Statistics 2021 Paper II 50 marks Solve

Q3

(a) Explain M|G|1 queuing system. Obtain Pollaczek-kinchine formula. (15 marks) (b) Use MODI method to solve the above transportation problem: Store I II III IV A 4 6 8 13 B 13 11 10 8 C 14 4 10 13 D 9 11 13 8 Supply 50 70 30 50 Demand 25 35 105 20 (15 marks) (c) Use two-phase method to solve: Maximize z = 2x₁ + x₂ + x₃ subject to the constraints 4x₁ + 6x₂ + 3x₃ ≤ 8 3x₁ - 6x₂ - 4x₃ ≤ 1 2x₁ + 3x₂ - 5x₃ ≥ 4 and x₁, x₂, x₃ ≥ 0. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) पंक्ति प्रणाली M|G|1 की व्याख्या कीजिए । पोलेकजेक-किंचिन सूत्र को प्राप्त कीजिए । (15 अंक) (b) निम्नलिखित परिवहन समस्या का MODI विधि का उपयोग करके हल निकालिए : भंडार I II III IV A 4 6 8 13 B 13 11 10 8 C 14 4 10 13 D 9 11 13 8 पूर्ति 50 70 30 50 मांग 25 35 105 20 (15 अंक) (c) द्विप्रावस्था विधि का उपयोग करके हल कीजिए : अधिकतमीकरण z = 2x₁ + x₂ + x₃ निम्न प्रतिबंधों के अंतर्गत 4x₁ + 6x₂ + 3x₃ ≤ 8 3x₁ - 6x₂ - 4x₃ ≤ 1 2x₁ + 3x₂ - 5x₃ ≥ 4 और x₁, x₂, x₃ ≥ 0. (20 अंक)

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Approach

Solve this three-part numerical problem by allocating approximately 30% time to part (a) for deriving the Pollaczek-Khinchine formula, 30% to part (b) for the MODI method transportation problem, and 40% to part (c) for the two-phase simplex method. Begin each part with clear problem setup, show all computational steps systematically, and conclude with verified final answers. For (a), explain M|G|1 characteristics before derivation; for (b), ensure initial basic feasible solution before MODI optimization; for (c), complete Phase I before proceeding to Phase II.

Key points expected

  • Part (a): Correct explanation of M|G|1 queuing system components (Poisson arrivals, General service time, single server) and derivation of Pollaczek-Khinchine formula for mean queue length Lq = λ²E(S²)/[2(1-ρ)] or equivalent forms
  • Part (b): Correct initial basic feasible solution using VAM or NWCR method, followed by MODI (UV method) iterations with proper stepping stone paths until optimality is reached with minimum transportation cost
  • Part (c): Proper conversion of inequalities to equations using slack, surplus and artificial variables; successful completion of Phase I to eliminate artificial variables; Phase II optimization yielding optimal solution
  • Verification of supply-demand balance (total supply = total demand = 200) in part (b) before solving, and correct handling of ≥ constraint in part (c) with surplus and artificial variables
  • Clear presentation of all simplex tableaus for part (c) showing entering and leaving variables, pivot operations, and final optimal value of objective function

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10For (a): correctly defines M|G|1 notation, arrival rate λ, service time distribution, traffic intensity ρ=λ/μ; for (b): verifies supply=demand balance and sets up transportation tableau correctly; for (c): converts all constraints properly with slack, surplus and artificial variables as needed, standard form correctBasic setup present but minor errors in constraint conversion for (c) or missing verification step for (b); notation in (a) partially correctMajor setup errors: unbalanced problem not recognized in (b), wrong variable types for constraints in (c), or incorrect queuing notation in (a)
Method choice20%10For (a): uses correct embedded Markov chain or supplementary variable technique for PK formula; for (b): applies MODI method with proper stepping stone paths; for (c): correctly implements two-phase method with Phase I minimization of artificial variablesCorrect method chosen but implementation has gaps; or uses alternative acceptable method (like stepping stone instead of MODI) with correct executionWrong method selected: attempts graphical method for (c), uses wrong optimization technique for (b), or incorrect derivation approach for (a)
Computation accuracy20%10All calculations accurate: correct integration for E(S²) in (a), accurate u+v calculations and cost improvements in (b), correct simplex pivots and ratios in (c) with no arithmetic errorsMinor computational errors (sign errors, arithmetic slips) that don't fundamentally alter solution approach or final answer significantlyMajor computational errors: wrong pivot selection, incorrect opportunity cost calculations, or algebraic mistakes leading to wrong final answers
Interpretation20%10Interprets PK formula implications for system performance; explains economic significance of transportation allocation; identifies binding constraints and shadow prices in LP; discusses practical relevance of queuing resultsBasic interpretation of results present but lacks depth; mentions optimality conditions without explaining significanceNo interpretation provided; stops at numerical answers without explaining what results mean for the system or decision problem
Final answer & units20%10Clear final answers: PK formula stated explicitly with all terms defined; transportation cost with allocation matrix; optimal x₁, x₂, x₃ values with maximum z for (c); all units specified where applicableFinal answers present but poorly formatted or missing some components; units inconsistent or missingIncomplete final answers; missing optimal values, or wrong format (no allocation shown for transportation, no simplex final tableau for LP)

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