Statistics 2021 Paper II 50 marks Solve

Q4

(a) Solve the following linear programming problem: Maximize z = 3x₁ + 5x₂ subject to the constraints 3x₁ + 2x₂ ≤ 18 x₁ ≤ 4 x₂ ≤ 6 and x₁, x₂ ≥ 0. Discuss the change in Cⱼ on the optimality of the optimal basic feasible solution. (15 marks) (b) A manufacturer has to supply his customers with 600 units of his product per year. Shortages are not allowed and storage amounts to 60 paise per unit per year. The set up cost per run is Rs. 80. Find (i) economic order quantity (ii) optimum period of supply per optimum order and (iii) increase in the total cost associated with ordering 20 per cent more and 40% less. (15 marks) (c) A machine is set to deliver the packets of a given weight. Ten samples of size 5 each were examined and the following results were obtained: | Sample No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |---|---|---|---|---|---|---|---|---|---|---| | Mean | 43 | 49 | 37 | 44 | 45 | 37 | 51 | 46 | 43 | 47 | | Range | 5 | 6 | 5 | 7 | 7 | 4 | 8 | 6 | 4 | 6 | Use mean and range charts to check whether process is under control. (Given for n = 5, d₂ = 2·326 and d₃ = 0·864) (20 marks)

हिंदी में प्रश्न पढ़ें

(a) निम्नलिखित रैखिक प्रोग्रामन समस्या को हल कीजिए : अधिकतमीकरण z = 3x₁ + 5x₂ निम्न प्रतिबंधों के अंतर्गत 3x₁ + 2x₂ ≤ 18 x₁ ≤ 4 x₂ ≤ 6 और x₁, x₂ ≥ 0. इष्टतम आधारी सुसंगत हल के इष्टतमत्व पर Cⱼ में परिवर्तन का वर्णन कीजिए । (15 अंक) (b) एक निर्माता को अपने ग्राहकों को प्रति वर्ष अपने उत्पाद की 600 इकाइयों की आपूर्ति करनी पड़ती है। अपयाप्तता की अनुमति नहीं है और गोदाम-भाड़ा 60 पैसे प्रति इकाई प्रति वर्ष है। झोंका लागत प्रति दौर 80 रुपये है। प्राप्त कीजिए (i) आर्थिक आदेश मात्रा (इकोनोमिक ऑर्डर क्वांटिटी) (ii) प्रति इष्टतम आदेश की आपूर्ति की इष्टतम अवधि और (iii) 20 प्रतिशत अधिक और 40% कम आदेश करने से संबंधित कुल लागत में वृद्धि। (15 अंक) (c) एक मशीन को दिये गये वजन के पैकेट देने के लिए सेट किया गया है। प्रत्येक आमाप 5 के दस प्रतिदर्शों की जाँच की गई और निम्नलिखित परिणाम प्राप्त हुए : | प्रतिदर्श संख्या | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |---|---|---|---|---|---|---|---|---|---|---| | माध्य | 43 | 49 | 37 | 44 | 45 | 37 | 51 | 46 | 43 | 47 | | परिसर | 5 | 6 | 5 | 7 | 7 | 4 | 8 | 6 | 4 | 6 | माध्य और परिसर सांचित्रों का उपयोग करके जाँच कीजिए कि क्या प्रक्रम नियंत्रण में है ? (n = 5 के लिए दिया है, d₂ = 2·326 और d₃ = 0·864) (20 अंक)

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Directive word: Solve

This question asks you to solve. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

Begin with the directive 'solve' for part (a), applying the simplex method or graphical method to find the optimal solution, then analyze sensitivity of Cⱼ coefficients. For part (b), apply the EOQ formula with given parameters (D=600, C₀=₹80, Cₕ=₹0.60) and calculate percentage cost variations. For part (c), construct X̄ and R control charts using given constants d₂=2.326 and d₃=0.864, computing center lines and control limits to assess process stability. Allocate approximately 30% time to (a), 25% to (b), and 45% to (c) given their mark distribution and computational complexity.

Key points expected

  • Part (a): Correct identification of feasible region vertices (0,0), (4,0), (4,3), (2,6), (0,6) and optimal solution at (2,6) with Z=36
  • Part (a): Sensitivity analysis showing range for C₁ as [0, 7.5] and C₂ as [4, ∞) maintaining optimality of current basis
  • Part (b): EOQ calculation as √(2×600×80/0.6) = 400 units; optimum period = 400/600 × 12 = 8 months
  • Part (b): Total cost at EOQ = ₹240; cost at 480 units = ₹244 (1.67% increase); cost at 240 units = ₹264 (10% increase)
  • Part (c): Grand mean X̄̄ = 44.2, average range R̄ = 5.8; X̄ chart limits: 44.2 ± 3×5.8/(2.326×√5) = 44.2 ± 3.34
  • Part (c): R chart limits: UCL = 5.8×(1+3×0.864/2.326) = 12.27, LCL = 5.8×(1-3×0.864/2.326) = 0 (adjusted to 0)
  • Part (c): Correct conclusion that Sample 3 (mean=37) and Sample 7 (mean=51) fall outside X̄ control limits, indicating process is not under statistical control

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies all constraints for (a) including non-negativity; properly lists EOQ parameters D, C₀, Cₕ for (b); accurately transcribes sample data and identifies n=5, m=10 for (c)Minor errors in constraint identification or parameter listing; one incorrect data point transcription in (c)Missing critical constraints in (a), wrong parameter values in (b), or fundamental misunderstanding of sample structure in (c)
Method choice20%10Uses simplex method or complete graphical analysis for (a); applies standard EOQ model correctly for (b); uses proper 3-sigma control chart formulas with given d₂, d₃ constants for (c)Correct method chosen but incomplete execution; uses approximate methods instead of exact formulasWrong method selection (e.g., transportation algorithm for LP); ignores given constants in (c); uses wrong inventory model
Computation accuracy20%10All arithmetic precise: optimal Z=36, sensitivity ranges correct; EOQ=400 exact, cost comparisons accurate; control limits calculated to 2 decimal places, all sample evaluations correctMinor calculation errors (1-2 marks lost) in one part; correct method but arithmetic slips in final valuesMultiple computational errors across parts; wrong optimal solution in (a); EOQ formula misapplied; control limits completely wrong
Interpretation20%10Clear explanation of why (2,6) is optimal in (a); explains economic meaning of EOQ and interprets cost sensitivity in (b); correctly identifies out-of-control samples with specific reasons (Sample 3 below LCL, Sample 7 above UCL) in (c)States results without full interpretation; identifies out-of-control points but lacks specific reasoningNo interpretation of results; fails to identify process status in (c); no economic insight in (b); no sensitivity discussion in (a)
Final answer & units20%10All answers explicitly stated with proper units: ₹ for costs, units for EOQ, months for period, clear out-of-control conclusion; sensitivity ranges clearly specified for (a)Correct answers but missing some units; incomplete final statementsMissing final answers; no units; incomplete parts; no conclusion on process control status

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