न | 1:15 PM 104 | 5:30 PM | 6:45 PM प्रत्येक जोड़ी के लिए उस शहर का भी उल्लेख कीजिए जहाँ चालक-दल को आधारित होना चाहिए। (15 अंक) (c) अनुक्रमिक प्रतिचयन आयोजनाएं क्या हैं? एक अनुक्रमिक प्रतिचयन आयोजना सुझाइए, जिसके लिए p₁ = 0·01, α = 0·05, p₂ = 0·06 और β = 0·10 हो। (15 अंक)

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Directive word: Solve

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How this answer will be evaluated

Approach

Begin with the directive 'solve' for part (a), applying the Big-M penalty method to convert equality constraints and maximize the objective. Allocate approximately 40% of time to part (a) given its 20 marks, 30% to part (b) for crew scheduling optimization, and 30% to part (c) for sequential sampling theory and design. Structure as: (a) complete LP solution with simplex iterations, (b) layover time matrix and optimal pairing, (c) definition followed by ASN and OC curve construction.

Key points expected

  • Part (a): Convert to standard form using Big-M penalty for equality constraints; introduce artificial variables A₁, A₂ with -M coefficient in objective
  • Part (a): Execute simplex iterations showing entering and leaving variables until optimality reached with Z_max = 15, x₁=2.5, x₂=5, x₃=2.5, x₄=0
  • Part (b): Construct 4×4 layover time matrix for Delhi-based and Jaipur-based crews; calculate layovers respecting 5-hour minimum
  • Part (b): Identify optimal pairings minimizing total layover: Flight 1-101 (Delhi base), 2-102 (Delhi), 3-103 (Jaipur), 4-104 (Jaipur)
  • Part (c): Define sequential sampling as item-by-item inspection with decision boundaries; state Wald's SPRT principles
  • Part (c): Calculate decision parameters h₁, h₂, s and construct acceptance/rejection lines; provide ASN ~ 40-50 and OC curve characteristics

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10For (a): correctly introduces artificial variables A₁, A₂ with -M penalty and slack for constraint 3; for (b): properly formats 24-hour clock times and builds complete layover matrix; for (c): accurately defines sequential sampling with ASN, OC function, and risk parametersSets up most variables correctly but misses one artificial variable or has minor time conversion errors; defines sequential sampling vaguely without mathematical structureFails to identify need for artificial variables, uses wrong penalty sign, or completely misinterprets layover calculation; no clear definition of sequential sampling
Method choice20%10For (a): applies Big-M method systematically with correct simplex tableau format; for (b): uses assignment problem/hungarian method or exhaustive comparison for optimal pairing; for (c): applies Wald's SPRT with correct boundary formulas and sampling plan designUses correct broad method but with inefficient calculations; attempts pairing without systematic approach; states SPRT but applies incorrect formulasUses wrong method (e.g., two-phase simplex when penalty asked, or graphical method); random pairing without optimization logic; confuses sequential with double sampling
Computation accuracy20%10For (a): all simplex iterations accurate with correct pivot operations, Z_max = 15; for (b): layover times accurate to minutes, total minimized layover identified; for (c): correct h₁ = 1.176, h₂ = 1.512, s = 0.0285 with proper ASN calculationMinor arithmetic errors in one or two iterations; layover calculations mostly correct but one pairing error; approximate boundary values with calculation mistakesMajor computational errors leading to wrong optimal solution; fundamentally incorrect layover times (e.g., ignoring 5-hour minimum); completely wrong sampling parameters
Interpretation20%10For (a): interprets final tableau showing all artificial variables eliminated and economic meaning; for (b): clearly states base city rationale comparing Delhi vs Jaipur layovers; for (c): interprets ASN as average sample size saved vs fixed sampling, and OC curve's discriminatory powerStates final answer without explaining why artificial variables must be zero; mentions base cities without clear comparison logic; limited interpretation of sampling efficiencyNo interpretation of why solution is optimal; no explanation of base city selection; no understanding of sequential sampling advantages or operating characteristics
Final answer & units20%10For (a): Z_max = 15, x₁ = 2.5, x₂ = 5, x₃ = 2.5, x₄ = 0 clearly boxed; for (b): four optimal pairings with base cities (Delhi/Jaipur) and total layover time; for (c): complete sampling plan with acceptance/rejection numbers at each stage n, and ASN ≈ 45Correct values but poorly formatted or missing some variables; pairings listed without base cities or total time; incomplete sampling plan missing some decision numbersMissing final answers, wrong values, or no units where applicable (hours for layover); incomplete or no sampling plan structure

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