Statistics 2023 Paper II 50 marks Solve

Q3

(a) Solve the following Linear Programming problem using Two Phase method : Maximize Z = 3x₁ - x₂ Subject to 2x₁ + x₂ ≥ 2, x₁ + 3x₂ ≤ 2, x₂ ≤ 4, x₁ ≥ 0, x₂ ≥ 0 (b)(i) Solve the above assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV. (b)(ii) Write down the dual for the given primal problem. Max Z = 6x₁ - 5x₂ + 7x₃ + x₄ Subject to 2x₁ + 4x₂ - x₃ + x₄ ≤ 4, x₁ - x₂ + 6x₃ + 7x₄ ≥ 5, 2x₁ + 2x₂ + 4x₃ + 5x₄ = 6, x₁ + 8x₂ + x₃ = 7; x₁ and x₄ unrestricted, x₂ ≥ 0, x₃ ≥ 0 (c) What is a basic Economic Order Quantity (EOQ) model in Inventory Control and state the assumption made. A Company estimates that it will sell 12000 units of its products for the forthcoming year. The ordering cost is ₹100 per order and the carrying cost per year is 20% of the purchase price per unit. The purchase price per unit is ₹50. Find (i) EOQ (ii) Number of orders per year (iii) Time between successive orders.

हिंदी में प्रश्न पढ़ें

(a) द्विप्रावस्था विधि का उपयोग करके निम्नलिखित रैखिक प्रोग्रामन समस्या को हल कीजिए : अधिकतमीकरण Z = 3x₁ - x₂ निम्न प्रतिबंधों के अंतर्गत 2x₁ + x₂ ≥ 2, x₁ + 3x₂ ≤ 2, x₂ ≤ 4, x₁ ≥ 0, x₂ ≥ 0 (b)(i) निम्नलिखित नियत समस्या को हल कीजिए । प्रत्येक मान मशीनों I, II, III और IV को कार्य A, B, C और D सौंपने की लागतों को दर्शाता है । (b)(ii) दी गई आद्य समस्या के लिए द्वैति लिखिए : अधिकतमीकरण Z = 6x₁ - 5x₂ + 7x₃ + x₄ निम्न प्रतिबंधों के अंतर्गत 2x₁ + 4x₂ - x₃ + x₄ ≤ 4, x₁ - x₂ + 6x₃ + 7x₄ ≥ 5, 2x₁ + 2x₂ + 4x₃ + 5x₄ = 6, x₁ + 8x₂ + x₃ = 7; x₁ और x₄ अप्रतिबंधित, x₂ ≥ 0, x₃ ≥ 0 (c) तालिका नियंत्रण में मूलभूत आर्थिक आदेश मात्रा (ई.ओ.क्यू.) मॉडल क्या है और इसमें ली गई अभिधारणा को बताइए । एक कंपनी का अनुमान है कि वह अपने उत्पादों की 12000 इकाइयाँ आगामी वर्ष में बेचेगी । आदेश लागत 100 रुपये प्रति आदेश है और प्रति वर्ष ले जाने की लागत खरीद मूल्य का 20 प्रतिशत प्रति इकाई है । खरीद मूल्य 50 रुपये प्रति इकाई है । ज्ञात कीजिए (i) आर्थिक आदेश मात्रा (ई.ओ.क्यू.) (ii) प्रति वर्ष आदेशों (ऑर्डरों) की संख्या (iii) क्रमिक आदेशों के बीच का समय ।

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Approach

Solve this multi-part numerical problem by allocating approximately 40% time to part (a) Two-Phase method as it requires extensive tableau iterations, 35% to part (c) EOQ calculations with clear formula application, and 25% to part (b) covering both assignment problem and dual formulation. Begin with clear problem identification for each part, show complete step-by-step working with proper tableaus for (a), cost matrix reduction for (b)(i), systematic dual conversion rules for (b)(ii), and standard EOQ model derivation followed by substitution for (c). Conclude each part with boxed final answers and appropriate units.

