Training Systems Engineering Decision Analysis & Trade Studies
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Decision Analysis & Trade Studies

24 min Systems Engineering

Decision Analysis & Trade Studies

Systems engineers use quantitative methods to make informed decisions among alternatives. Trade studies evaluate options against weighted criteria.

Weighted Decision Matrix

For $m$ alternatives and $n$ criteria with weights $w_j$ and scores $s_{ij}$:

$$S_i = \sum_{j=1}^{n} w_j \cdot s_{ij}$$

The alternative with the highest $S_i$ is preferred. Weights must satisfy $\sum w_j = 1$.

Expected Monetary Value (EMV)

$$EMV = \sum_{i} P_i \times V_i$$

where $P_i$ = probability of outcome $i$, $V_i$ = value of that outcome. Choose the option with the highest EMV.

Analytic Hierarchy Process (AHP) — Consistency

Consistency ratio $CR = CI/RI$ where $CI = (\lambda_{\max} - n)/(n-1)$. Acceptable if $CR < 0.10$.

Example 1

Three designs evaluated on cost (weight 0.4), performance (0.35), and reliability (0.25). Scores (1–10):
Design A: 8, 6, 7
Design B: 5, 9, 8
Design C: 7, 7, 6

$S_A = 0.4(8) + 0.35(6) + 0.25(7) = 3.2 + 2.1 + 1.75 = 7.05$

$S_B = 0.4(5) + 0.35(9) + 0.25(8) = 2.0 + 3.15 + 2.0 = 7.15$

$S_C = 0.4(7) + 0.35(7) + 0.25(6) = 2.8 + 2.45 + 1.5 = 6.75$

Design B wins with score 7.15.

Example 2

A project can succeed (P = 0.7, profit = \$500K) or fail (P = 0.3, loss = \$200K). Find EMV.

$$EMV = 0.7(500{,}000) + 0.3(-200{,}000) = 350{,}000 - 60{,}000 = \$290{,}000$$

Since EMV > 0, proceed with the project.

Example 3

A sensitivity analysis: if the weight on cost changes from 0.4 to 0.5, does Design B still win?

$S_A = 0.5(8) + 0.3(6) + 0.2(7) = 4.0 + 1.8 + 1.4 = 7.2$

$S_B = 0.5(5) + 0.3(9) + 0.2(8) = 2.5 + 2.7 + 1.6 = 6.8$

Now Design A wins, showing the result is sensitive to the cost weight.

Practice Problems

1. Two options, 3 criteria: weights (0.5, 0.3, 0.2). Option X scores (9, 5, 7). Option Y scores (6, 8, 9). Which wins?
2. A lottery: 20% chance of \$1,000, 80% chance of \$0. Entry fee \$150. Should you play?
3. A company can invest \$1M. Success (60%, return \$3M) or failure (40%, return \$0.5M). Find EMV.
4. In #3, what is the expected profit (subtract the \$1M investment)?
5. Normalize the criteria weights: raw weights 5, 3, 2. Find normalized weights.
6. If all criteria are equally weighted for 4 criteria, what is each weight?
7. Decision tree: invest or don't invest. If invest, 70% chance of \$500K, 30% chance of -\$100K. If don't invest, guaranteed \$0. Find optimal decision.
8. Pugh matrix: a baseline and 3 alternatives are compared on 5 criteria with +, -, or S (same). Alt A: (+,+,S,-,+), Alt B: (-,+,+,S,-), Alt C: (+,+,+,-,-). Score each.
9. Cost-benefit ratio: project costs \$200K and delivers \$500K in benefits. Find the ratio.
10. Pareto efficiency: define when option A dominates option B.
11. Multi-attribute utility: $U = 0.6 \cdot u_{\text{cost}} + 0.4 \cdot u_{\text{perf}}$. If cost utility = 0.8 and performance utility = 0.7, find $U$.
12. Risk = probability × impact. Event A: P = 0.1, impact = \$500K. Event B: P = 0.4, impact = \$100K. Which has higher risk?
Show Answer Key

1. $S_X = 4.5+1.5+1.4 = 7.4$; $S_Y = 3.0+2.4+1.8 = 7.2$. Option X wins.

2. $EMV = 0.2(1{,}000) + 0.8(0) = \$200$. Profit $= 200-150 = \$50 > 0$; yes, play.

3. $EMV = 0.6(3{,}000{,}000) + 0.4(500{,}000) = 1{,}800{,}000 + 200{,}000 = \$2{,}000{,}000$

4. Profit $= 2{,}000{,}000 - 1{,}000{,}000 = \$1{,}000{,}000$

5. Sum $= 10$; weights: $0.5, 0.3, 0.2$

6. $1/4 = 0.25$ each

7. EMV(invest) $= 0.7(500K) + 0.3(-100K) = 350K - 30K = \$320K > \$0$. Invest.

8. Net scores: A $= 3(+) - 1(-) = +2$; B $= 2-2 = 0$; C $= 3-2 = +1$. Alt A is best.

9. $BCR = 500/200 = 2.5$

10. A dominates B if A is at least as good on all criteria and strictly better on at least one.

11. $U = 0.6(0.8) + 0.4(0.7) = 0.48 + 0.28 = 0.76$

12. Risk A $= 0.1 \times 500K = \$50K$; Risk B $= 0.4 \times 100K = \$40K$. Event A has higher risk.

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