CAT 2022 Slot 2 · DILR · Set 2: Widget Deliveries from a Warehouse

The full set and all 5 questions (Q6–Q10) from CAT 2022 Slot 2, with answers and solutions.

Every day a widget supplier supplies widgets from the warehouse (W) to four locations - Ahmednagar (A), Bikrampore (B), Chitrachak (C), and Deccan Park (D). The daily demand for widgets in each location is uncertain and independent of each other. Demands and corresponding probability values (in parenthesis) are given against each location (A, B, C, and D) in the figure below. For example, there is a 40% chance that the demand in Ahmednagar will be 50 units and a 60% chance that the demand will be 70 units. The lines in the figure connecting the locations and warehouse represent two-way roads connecting those places with the distances (in km) shown beside the line. The distances in both the directions along a road are equal. For example, the road from Ahmednagar to Bikrampore and the road from Bikrampore to Ahmednagar are both 6 km long.

Every day the supplier gets the information about the demand values of the four locations and creates the travel route that starts from the warehouse and ends at a location after visiting all the locations exactly once. While making the route plan, the supplier goes to the locations in decreasing order of demand. If there is a tie for the choice of the next location, the supplier will go to the location closest to the current location. Also, while creating the route, the supplier can either follow the direct path (if available) from one location to another or can take the path via the warehouse. If both paths are available (direct and via warehouse), the supplier will choose the path with minimum distance.

Question 6 · Caselet DI

If the last location visited is Ahmednagar, then what is the total distance covered in the route (in km)?

Answer & solution

Answer: 35

Solving the set

Step 1 – Shortest allowed distances between locations (direct road, or via the warehouse; the smaller one is used).

From / To	A	B	C	D
W	5	10	12	2
A	—	6 (direct)	17 (via W)	7 (via W; direct is 8)
B	6	—	4 (direct)	12 (via W)
C	17	4	—	6 (direct)
D	7	12	6	—

Step 2 – Routes. Visit in decreasing order of demand; for a tie, go to the nearer location from where the supplier is. The 16 demand combinations, their probabilities (product of the four probabilities), routes and distances are:

A	B	C	D	Probability	Route	Distance (km)	Total units
50	40	70	30	1.44%	W-C-A-B-D	47	190
50	40	70	50	2.16%	W-C-D-A-B	31	210
50	40	100	30	3.36%	W-C-A-B-D	47	220
50	40	100	50	5.04%	W-C-D-A-B	31	240
50	60	70	30	3.36%	W-C-B-A-D	29	210
50	60	70	50	5.04%	W-C-B-A-D	29	230
50	60	100	30	7.84%	W-C-B-A-D	29	240
50	60	100	50	11.76%	W-C-B-A-D	29	260
70	40	70	30	2.16%	W-A-C-B-D	38	210
70	40	70	50	3.24%	W-A-C-D-B	40	230
70	40	100	30	5.04%	W-C-A-B-D	47	240
70	40	100	50	7.56%	W-C-A-D-B	48	260
70	60	70	30	5.04%	W-A-C-B-D	38	230
70	60	70	50	7.56%	W-A-C-B-D	38	250
70	60	100	30	11.76%	W-C-A-B-D	47	260
70	60	100	50	17.64%	W-C-A-B-D	47	280

This question

A can be last only if it has the smallest demand together with ties going against it: A = 50, B = 60, D = 50 (C = 70 or 100). The route is W → C → B, and then A and D tie at 50.

Under the tie rule as stated, the supplier at B would go to A (6 km) before D (12 km), giving W-C-B-A-D, which ends at D. The official answer key, however, takes the route W-C-B-D-A, which ends at A: 12 + 4 + 12 + 7 = 35 km.

Hence, the answer is 35.

Question 7 · Caselet DI

If the total number of widgets delivered in a day is 250 units, then what is the total distance covered in the route (in km)?

Answer & solution

Answer: 38

Solving the set

Step 1 – Shortest allowed distances between locations (direct road, or via the warehouse; the smaller one is used).

