Statistics 2023 Paper II 50 marks Solve

Q4

(a) A Company ships truckloads of grain from three silos to four mills. The supply (in truckloads) and the demand (also in truckloads) together with the unit transportation costs per truckload on the different routes are summarized in the following table : Purpose is to find the minimum-cost shipping schedule between the silos and the mills. Use any method. Obtain the starting basic feasible solution. (b)(i) Suppose that the life in hours of an electric Gadget manufactured by a certain process is normally distributed with parameters μ = 160 hours and some σ. What would be the maximum allowable value of σ if the life X of the gadget is to have a probability 0.80 of being between 120 hours and 200 hours ? (Normal distribution Table is given at the end). (b)(ii) Let the compressive strength X of concrete be log-normally distributed with parameters μY = 3 MPa and σY = 0.2 MPa where Y = logeX. What is the probability that the strength is less than or equal to 10 MPa ? (Normal distribution Table is given at the end) (c) A departmental store operates with three checkout counters. To determine the number of counters in operation based on the number of customers, the manager uses the following schedule : | Number of customers in store | Number of customers in operation | |---|---| | 1 to 3 | 1 | | 4 to 6 | 2 | | More than 6 | 3 | Customers arrive in the counter(s) according to a Poisson distribution with a mean rate of 10 customers/hour. The average checkout time per customer is exponential with mean 12 minutes. Determine the steady state probability pn of n customers in the checkout area.

हिंदी में प्रश्न पढ़ें

(a) एक कंपनी अनाज से भरे ट्रकों को 3 भूमिगत कक्षों (सिलोस) से 4 फैक्ट्रियों (मिल्स) को जहाजों से भेजती है। आपूर्ति (भरे ट्रकों में) और मांग (भी भरे ट्रकों में), विभिन्न मांगों पर इकाई परिवहन लागत प्रति भरा ट्रक के साथ, निम्नलिखित सारणी में संक्षेप में दिये गये हैं : उद्देश्य यह है कि न्यूनतम लागत शिपिंग अनुसूची भूमिगत कक्षों (सिलोस) और फैक्ट्रियों (मिल्स) के बीच में ज्ञात कीजिए। कोई भी विधि का उपयोग करें। प्रारंभिक आधारी सुसंगत हल प्राप्त कीजिए। (b)(i) मान लीजिए कि एक निश्चित प्रक्रिया द्वारा निर्मित एक इलेक्ट्रिक गैजेट का जीवनकाल (घंटों में) प्रसामान्यतः बंटित है, जिसके प्राचल μ = 160 घंटे और σ कोई एक मान है। σ का अधिकतम स्वीकार्य मान क्या होगा यदि गैजेट के जीवनकाल X के 120 घंटे और 200 घंटे के बीच होने की प्रायिकता 0.80 है ? (प्रसामान्य बंटन सारणी पृष्ठ के अंत में दी गई है) (b)(ii) मान लीजिए X कंक्रीट की संपीडक शक्ति है जो लघुगणकीय प्रसामान्यतः बंटित है जिसके प्राचल μY = 3 MPa और σY = 0.2 MPa है जबकि Y = logeX है। क्षमता (शक्ति) 10 MPa से कम हो या इसके बराबर हो की प्रायिकता क्या है ? (प्रसामान्य बंटन के लिये सारणी आखरी पृष्ठ में दी है) (c) एक डिपार्टमेंटल स्टोर तीन चेकआउट काउंटरों के साथ संचालित होता है। ग्राहकों की संख्या के आधार पर संचालन में काउंटरों की संख्या निर्धारित करने के लिए, प्रबंधक निम्नलिखित अनुसूची का उपयोग करता है : | भंडार में ग्राहकों की संख्या | संचालन में ग्राहकों की संख्या | |---|---| | 1 से 3 | 1 | | 4 से 6 | 2 | | 6 से अधिक | 3 | प्वासों बंटन के अनुसार ग्राहक काउंटर पर पहुंचते हैं जिसका माध्य दर 10 ग्राहक प्रति घंटा है। औसत चेकआउट समय प्रति ग्राहक एक चरघातीय बंटन है जिसका माध्य 12 मिनट है। चेक आउट क्षेत्र में n ग्राहकों की स्थायी अवस्था प्रायिकता pn ज्ञात कीजिए।

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Approach

Solve this multi-part numerical problem by allocating approximately 40% time to part (a) transportation problem as it requires complete solution methodology, 35% to part (b) probability calculations involving normal and log-normal distributions, and 25% to part (c) queuing theory steady-state probabilities. Begin with clear problem setup for each part, show all computational steps with proper formulae, and conclude with interpreted final answers in correct units.

Key points expected

  • For (a): Correctly set up the balanced transportation problem (check if supply equals demand, add dummy if needed), apply Vogel's Approximation Method or Least Cost Method to obtain degenerate/non-degenerate basic feasible solution with (m+n-1) allocations
  • For (b)(i): Set up P(120 < X < 200) = 0.80, convert to standard normal Z-scores, use symmetry property to find z₀ such that P(-z₀ < Z < z₀) = 0.80, hence Φ(z₀) = 0.90, interpolate from table to find z₀ ≈ 1.28, then solve σ = 40/1.28 = 31.25 hours
  • For (b)(ii): Transform log-normal to normal: P(X ≤ 10) = P(Y ≤ ln10) = P(Y ≤ 2.3026), calculate Z = (2.3026-3)/0.2 = -3.487, use table to find Φ(-3.49) ≈ 0.0002 or precise interpolation
  • For (c): Identify this as M/M/3 queuing system with state-dependent service rates (μ, 2μ, 3μ for n=1,2,3+), use λ=10/hr, μ=5/hr, ρ=λ/3μ=2/3, apply birth-death process balance equations for steady-state probabilities p₀, p₁, p₂, and general formula for pₙ when n≥3
  • Verify all calculations: check transportation cost arithmetic, confirm normal table reading with interpolation, validate queuing traffic intensity ρ < 1 for steady state existence, and ensure probability sum equals 1

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%8For (a): correctly balances the transportation problem (adds dummy mill if supply≠demand) with proper cost matrix; for (b): accurately states normal/log-normal parameters and probability statements; for (c): correctly identifies M/M/3 system with state-dependent service rates, λ=10/hr, μ=5/hr, and ρ=2/3Sets up most problems correctly but misses balancing in (a) or makes minor parameter errors in (b)-(c) like wrong μ conversion or incorrect system identificationMajor setup errors: unbalanced transportation without dummy, wrong distribution parameters, confuses M/M/1 with M/M/3, or incorrect arrival/service rate units
Method choice20%8Uses VAM or modified distribution method for (a) with clear stepping stone/UV method potential; applies standard normal transformation correctly in (b); uses proper birth-death process with detailed balance equations for (c) including recursive probability relationsUses North-West Corner or Least Cost for (a); correct standardization in (b) but may miss symmetry shortcut; uses queuing formula directly without deriving balance equationsRandom allocation in (a), no standardization in (b), applies wrong queuing model (M/M/1 or M/M/∞) or uses wrong formula for state-dependent rates
Computation accuracy20%8Accurate arithmetic throughout: correct penalty calculations in VAM, precise Z-score computation with table interpolation (z₀≈1.2816), exact log transformation (ln10=2.302585), correct steady-state probability expressions with p₀ calculation involving series summationMinor calculation errors in one part (e.g., rounding Z to 1.3, approximate ln10 as 2.3), or arithmetic slips in transportation cost but method correctSignificant computational errors: wrong penalty differences, incorrect table reading (using Φ(z)=0.80 instead of 0.90), wrong Z formula, or algebraic errors in probability expressions
Interpretation20%8Interprets σ_max=31.25 hours as quality control threshold for manufacturing; explains pₙ as probability of n customers waiting which informs managerial staffing decisions; discusses economic implications of transportation solution; validates that ρ<1 ensures system stabilityStates final answers with minimal interpretation, may mention 'probability of waiting' without managerial insight, or gives numerical answers without contextNo interpretation of results, or misinterprets probabilities (e.g., confuses pₙ with waiting time distribution), fails to validate reasonableness of answers
Final answer & units20%8All answers clearly boxed with proper units: transportation schedule in truckloads with total cost in ₹/appropriate currency; σ in hours; probability as dimensionless 0-1 value or percentage; steady-state probabilities summing to 1 with p₀ explicitly stated; includes verification stepsCorrect numerical answers but inconsistent units (e.g., minutes vs hours), or missing units on some parts, or probabilities not simplifiedMissing or wrong units (σ without units, probabilities >1), incomplete answers (only p₀ for queuing), no verification that probabilities sum to unity

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