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: 1
If the last location visited is Ahmednagar, then what is the total distance covered in the route (in km)?
The route starts from the warehouse and ends at a location after visiting all locations exactly once.
The route plan follows a decreasing order of demand, and in case of equal demand, it prioritizes the nearest locations first.
The supplier can choose either a direct path or a path via the warehouse, preferring the minimum distance.
Additionally, it is given in the question that the last location is A. The demand in the remaining places should be greater than A, meaning A's demand cannot be 70 units. Therefore, it is 50 units.
Regarding the demand of location D, it can be either 30 or 50 units. This implies that D should be placed before the location with a demand greater than or equal to 50 units.
The location visited before D is B, which has a demand of 60 units. It cannot be C because the values of C are greater than the values of B. Therefore, the order is C - B - D - A.
The distance covered:
Warehouse to C: 12 km
C to B: 4 km
B to D: 12 km
D to A (through the warehouse): 7 km
Total distance covered: 12 + 4 + 12 + 7 = 35 km.
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Question: 2
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)?
The route starts from the warehouse and ends at a location after visiting all locations exactly once.
The route plan follows a decreasing order of demand, and in case of equal demand, it prioritizes the nearest locations first.
The supplier can choose either a direct path or a path via the warehouse, preferring the minimum distance.
To summarize:
The total number of units delivered is 250 units.
The maximum number of widgets that can be delivered is 280 units, considering the maximum demand for each location (70 units for A, 50 units for D, 60 units for B, and 100 units for C).
To meet the constraint of delivering 250 units, the demand for location C should be reduced to 70 units.
Therefore, the delivery plan is A - C - B - D, with the demand values: 70 units for A, 50 units for D, 60 units for B, and 70 units for C.
From statement 1, the order should be A - C - B - D, as A is nearer to the warehouse than C.
The distances covered are as follows:
Warehouse to A: 5 km
A to C: 17 km
C to B: 4 km
B to D: 12 km
Total distance covered: 5 + 17 + 4 + 12 = 38 km.
This plan ensures the delivery of 250 units with the specified constraints and the minimum total distance covered.
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Question: 3
What is the chance that the total number of widgets delivered in a day is 260 units and the route ends at Bikrampore?
The first location visited from the warehouse is A, and if A's demand is 50 units, C's demand should be less than 50 units, which is not possible. Therefore, the demand for both A and C is 70 units.
The order of locations visited is A(70 units) -> C(70 units), with distances covered: Warehouse to A (5 km), A to C (17 km), total distance covered = 5 + 17 = 22 km.
The remaining distance to be covered is 40 km - 22 km = 18 km.
The next leg of the route is C to B, covering 4 km, and B to D, covering 12 km, for a total of 16 km.
The remaining distance to be covered is 18 km, which can be achieved by going from C to D (6 km) and D to B (12 km).
To fulfill the conditions, D's demand is 50 units (60% probability) and B's demand is 40 units (30% probability).
The required value is calculated as 0.6 * 0.3 = 0.18, which is equivalent to 18%.
Therefore, the probability of the specified conditions being met is 18%.
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Question: 5
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?