From / To	A	B	C	D
W	5	10	12	2
A	—	6 (direct)	17 (via W)	7 (via W; direct is 8)
B	6	—	4 (direct)	12 (via W)
C	17	4	—	6 (direct)
D	7	12	6	—

A	B	C	D	Probability	Route	Distance (km)	Total units
50	40	70	30	1.44%	W-C-A-B-D	47	190
50	40	70	50	2.16%	W-C-D-A-B	31	210
50	40	100	30	3.36%	W-C-A-B-D	47	220
50	40	100	50	5.04%	W-C-D-A-B	31	240
50	60	70	30	3.36%	W-C-B-A-D	29	210
50	60	70	50	5.04%	W-C-B-A-D	29	230
50	60	100	30	7.84%	W-C-B-A-D	29	240
50	60	100	50	11.76%	W-C-B-A-D	29	260
70	40	70	30	2.16%	W-A-C-B-D	38	210
70	40	70	50	3.24%	W-A-C-D-B	40	230
70	40	100	30	5.04%	W-C-A-B-D	47	240
70	40	100	50	7.56%	W-C-A-D-B	48	260
70	60	70	30	5.04%	W-A-C-B-D	38	230
70	60	70	50	7.56%	W-A-C-B-D	38	250
70	60	100	30	11.76%	W-C-A-B-D	47	260
70	60	100	50	17.64%	W-C-A-B-D	47	280

This question

Total 250 occurs only for A = 70, B = 60, C = 70, D = 50 (70 + 60 + 70 + 50). From W, A and C tie at 70; A is closer (5 km vs 12 km). Then C (17 km via W), then B (60, 4 km), then D (50, 12 km via W). Route W-A-C-B-D = 5 + 17 + 4 + 12 = 38 km.

Hence, the answer is 38.

Question 8 · Caselet DI

What is the chance that the total number of widgets delivered in a day is 260 units and the route ends at Bikrampore?

A
33.33%
B
10.80%
C
17.64%
D
7.56%

Pick an option, then check.

Answer & solution

Answer: Option D

Solving the set

Step 1 – Shortest allowed distances between locations (direct road, or via the warehouse; the smaller one is used).

From / To	A	B	C	D
W	5	10	12	2
A	—	6 (direct)	17 (via W)	7 (via W; direct is 8)
B	6	—	4 (direct)	12 (via W)
C	17	4	—	6 (direct)
D	7	12	6	—

A	B	C	D	Probability	Route	Distance (km)	Total units
50	40	70	30	1.44%	W-C-A-B-D	47	190
50	40	70	50	2.16%	W-C-D-A-B	31	210
50	40	100	30	3.36%	W-C-A-B-D	47	220
50	40	100	50	5.04%	W-C-D-A-B	31	240
50	60	70	30	3.36%	W-C-B-A-D	29	210
50	60	70	50	5.04%	W-C-B-A-D	29	230
50	60	100	30	7.84%	W-C-B-A-D	29	240
50	60	100	50	11.76%	W-C-B-A-D	29	260
70	40	70	30	2.16%	W-A-C-B-D	38	210
70	40	70	50	3.24%	W-A-C-D-B	40	230
70	40	100	30	5.04%	W-C-A-B-D	47	240
70	40	100	50	7.56%	W-C-A-D-B	48	260
70	60	70	30	5.04%	W-A-C-B-D	38	230
70	60	70	50	7.56%	W-A-C-B-D	38	250
70	60	100	30	11.76%	W-C-A-B-D	47	260
70	60	100	50	17.64%	W-C-A-B-D	47	280

This question

Rows of the table with 260 units: (50, 60, 100, 50) ends at D; (70, 40, 100, 50) ends at B; (70, 60, 100, 30) ends at D. Only the middle one ends at B.

Probability = 0.6 (A = 70) × 0.3 (B = 40) × 0.7 (C = 100) × 0.6 (D = 50) = 0.0756 = 7.56%.

Hence, the answer is option 4.

Question 9 · Caselet DI

If the first location visited from the warehouse is Ahmednagar, then what is the chance that the total distance covered in the route is 40 km?

