Paper I — Q6
(a) Five jobs A, B, C, D and E need to be processed on a machine. Processing time (in days) and due date (from now) are given…
Five jobs A, B, C, D and E need to be processed on a machine. Processing time (in days) and due date (from now) are given below :
| Jobs | Processing Time (in days) | Due Date (from now) |
|---|---|---|
| A | 6 | 5 |
| B | 4 | 10 |
| C | 5 | 15 |
| D | 8 | 20 |
| E | 7 | 30 |
Determine the total completion time (in days) for all jobs, average flow time, average number of jobs in system per day and average tardiness using Shortest Processing Time (SPT) rule so as to establish appropriate sequence for processing of jobs. 15 marks
Three cities namely Chennai, Delhi and Kolkata are being considered as potential locations for a new plant. Estimated data of annual fixed cost, variable cost and revenue per unit for each potential location are given below in the table. Determine the most attractive location for the plant if the estimated annual production volume desired is 40000 units. Also determine break-even production volume for each location. 15 marks
| Head | City | ||
|---|---|---|---|
| Chennai | Delhi | Kolkata | |
| Fixed cost (₹) | 800000 | 600000 | 500000 |
| Variable cost per unit (₹) | 30 | 40 | 50 |
| Revenue per unit (₹) | 60 | 60 | 60 |
Orthogonal turning of a metal is performed using single point cutting tool having rake angle of 10°. The turning operation produces chip-thickness ratio of 0·2, horizontal cutting force (F_H) of 1400 N and vertical cutting force (F_V) of 2000 N. Determine the shear plane angle, normal force, friction force on rake face and friction angle. 20 marks
हिंदी में प्रश्न पढ़ें
पाँच कृत्यों A, B, C, D व E का मशीन पर प्रक्रमण होना है । प्रक्रमण समय (दिनों में) व नियत तिथि (अभी से) निम्नवत हैं :
| कृत्यक | प्रक्रमण समय (दिनों में) | नियत तिथि (अभी से) |
|---|---|---|
| A | 6 | 5 |
| B | 4 | 10 |
| C | 5 | 15 |
| D | 8 | 20 |
| E | 7 | 30 |
सभी कृत्यों के लिए कुल समापन समय (दिनों में), औसत प्रवाह समय, प्रति दिन निकाय में कृत्यों की माध्य संख्या तथा न्यूनतम प्रसंस्करण समय (SPT) नियम से माध्य विलंबन ज्ञात कीजिए, जिससे कि कृत्यों के प्रक्रमण के लिए सटीक क्रम निर्धारित किया जा सके । (15 अंक)
तीन शहरों चेन्नई, दिल्ली व कोलकाता को एक नए संयंत्र के लिए संभावित लोकेशन माना जा रहा है । प्रत्येक संभावित लोकेशन के लिए वार्षिक स्थिर लागत, परिवर्त्य लागत तथा आय प्रति इकाई के अनुमानित आँकड़े नीचे सारणी में दिए गए हैं । संयंत्र के लिए अनुमानित 40000 इकाइयों की वार्षिक उत्पादन मात्रा के लिए सबसे आकर्षक लोकेशन का निर्धारण कीजिए । प्रत्येक लोकेशन के लिए उत्पादन की संतुलन स्तर मात्रा भी ज्ञात कीजिए । (15 अंक)
| शीर्ष | शहर | ||
|---|---|---|---|
| चेन्नई | दिल्ली | कोलकाता | |
| स्थिर लागत (₹) | 800000 | 600000 | 500000 |
| परिवर्त्य लागत प्रति इकाई (₹) | 30 | 40 | 50 |
| आय प्रति इकाई (₹) | 60 | 60 | 60 |
एक एकल बिंदु कतन औजार जिसका रेक कोण 10° है, से एक धातु का लंबकोणीय खरादन किया जाता है । खरादन प्रक्रम छीलन मोटाई अनुपात 0·2, क्षैतिज कतन बल (F_H) 1400 N तथा उद्वधार कतन बल (F_V) 2000 N उत्पन्न करता है । अपरूपण तल कोण, अभिलंब बल, रेक फलक पर घर्षण बल तथा घर्षण कोण ज्ञात कीजिए । (20 अंक)
Model answer
Drafted with AI by UPSC Answer Check against this question's marking rubric, to the expected length. UPSC does not publish answers for Mains — this is one way to score well, not an official key. How we build rubrics and answers.
(a) SPT sequencing of five jobs
SPT arranges jobs by increasing processing time: B (4), C (5), A (6), E (7), D (8). The sequence is B–C–A–E–D.
Completion (flow) times: B = 4, C = 9, A = 15, E = 22, D = 30 days.
Total completion time for all jobs = 30 days; the sum of flow times is 4 + 9 + 15 + 22 + 30 = 80 days.
Average flow time = 80/5 = 16 days.
