An automated algorithmic system updates a portfolio's Index (V) and Stabilization Pool (P)… — GMAT

An automated algorithmic system updates a portfolio's Index ( $V$ ) and Stabilization Pool ( $P$ ) across sequential stages $k = 1, 2, 3, 4$ .

Initial State (Stage 0): $V_0 = 100$ and $P_0 = 20$ .

For each Stage $k$ (where $k = 1, 2, 3, 4$ ):
1. Sub-stage A (Temporary Index Calculation): Calculate $V_k^{\text{temp}} = V_{k-1} + 0.5 \times P_{k-1}$ .
2. Sub-stage B (State Adjustment):
- If $V_k^{\text{temp}} \le 110$ : set $P_k = P_{k-1} + 15$ and final stage index $V_k = V_k^{\text{temp}} + 10$ .
- If $V_k^{\text{temp}} > 110$ : set $P_k = P_{k-1} - 10$ and final stage index $V_k = V_k^{\text{temp}} - 5$ .

Arrange the following four state transition events in chronological order from earliest to latest occurrence.

1The stabilization pool $P$ reaches its peak value of $35$ .
2The temporary index $V^{\text{temp}}$ first exceeds $130$ .
3The stabilization pool $P$ drops to $25$ .
4The final stage index $V$ reaches $140$ .

Answer

The correct chronological order of events is: (1) The stabilization pool

P

reaches its peak value of

35

, (2) The temporary index

V^{\text{temp}}

first exceeds

130

, (3) The stabilization pool

P

drops to

25

, and (4) The final stage index

V

reaches

140

Tracing the state transitions step-by-step reveals:
- At Sub-stage 1B,

P_1 = 35

, which is the peak value of

P

.
- At Sub-stage 2A,

V_2^{\text{temp}} = 137.5

, marking the first time

V^{\text{temp}}

exceeds

130

.
- At Sub-stage 2B,

P_2 = 25

.
- At Sub-stage 3B,

V_3 = 140.0

.

Thus, the event where the pool reaches

35

happens first (Sub-stage 1B), followed by

V^{\text{temp}}

exceeding

130

(Sub-stage 2A), then the pool dropping to

25

(Sub-stage 2B), and finally

V

reaching

140

(Sub-stage 3B).

Step-by-Step Solution

Evaluate Stage 1 state transitions

Sub-stage 1A:

V_1^{\text{temp}} = 100 + 0.5(20) = 110

. Since

V_1^{\text{temp}} \le 110

, Sub-stage 1B sets

P_1 = 20 + 15 = 35

and

V_1 = 110 + 10 = 120

This establishes that the event 'The stabilization pool

P

reaches its peak value of

35

' occurs at Sub-stage 1B.

Evaluate Stage 2 state transitions

Sub-stage 2A:

V_2^{\text{temp}} = 120 + 0.5(35) = 137.5

. Here,

V^{\text{temp}}

first exceeds

130

. Sub-stage 2B: Since

V_2^{\text{temp}} = 137.5 > 110

, set

P_2 = 35 - 10 = 25

and

V_2 = 137.5 - 5 = 132.5

This establishes that 'The temporary index

V^{\text{temp}}

first exceeds

130

' occurs at Sub-stage 2A, followed immediately by 'The stabilization pool

P

drops to

25

' at Sub-stage 2B.

Evaluate Stage 3 state transitions

Sub-stage 3A:

V_3^{\text{temp}} = 132.5 + 0.5(25) = 145.0

. Sub-stage 3B: Since

V_3^{\text{temp}} = 145.0 > 110

, set

P_3 = 25 - 10 = 15

and

V_3 = 145.0 - 5 = 140.0

This confirms that 'The final stage index

V

reaches

140

' occurs at Sub-stage 3B.

Synthesize chronological sequence of all four events

Sequence: Sub-stage 1B (

P = 35

)

\rightarrow

Sub-stage 2A (

V^{\text{temp}} = 137.5

)

\rightarrow

Sub-stage 2B (

P = 25

)

\rightarrow

Sub-stage 3B (

V = 140

Matching the chronological execution order yields the correct item ordering.

Key Concept

Sequential Processes and State Transitions

Question

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