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CAT 2025 Slot 1 · DILR · Allocation and Selection · hard

CAT 2025 Slot 1 DILR: Assigning projects to managers

A company must assign five new projects — Alpha, Beta, Gamma, Delta and Epsilon — to five managers — M1, M2, M3, M4 and M5 — with exactly one project per manager and exactly one manager per project. Each manager has been rated (on a scale of 1 to 10, higher is better) on how suitable they are for each project, as shown in the table below. The company wants to choose the one-to-one assignment of projects to managers that maximises the total (sum) of the suitability ratings across all five manager-project pairs; call this the "optimal assignment."

Suitability ratings

| Manager | Alpha | Beta | Gamma | Delta | Epsilon |
|---|---|---|---|---|---|
| M1 | 8 | 6 | 7 | 5 | 9 |
| M2 | 7 | 9 | 6 | 8 | 5 |
| M3 | 6 | 7 | 9 | 6 | 8 |
| M4 | 9 | 5 | 8 | 7 | 6 |
| M5 | 5 | 8 | 6 | 9 | 7 |

Question 1

Single correct option

In the optimal assignment, which project is assigned to M1?
  1. A. Alpha
  2. B. Beta
  3. C. Gamma
  4. D. Epsilon
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Answer: (D) Epsilon

Checking all 120 possible one-to-one assignments (or by inspection, since each manager's personal-best project turns out to be distinct across managers): the unique maximum-total assignment is M1–Epsilon (9), M2–Beta (9), M3–Gamma (9), M4–Alpha (9), M5–Delta (9), for a total of 45. So M1 is assigned Epsilon.

Question 2

Numerical answer

What is the maximum total suitability rating achievable across all five manager-project pairs, over all possible one-to-one assignments?
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Answer: 45

The optimal (maximum-sum) assignment is M1–Epsilon, M2–Beta, M3–Gamma, M4–Alpha, M5–Delta, with ratings 9, 9, 9, 9 and 9 respectively. Total = 9×5 = 45. No other one-to-one assignment scores higher (verified by checking all permutations).

Question 3

Single correct option

In the optimal assignment, which manager is assigned to project Alpha?
  1. A. M1
  2. B. M2
  3. C. M3
  4. D. M4
Show answer and solution

Answer: (D) M4

In the optimal assignment (M1–Epsilon, M2–Beta, M3–Gamma, M4–Alpha, M5–Delta), project Alpha goes to M4, who rates it 9 — the highest rating any manager gives to Alpha.

Question 4

Numerical answer

In the optimal assignment, what is the sum of the ratings of M2 and M3 for their respectively assigned projects?
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Answer: 18

In the optimal assignment, M2 is assigned Beta (rating 9) and M3 is assigned Gamma (rating 9). Sum = 9+9 = 18.

Question 5

Single correct option

If manager M4 is unexpectedly unavailable, and the remaining four managers (M1, M2, M3, M5) are optimally assigned to four of the five projects (leaving exactly one project unassigned) so as to maximise the total rating, which project is left unassigned?
  1. A. Alpha
  2. B. Beta
  3. C. Gamma
  4. D. Epsilon
Show answer and solution

Answer: (A) Alpha

With M4 removed, checking all ways of assigning the 4 remaining managers to 4 of the 5 projects, the best total (36) is achieved by keeping each manager's own best pick where possible: M1–Epsilon (9), M2–Beta (9), M3–Gamma (9), M5–Delta (9), total 36 — the same four pairs as before, just without M4–Alpha. So Alpha, which only M4 rated a 9, is the project left unassigned.
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Original question written in the CAT 2025 Slot 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake