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For the standard transportation linear programme with m sources and n destinations and total supply equaling total demand, an optimal solution (lowest cost) with the smallest number of non-zero xij values (amounts from source i to destinationj) is desired. The best upper bound for this number is:
- a)mn
- b)2(m + n)
- c)m + n
- d)m + n - 1
Correct answer is option 'D'. Can you explain this answer?
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For the standard transportation linear programme with m sources and n ...
Explanation:
To find the best upper bound for the number of non-zero xij values, we need to understand the structure of the standard transportation linear program.
1. The Structure of the Transportation Linear Program:
The standard transportation linear program consists of m sources and n destinations, with total supply equaling total demand. The goal is to minimize the total cost of transportation.
The decision variables, xij, represent the amount of goods transported from source i to destination j. These variables are subject to the following constraints:
- Supply constraints: The total amount of goods transported from each source i should not exceed the supply at that source.
- Demand constraints: The total amount of goods received at each destination j should equal the demand at that destination.
- Non-negativity constraints: The amount of goods transported cannot be negative.
The objective function represents the total cost of transportation, which is the sum of the cost of transporting goods from each source i to each destination j, multiplied by the corresponding amount xij.
2. Finding the best upper bound for the number of non-zero xij values:
To find the best upper bound for the number of non-zero xij values, we need to understand the conditions under which a non-zero value is required.
In the transportation linear program, a non-zero value of xij is required when there is a positive amount of goods to be transported from source i to destination j. This means that both the supply at source i and the demand at destination j are positive.
3. Deriving the best upper bound:
To minimize the number of non-zero xij values, we need to minimize the number of positive supplies and positive demands.
- The minimum number of positive supplies is m, as there are m sources.
- The minimum number of positive demands is n, as there are n destinations.
Therefore, the best upper bound for the number of non-zero xij values is the minimum between m and n.
Since the question states that the total supply is equal to the total demand, we can conclude that m = n.
Thus, the best upper bound for the number of non-zero xij values is given by m * n - 1, which corresponds to option D. This is because when m = n, the minimum number of positive supplies and positive demands is m = n, so we subtract 1 to account for the one non-zero value that is already known (the supply equals the demand).
Therefore, the correct answer is option D, m * n - 1.
To find the best upper bound for the number of non-zero xij values, we need to understand the structure of the standard transportation linear program.
1. The Structure of the Transportation Linear Program:
The standard transportation linear program consists of m sources and n destinations, with total supply equaling total demand. The goal is to minimize the total cost of transportation.
The decision variables, xij, represent the amount of goods transported from source i to destination j. These variables are subject to the following constraints:
- Supply constraints: The total amount of goods transported from each source i should not exceed the supply at that source.
- Demand constraints: The total amount of goods received at each destination j should equal the demand at that destination.
- Non-negativity constraints: The amount of goods transported cannot be negative.
The objective function represents the total cost of transportation, which is the sum of the cost of transporting goods from each source i to each destination j, multiplied by the corresponding amount xij.
2. Finding the best upper bound for the number of non-zero xij values:
To find the best upper bound for the number of non-zero xij values, we need to understand the conditions under which a non-zero value is required.
In the transportation linear program, a non-zero value of xij is required when there is a positive amount of goods to be transported from source i to destination j. This means that both the supply at source i and the demand at destination j are positive.
3. Deriving the best upper bound:
To minimize the number of non-zero xij values, we need to minimize the number of positive supplies and positive demands.
- The minimum number of positive supplies is m, as there are m sources.
- The minimum number of positive demands is n, as there are n destinations.
Therefore, the best upper bound for the number of non-zero xij values is the minimum between m and n.
Since the question states that the total supply is equal to the total demand, we can conclude that m = n.
Thus, the best upper bound for the number of non-zero xij values is given by m * n - 1, which corresponds to option D. This is because when m = n, the minimum number of positive supplies and positive demands is m = n, so we subtract 1 to account for the one non-zero value that is already known (the supply equals the demand).
Therefore, the correct answer is option D, m * n - 1.
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For the standard transportation linear programme with m sources and n destinations and total supply equaling total demand, an optimal solution (lowest cost) with the smallest number of non-zero xij values (amounts from source i to destinationj) is desired. The best upper bound for this number is:a)mnb)2(m + n)c)m + nd)m + n - 1Correct answer is option 'D'. Can you explain this answer?
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