(a) A company manufactures 30 items per day. The sale of those items depends upon demand which has the following distribution :

| Sale (units) | 27 | 28 | 29 | 30 | 31 | 32 |
|-------------|----|----|----|----|----|----|
| Probability | 0·10 | 0·15

Question

(a) A company manufactures 30 items per day. The sale of those items depends upon demand which has the following distribution :

| Sale (units) | 27 | 28 | 29 | 30 | 31 | 32 |
|-------------|----|----|----|----|----|----|
| Probability | 0·10 | 0·15 | 0·20 | 0·35 | 0·15 | 0·05 |

The production cost and selling price of each unit are ₹ 400 and ₹ 500 respectively. Any unsold product is to be disposed off at a loss of ₹ 150 per unit. There is a penalty of ₹ 50 per unit if the demand is not met.

Use the following random numbers to estimate total profit/loss for the company for the next 10 days :

23, 99, 65, 99, 95, 01, 79, 11, 16, 10

If the company decides to produce 20 items per day, what is the advantage or disadvantage to the company? (15 marks)

(b) A company has four plants P₁, P₂, P₃ and P₄ from which it supplies to three markets M₁, M₂ and M₃. Determine the optimal transportation plan from the following data giving the plant to market shifting costs, quantities available at each plant and quantities required at each market :

| Market ↓ | P₁ | P₂ | P₃ | P₄ | Required at market |
|:---|:---:|:---:|:---:|:---:|:---:|
| M₁ | 19 | 14 | 23 | 11 | 11 |
| M₂ | 15 | 16 | 12 | 21 | 13 |
| M₃ | 30 | 25 | 16 | 39 | 19 |
| Available at plant | 6 | 10 | 12 | 15 | 43 |

(15 marks)

(c) On January 1 (this year), brands A, B and C of a commodity had 40, 40 and 20 percent of the market share. Basing upon a market research, it is compiled that brand A retains 90 percent of its customers, while gaining 5 percent of B's customers and 10 percent of C's customers. Brand B retains 85 percent of its customers, while gaining 5 percent of A's customers and 7 percent of C's customers. Brand C retains 83 percent of its customers and gains 5 percent of A's customers and 10 percent of B's customers. What will be each brand's share on January 1 (next year) and what will be each brand's share in the market at equilibrium? (20 marks)

UPSC Answer Check · Accepted Answer

Solve all three sub-parts systematically: for (a) set up the Monte Carlo simulation with correct random number intervals and profit/loss calculations; for (b) apply the transportation algorithm (VAM for IBFS then MODI/UV method for optimization); for (c) construct the transition probability matrix and compute next year's shares, then solve for steady-state equilibrium using πP = π. Allocate approximately 30% time to (a), 30% to (b), and 40% to (c) given the 20 marks weightage for part (c). Present all working clearly with tabular formats where appropriate.
- For (a): Correctly establish random number intervals for demand simulation (00-09→27, 10-24→28, 25-44→29, 45-79→30, 80-94→31, 95-99→32) and calculate profit/loss for each of 10 days using given random numbers
- For (a): Compute total profit for 10 days with production=30, then recompute for production=20 to compare advantage/disadvantage with clear numerical conclusion
- For (b): Obtain initial basic feasible solution using Vogel's Approximation Method (VAM) and verify degeneracy condition (m+n-1=6 basic cells)
- For (b): Apply MODI/UV method to test optimality and iterate if needed to reach optimal transportation schedule with minimum total cost
- For (c): Construct correct transition probability matrix from customer retention and switching data, then compute January 1 next year shares by matrix multiplication
- For (c): Set up and solve system of linear equations πA=π, πB=π, πC=π with πA+πB+πC=1 to find equilibrium market shares

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	For (a): Correct random number intervals mapped to demand distribution; for (b): Properly balanced transportation problem with dummy if needed, correct cost matrix; for (c): Accurate transition probability matrix with rows summing to 1	Minor errors in interval mapping or one/two transition probabilities incorrect, but overall structure recognizable	Fundamental errors in problem setup: wrong probability intervals, unbalanced transportation problem without adjustment, or major errors in transition matrix
Method choice	20%	10	For (a): Monte Carlo simulation correctly applied; for (b): VAM for IBFS followed by MODI/UV for optimality test; for (c): Matrix multiplication for next year, proper eigenvector approach for equilibrium	Correct method chosen but incomplete application (e.g., VAM used but optimality test missing, or equilibrium found by iteration without algebraic solution)	Wrong methods chosen (e.g., LCM/NWC instead of VAM, no optimality test, incorrect approach to equilibrium)
Computation accuracy	20%	10	All calculations accurate: correct demands from random numbers, precise profit/loss figures, correct IBFS and optimal cost, accurate matrix operations and equilibrium solution	Minor arithmetic errors (1-2 calculation mistakes) but method and approach correct	Multiple calculation errors affecting final answers, or systematic errors in applying algorithms
Interpretation	20%	10	Clear interpretation: for (a) explicit comparison showing advantage/disadvantage with ₹ difference; for (b) optimal allocation stated with economic insight; for (c) meaningful discussion of market dynamics and convergence to equilibrium	Numerical answers present but interpretation superficial or missing economic/business context	No interpretation provided, or incorrect interpretation of results (e.g., wrong conclusion about advantage/disadvantage)
Final answer & units	20%	10	All final answers clearly boxed/stated with proper units: total profit in ₹ for (a), minimum transportation cost in ₹ for (b), market share percentages for (c) next year and equilibrium	Answers present but units missing or inconsistent, or not clearly distinguished as final answers	Missing final answers, wrong units, or answers not identifiable from working

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