A
18%
B
5.4%
C
3.24%
D
30%

Pick an option, then check.

Answer & solution

Answer: Option A

Solving the set

Step 1 – Shortest allowed distances between locations (direct road, or via the warehouse; the smaller one is used).

From / To	A	B	C	D
W	5	10	12	2
A	—	6 (direct)	17 (via W)	7 (via W; direct is 8)
B	6	—	4 (direct)	12 (via W)
C	17	4	—	6 (direct)
D	7	12	6	—

A	B	C	D	Probability	Route	Distance (km)	Total units
50	40	70	30	1.44%	W-C-A-B-D	47	190
50	40	70	50	2.16%	W-C-D-A-B	31	210
50	40	100	30	3.36%	W-C-A-B-D	47	220
50	40	100	50	5.04%	W-C-D-A-B	31	240
50	60	70	30	3.36%	W-C-B-A-D	29	210
50	60	70	50	5.04%	W-C-B-A-D	29	230
50	60	100	30	7.84%	W-C-B-A-D	29	240
50	60	100	50	11.76%	W-C-B-A-D	29	260
70	40	70	30	2.16%	W-A-C-B-D	38	210
70	40	70	50	3.24%	W-A-C-D-B	40	230
70	40	100	30	5.04%	W-C-A-B-D	47	240
70	40	100	50	7.56%	W-C-A-D-B	48	260
70	60	70	30	5.04%	W-A-C-B-D	38	230
70	60	70	50	7.56%	W-A-C-B-D	38	250
70	60	100	30	11.76%	W-C-A-B-D	47	260
70	60	100	50	17.64%	W-C-A-B-D	47	280

This question

A is visited first only when A = 70 and C = 70 (A beats C on the tie as it is closer to W). P(A = 70 and C = 70) = 0.6 × 0.3 = 0.18.

Within this, the 40 km route W-A-C-D-B needs B = 40 and D = 50: probability 0.3 × 0.6 = 0.18 of those cases. (The other three B, D combinations give 38 km.)

So the chance is 18%.

Hence, the answer is option 1.

Question 10 · Caselet DI

If Ahmednagar is not the first location to be visited in a route and the total route distance is 29 km, then which of the following is a possible number of widgets delivered on that day?

A
210
B
220
C
200
D
250

Pick an option, then check.

Answer & solution

Answer: Option A

Solving the set

Step 1 – Shortest allowed distances between locations (direct road, or via the warehouse; the smaller one is used).

From / To	A	B	C	D
W	5	10	12	2
A	—	6 (direct)	17 (via W)	7 (via W; direct is 8)
B	6	—	4 (direct)	12 (via W)
C	17	4	—	6 (direct)
D	7	12	6	—

A	B	C	D	Probability	Route	Distance (km)	Total units
50	40	70	30	1.44%	W-C-A-B-D	47	190
50	40	70	50	2.16%	W-C-D-A-B	31	210
50	40	100	30	3.36%	W-C-A-B-D	47	220
50	40	100	50	5.04%	W-C-D-A-B	31	240
50	60	70	30	3.36%	W-C-B-A-D	29	210
50	60	70	50	5.04%	W-C-B-A-D	29	230
50	60	100	30	7.84%	W-C-B-A-D	29	240
50	60	100	50	11.76%	W-C-B-A-D	29	260
70	40	70	30	2.16%	W-A-C-B-D	38	210
70	40	70	50	3.24%	W-A-C-D-B	40	230
70	40	100	30	5.04%	W-C-A-B-D	47	240
70	40	100	50	7.56%	W-C-A-D-B	48	260
70	60	70	30	5.04%	W-A-C-B-D	38	230
70	60	70	50	7.56%	W-A-C-B-D	38	250
70	60	100	30	11.76%	W-C-A-B-D	47	260
70	60	100	50	17.64%	W-C-A-B-D	47	280

This question

From the table, 29 km occurs only for W-C-B-A-D (12 + 4 + 6 + 7), with A = 50 and B = 60 (C and D free). The totals are 210, 230, 240 and 260. Of the options, 210 is possible.

Hence, the answer is option 1.

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