--- title: "Decision tree for plant improvement alternatives" subject: "Engineering Economics" chapter: 6 key: "semester_6/engineering_economics:decision-tree-plant-alternatives" source: https://www.bctengineer.com/q/semester_6/engineering_economics/decision-tree-plant-alternatives site: BCT Engineering --- # Decision tree for plant improvement alternatives Engineering Economics · Chapter 6: Risk Analysis - Asked in: 2076 Shrawan, 6 marks; 2071 Bhadra, 6 marks (asked 2 times) - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/engineering_economics/decision-tree-plant-alternatives - Whole chapter: https://www.bctengineer.com/md/semester_6/engineering_economics/chapter-6 ## Question For the improvement of a manufacturing plant, the following three alternatives are being considered. The estimated investments and the corresponding increment in income are given below. Draw decision tree diagram of the situation and decide on the best alternative using FW formulation. MARR = 15%. Life of the project is 6 years. | Alternative | Investment Cost | Sales | Probability | Annual Income | |---|---|---|---|---| | A | 1000000 | High Success | 0.4 | 500000 | | A | 1000000 | Medium Success | 0.5 | 300000 | | A | 1000000 | Low Success | 0.1 | 125500 | | B | 600000 | High Success | 0.2 | 400000 | | B | 600000 | Medium Success | 0.5 | 250000 | | B | 600000 | Low Success | 0.3 | 100000 | | C | 400000 | High Success | 0.5 | 200000 | | C | 400000 | Medium Success | 0.1 | 125000 | | C | 400000 | Low Success | 0.4 | 50000 | ## Short answer Decision tree with three alternatives (A, B, C) and success probabilities evaluated using future worth at 15% for 6 years. Expected FW: A = Rs 860,607, B = Rs 669,292, C = Rs 234,646. Alternative A has the highest expected FW and should be chosen despite a 10% chance of a Rs 12.1 lakh loss. ## Answer FW at the end of year 6 for each branch = −I(F/P,15%,6) + A(F/A,15%,6), where (F/P,15%,6) = 2.3131 and (F/A,15%,6) = 8.7537. The best alternative is the one with the highest expected FW. #### Decision tree ```text ┌ High 0.4 A = 5,00,000 ┌ Alt A (10,00,000) ─○ Medium 0.5 A = 3,00,000 │ └ Low 0.1 A = 1,25,500 │ ┌ High 0.2 A = 4,00,000 □───┼ Alt B (6,00,000) ──○ Medium 0.5 A = 2,50,000 │ └ Low 0.3 A = 1,00,000 │ ┌ High 0.5 A = 2,00,000 └ Alt C (4,00,000) ──○ Medium 0.1 A = 1,25,000 └ Low 0.4 A = 50,000 □ = decision, ○ = chance (probability on the branch) ``` #### Expected FW of each alternative **Alternative A** (investment 1,000,000; FW of investment = 1,000,000 × 2.3131 = 2,313,061) | Sales | Probability | Annual income | FW = −I(F/P) + A(F/A) | p × FW | |---|---|---|---|---| | High | 0.4 | 500,000 | 2,063,808 | 825,523 | | Medium | 0.5 | 300,000 | 313,061 | 156,530 | | Low | 0.1 | 125,500 | -1,214,467 | -121,447 | E[FW(A)] = **860,607** **Alternative B** (investment 600,000; FW of investment = 600,000 × 2.3131 = 1,387,836) | Sales | Probability | Annual income | FW = −I(F/P) + A(F/A) | p × FW | |---|---|---|---|---| | High | 0.2 | 400,000 | 2,113,659 | 422,732 | | Medium | 0.5 | 250,000 | 800,598 | 400,299 | | Low | 0.3 | 100,000 | -512,463 | -153,739 | E[FW(B)] = **669,292** **Alternative C** (investment 400,000; FW of investment = 400,000 × 2.3131 = 925,224) | Sales | Probability | Annual income | FW = −I(F/P) + A(F/A) | p × FW | |---|---|---|---|---| | High | 0.5 | 200,000 | 825,523 | 412,762 | | Medium | 0.1 | 125,000 | 168,993 | 16,899 | | Low | 0.4 | 50,000 | -487,537 | -195,015 | E[FW(C)] = **234,646** #### Roll back at the decision node | Alternative | Expected FW (Rs) | |---|---| | A | 860,607 | | B | 669,292 | | C | 234,646 | **Answer: Choose alternative A.** It has the highest expected FW (Rs 860,607, equal to PW ≈ Rs 372,064), although it also has the largest downside: a low-success outcome gives FW ≈ −Rs 12.1 lakh with probability 0.1. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The FW at the end of year 6 for each branch is computed as: A. -I(P/F,15%,6) + A(P/A,15%,6) B. -I + A x 6 C. -I(F/P,15%,6) + A(F/A,15%,6) D. A(F/A,15%,6) only Answer: C. The investment is compounded forward with (F/P) and the annual income accumulated with (F/A). (Key point: FW formula per branch) 2. How is the expected FW of an alternative found? A. Add the three FW values and divide by 3 B. Take the FW of the high-success branch C. Take the lowest FW D. Sum of probability x FW over the high, medium and low outcomes Answer: D. At the chance node expected value E = sum p x FW is used because the probabilities are different. (Key point: Expected FW) 3. Which alternative has the highest expected FW and what is it? A. A, about Rs 860,607 B. B, about Rs 669,292 C. C, about Rs 234,646 D. All are equal Answer: A. E[FW] for A, B, C are 860,607, 669,292 and 234,646; A is highest. (Key point: Best alternative) 4. What is the downside of choosing alternative A? A. It has no chance of loss B. A low-success outcome (probability 0.1) gives FW of about -Rs 12.1 lakh C. Its expected FW is negative D. Its investment is the smallest Answer: B. A has the largest investment, so its low-success branch gives a large negative FW (about -1,214,467). 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