Question

Problem 2. NASA In designing a new space vehicle, NASA needs to decide whether to provide 0, 1, or 2 backup systems for a critical component of the vehicle. The first backup system, if included, comes into use only if the original system fails. The second backup system, if included, comes into use only if the original system and the first backup system both fail. NASA engineers claim that each system, independently of the others, has 1% chance of failing if called into use. Each backup system costs $70,000 to produce and install within the vehicle. Once the vehicle is in flight, the mission will be scrubbed only if all systems (the original and backups) fail. The cost of a scrubbed mission, in addition to production cost, is assessed to be $8,000,000. Create a decision tree and identify the strategy that minimizes NASA's expected total cost.

Answer 1

Nasa has to decide between 3 choice

1. Have 0 backup systems
   1. Fail with probability 0.01 and incur $8,000,000 additional cost (in addition to production cost)
   2. success with probability 0.99 and incur $0 additional cost (in addition to production cost)
2. Have 1 backup systems
   1. Original Fail with probability 0.01
      1. Backup fails with probability 0.01 and incur $8,000,000+70,000 additional cost (in addition to production cost)
      2. Backup succeeds with probability 0.99 and incur $70,000 additional cost (in addition to production cost)
   2. success with probability 0.99 and incur $70,000 additional cost (in addition to production cost)
3. Have 2 backup systems (Cost of 2 backup systems $140,000)
   1. Original Fail with probability 0.01
      1. Backup fails with probability 0.01
         1. The second backup fails with probability 0.01 and incur $8,000,000+140,000 additional cost (in addition to production cost)
         2. The second backup succeeds with probability 0.99 and incur $140,000 additional cost (in addition to production cost)
      2. Backup succeeds with probability 0.99 and incur $140,000 additional cost (in addition to production cost)
   2. success with probability 0.99 and incur $140,000 additional cost (in addition to production cost)

The decision tree is below

Original fail with p(f)-0.01 EV-$80,000 Additional Cost-$8,000,000 0 backu systems 2 Original Succeed with probabili P(s) 0.99 First backup fail Additional Cost-$8,070,000 Additional Costcost-$o p(f)-0.01 Original fail with EV-$150,000 p(f) 0.01 First backup succeed with probability p(s)-0.99 1 backup system EV-$70,800 Additional Costcost $70,000 Original Succeed with probability Additional Costcost $70,000 second backup fail Original Succeed with probability p SF0.99 Additional Costcost $140,000 First backup fail p(f) 0.01 Additional Costcost-$8,140,000 p(f)-0.01 EV-$140,008 2 backup EV-$220,000 4 7 second backup succeed with probability P(s)-0.99 Original fail with probability p(f)-0.01 Additional Costcost $140,000 EV $140,800 First backup succeed with probability p(s)-0.99 6 Additional Costcost-$140,000

Moving from the right to the left

Chance node 7: 2 backup systems

The expected cost is

EV (7) Probability of failure of 2nd backup × Cost of failure + Probability of success of 2nd backup = 0.01 × 8140000+0.99 × 140000 = $220,000 Cost of success

chance node: 6: 2 backup systems

The expected cost is

Probability of failure of 1st backup × expected cost of failure of 1st backup + Probability of success of 1st backup × Cost of success = 0.01 × 220000+0.99× 140000 = $140,800 EV 6

chance node 5: 1 backup system

The expected cost is

EV(5) Probability of failure of 1st backup x Cost of failure of 1st backup + Probability of success of 1st backup x Cost of success = 0.01 × 8070000 + 0.99 × 70000 $150, 000

chance node 2: 0 backups

The expected cost is

EV 2 Probability of failure of original × Cost of failụre of original + Probability of success of original x Cost of success 0.01 × 8000000 + 0.99 × 0 = $80、000

chance node 3: 1 backup

The expected cost is

EN 3 Probability of failure of original × expected Cost of failure of original Probability of success of original x Cost of success = 0.01 × 150000+0.99 × 70000 $70,800

chance node 4:

The expected cost is

EN 4 Probability of failure of original × expected Cost of failure of original Probability of success of original x Cost of success = 0.01 × 140800+0.99 × 140000 $140, 008

At the decision node 1 NASA has 3 options

1. 0 backup systems at an expected total cost of $80,000+Product cost
2. 1 backup system at an expected total cost of $70,800+Product cost
3. 2 backup system sat an expected total cost of $140,008+Product cost

The lowest expected total cost is when NASA goes for 1 backup systems

ans: The strategy that minimizes NASA's expected total cost is to go for 1 backup system.