PSDA.6:5.5 - Obtain an exchange family and use a new judgement
A service manager supplies one legitimate value account for a four-week arrangement. Known costs are in thousands of resource units, waiting times in hours. The relevant ranges are 20–60 and 1–5. Lower values are preferred. The manager explicitly accepts constant value increments and a constant cost-for-time exchange within these ranges. Those are the form assumptions; the following comparisons constrain its parameters.
| Alternative | Cost c | Waiting time t | Eligibility for this comparison |
|---|---|---|---|
| A | 60 | 2 | Admissible. |
| B | 30 | 5 | Admissible. |
| C | 45 | 3 | Admissible. |
| D | — | — | Fails a currently binding access condition; no timely amendment is available. |
The manager says that (60,1) is no worse than (20,5), and (20,4) is no worse than (60,2). Applying §4.5.2 gives:
| Stated judgement | Joint constraint | Consequence for r |
|---|---|---|
(60,1) no worse than (20,5) | 60 + r <= 20 + 5r | r >= 10 |
(20,4) no worse than (60,2) | 20 + 4r <= 60 + 2r | r <= 20 |
Thus L_r(c,t) = c + r t with 10 <= r <= 20 is the compatible family. It is nonempty: both endpoints satisfy both weak comparisons. No interview is available to choose a narrower exchange. In normalized range weights the same family has 1/3 <= w_c <= 1/2, with w_t = 1-w_c; those numbers were derived from the profiles, not assigned to criterion labels.
The receiving PSDA.8 comparison keeps one shared r:
| Difference over the whole family | Consequence |
|---|---|
L_B - L_C = 2r - 15, between 5 and 25 | C is strictly preferred to B throughout. |
L_B - L_A = 3r - 30, between 0 and 30 | A is no worse than B throughout, with a tie at r = 10. |
L_A - L_C = 15 - r, between -5 and 5 | C is better below 15; A is better above 15; they tie at 15. |
These linear differences establish the signs over the entire interval. At r = 10, the losses for A, B and C are 80, 80 and 75; at r = 20, they are 100, 130 and 105. Both exchanges satisfy the elicited information but give different best alternatives. The PSDA.9 sensitivity result is therefore the threshold 15 thousand per hour, and the PSDA.14 return is the conditional set {A,C}. B cannot be best in any compatible model; D remains ineligible. Choosing r = 15 merely because it is the interval’s midpoint would add unsupported indifference.
Later the same manager states that C is no worse than A. Its new row is 45 + 3r <= 60 + 2r, hence r <= 15. Intersecting gives [10,15]. Now L_A-L_C lies in [0,5] and L_B-L_C in [5,15]: C is no worse than A and strictly better than B for every compatible exchange. C is a necessary best, while {A,C} remains the set of possible best alternatives because A ties at 15. Return C as a supported direction with that tie intact. The new judgement changed the value family and its comparison; it changed neither the consequences nor D’s access failure. The recommendation still leaves the authorized recipient’s later choice distinct.