Average number of jobs in the system = 80/30 = 2.67 jobs.
Tardiness = max(0, completion − due date): B = 0, C = 0, A = 15 − 5 = 10, E = 0, D = 30 − 20 = 10. Total = 20 days, so average tardiness = 20/5 = 4 days.
(b) Plant location by cost–volume analysis
Profit = (60 − v)Q − F, and break-even volume Q_BE = F/(60 − v).
Chennai: contribution ₹30 per unit; profit at 40 000 units = 30 × 40 000 − 8 00 000 = ₹4 00 000; Q_BE = 8 00 000/30 = 26 667 units.
Delhi: contribution ₹20; profit = 20 × 40 000 − 6 00 000 = ₹2 00 000; Q_BE = 6 00 000/20 = 30 000 units.
Kolkata: contribution ₹10; profit = 10 × 40 000 − 5 00 000 = −₹1 00 000 (a loss); Q_BE = 5 00 000/10 = 50 000 units.
Chennai is the most attractive location at 40 000 units: it has the highest profit and the lowest break-even volume. Kolkata, with the lowest fixed cost, loses money at this volume because its variable cost is highest.
(c) Orthogonal cutting forces
Take α = 10°, r = t₁/t₂ = 0.2, cutting force F_c = F_H = 1400 N and thrust force F_t = F_V = 2000 N.
Shear plane angle: tan φ = r cos α/(1 − r sin α) = 0.2 × 0.9848/(1 − 0.2 × 0.1736) = 0.1970/0.9653 = 0.2040, so φ = 11.5°.
Friction force on the rake face: F = F_c sin α + F_t cos α = 1400(0.1736) + 2000(0.9848) = 243.1 + 1969.6 = 2212.7 N.
Normal force on the rake face: N = F_c cos α − F_t sin α = 1400(0.9848) − 2000(0.1736) = 1378.7 − 347.3 = 1031.4 N.
Friction angle: β = tan⁻¹(F/N) = tan⁻¹(2.145) = 65.0°. Check: β = α + tan⁻¹(F_t/F_c) = 10° + 55.0° = 65.0°.
Answer: φ = 11.5°, N = 1031 N, F = 2213 N, β = 65.0°. The coefficient of friction is tan β = 2.15, which is high but follows from the data given.
How this answer will be evaluated
Framework: Mechanical Engineering, Paper 1
What earns full marks: Correct sequence/logic, all formulas stated, accurate calculations, clear diagrams/tables, and physical interpretation of results.
Evaluation rubric
Each sub-part is marked on its own, against the marks and word limit printed on the paper.
- (a) Sequence jobs by SPT and find total completion time, average flow time, jobs in system and average tardiness 15 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- SPT sequence B–C–A–E–D with completion times 4, 9, 15, 22, 30
- Average flow time 16 days and jobs in system 2.67
- Average tardiness 4 days
Loses marks
- Sorting by due date instead of processing time
- Dividing flow time by the wrong denominator
Earns more
- Total completion time 30 days and flow-time sum 80
- Tardiness computed job by job
Extra mark
- Notes SPT minimises mean flow time but may not minimise tardiness
- (b) Choose the best plant location at 40 000 units and find break-even volumes 15 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- Profits ₹4 00 000, ₹2 00 000 and −₹1 00 000 for Chennai, Delhi and Kolkata
- Break-even 26 667, 30 000 and 50 000 units
- Chennai selected with reasons
Loses marks
- Choosing by fixed cost alone
- Break-even formula with wrong denominator
Earns more
- Contribution per unit computed for each city
- Comparison at the stated volume
Extra mark
- Explains why low fixed cost does not make Kolkata best
- (c) Find the shear plane angle, normal force, friction force and friction angle in orthogonal cutting 20 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- Shear angle φ = 11.5° from tan φ = r cos α/(1 − r sin α)
- F = 2212.7 N and N = 1031.4 N
- Friction angle β = 65.0°
Loses marks
- Swapping F_H and F_V in the resolution
- Using the wrong shear-angle formula
Earns more
- Cutting and thrust forces resolved correctly on the rake face
- Cross-check β = α + tan⁻¹(F_t/F_c)
Extra mark
- Comments that tan β is high
What "Calculate" is asking you to do
Apply the standard formula or schedule to data the question has already supplied — a table of readings, cost records, a balance sheet — and produce the number. The method is rarely in doubt; the marks sit in the named intermediate quantities, each of which has to appear as a labelled line.
Structure that answers it
Data as given → formula or standard treatment, named → substitution → each intermediate, labelled → result with units
Where marks are lost
Omitting an intermediate the marking scheme pays for separately, or rounding at an intermediate line so the final figure drifts. In commerce and accountancy, any figure in a statement that no numbered working note supports is treated as unearned.
Practice this exact question
Write your answer and it is marked point by point against the model answer above — what you covered, what you missed, what you got wrong.
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