Key points expected

  • Part (a): Convert LPP to standard form by introducing surplus, slack and artificial variables; set up Phase I objective to minimize artificial variable; execute simplex iterations till feasibility; proceed to Phase II with original objective; identify optimal solution at x₁ = 2, x₂ = 0, Z = 6
  • Part (b)(i): Apply Hungarian algorithm to 4×4 cost matrix — row reduction, column reduction, minimum lines to cover zeros, adjust matrix, make optimal assignments; state final assignment with minimum total cost
  • Part (b)(ii): Convert primal to dual by transforming maximization to minimization, reversing inequality signs for ≥ constraints, handling equality with unrestricted dual variables, and noting primal unrestricted variables become dual equality constraints
  • Part (c): Define EOQ as optimal order quantity minimizing total inventory cost; list assumptions: constant demand, instantaneous replenishment, no stockouts, fixed ordering cost, constant carrying cost percentage
  • Calculate EOQ = √(2×12000×100)/(0.20×50) = 490 units (or √480000 ≈ 693 if using 20% as 0.20 directly); number of orders = 12000/490 ≈ 24.5; time between orders = 365/24.5 ≈ 14.9 days

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%9Correctly identifies all variable types for (a): surplus for ≥, slack for ≤, artificial for Phase I; properly sets up 4×4 cost matrix for (b)(i) with clear row/column labels; accurately transforms all primal constraints to dual form handling unrestricted variables and equalities correctly; states all EOQ assumptions comprehensivelyMinor errors in variable identification such as missing surplus or artificial variables; incomplete cost matrix setup; some dual conversion errors like wrong inequality direction; partial list of EOQ assumptionsMajor setup errors: wrong method selected (e.g., Big-M instead of Two-Phase), incorrect dual formulation, missing constraints, or fundamentally wrong EOQ model identification
Method choice20%9Explicitly applies Two-Phase method with clear Phase I and Phase II separation; uses Hungarian algorithm with systematic row/column reduction for assignment; applies standard dual conversion rules with proper handling of unrestricted variables; uses classical EOQ formula with correct cost parameter identificationCorrect method chosen but with some procedural shortcuts or missing justification; Hungarian algorithm applied but with arithmetic shortcuts visible; dual conversion mostly correct but with inconsistent variable handling; EOQ formula correct but with parameter confusionWrong method entirely (e.g., graphical method for 2-variable LPP, transportation algorithm for assignment); incorrect dual approach; wrong inventory model selected
Computation accuracy20%9All simplex tableaus in (a) show correct pivot operations, accurate ratios, and proper basis updates leading to correct optimal solution; (b)(i) Hungarian steps show correct minimum subtractions and matrix adjustments; (b)(ii) dual coefficients correctly transcribed; (c) EOQ = 490 units (or 693 if 20% treated differently), orders = 24-25, cycle time ≈ 15 days with precise arithmeticOne or two arithmetic errors in tableaus that don't affect final answer; minor calculation errors in Hungarian algorithm covered by later corrections; small arithmetic slips in EOQ formula yielding approximately correct answersMultiple computational errors affecting feasibility or optimality; incorrect final answers due to systematic calculation mistakes; wrong formula application yielding nonsensical results
Interpretation20%9Correctly interprets Phase I termination (w=0 indicates feasibility) and Phase II optimality conditions; explains assignment pattern and economic meaning of minimum cost; interprets dual variables as shadow prices; explains EOQ trade-off between ordering and carrying costs with managerial implications for inventory policyStates final answers without explaining optimality conditions; mentions dual variables without economic interpretation; basic EOQ explanation without cost trade-off analysisNo interpretation of simplex indicators, no explanation of why solution is optimal, no managerial insights from EOQ results
Final answer & units20%9Clear boxed final answers: (a) x₁ = 2, x₂ = 0, Z_max = 6; (b)(i) specific A-I, B-II etc. assignment with total cost; (b)(ii) complete dual problem with all constraints; (c) EOQ = 490 units (or √480000), 24.5 orders/year, 14.9 days/₹ ordering cycle with all units specifiedCorrect final answers but poorly formatted or missing some units; incomplete dual statement; EOQ answer correct but missing time between orders or number of ordersMissing final answers, wrong values stated as final, no units throughout, or answers without any conclusion

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