Chat with AIExam 19: Solution Procedures for Transportation and Assignment Problems
For an assignment problem where the number of agents does not equal the number of tasks, what adjustments must be made to allow the problem to be solved using the Hungarian method?
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Answered by Examlex AI CopilotThe Hungarian method is an algorithm used to solve assignment problems where the goal is to assign agents to tasks in a one-to-one manner while minimizing the total cost or maximizing the total efficiency. However, the standard Hungarian method requires that the number of agents be equal to the number of tasks. When the number of agents does not equal the number of tasks, adjustments must be made to the problem to use the Hungarian method. Here's how you can adjust the problem:
1. **Create Dummy Rows or Columns**: If there are more tasks than agents, add dummy rows to the cost matrix to make the matrix square. These dummy rows represent fictitious agents and should be filled with zeros (if minimizing cost) or a value that is high but not prohibitively so (if maximizing efficiency). Conversely, if there are more agents than tasks, add dummy columns to represent fictitious tasks with the same logic for the values.
2. **Adjust the Cost Matrix**: After adding dummy rows or columns, you need to adjust the cost matrix. For a minimization problem, the added dummy elements should have a cost of zero, as they do not contribute to the actual cost. For a maximization problem, the dummy elements should have a cost that is sufficiently low (or high negative value) so that they are not preferred over real assignments.
3. **Apply the Hungarian Method**: Once the matrix is square, apply the Hungarian method as usual. This involves subtracting the smallest value in each row and then each column from all other elements in the respective rows and columns, marking zeros to find the minimum number of lines to cover them, and then adjusting the matrix until an optimal assignment can be made.
4. **Interpret the Results**: After solving the assignment problem using the Hungarian method, interpret the results by ignoring the assignments to dummy agents or tasks. The real agents should be assigned to real tasks based on the optimal assignment found, and any assignments to dummy rows or columns should be disregarded as they do not correspond to actual assignments.
By making these adjustments, you can effectively use the Hungarian method for assignment problems where the number of agents and tasks are unequal, ensuring that the algorithm can still find an optimal solution.
The transportation simplex method can be used to solve the assignment problem.
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VerifiedTrue
The stepping-stone method requires that one or more artificially occupied cells with a flow of zero be created in the transportation tableau when the number of occupied cells is fewer than
(30) VerifiedB
After some special presentations, the employees of the AV Center have to move overhead projectors back to classrooms. The table below indicates the buildings where the projectors are now (the origins), where they need to go (the destinations), and a measure of the distance between sites. Determine the transport arrangement that minimizes the total transport distance. Destination Origin Business Education Parsons Hall Holmstedt Hall Supply Baker Hall 10 9 5 2 35 Tirey Hall 12 11 1 6 10 Arena 15 14 7 6 20 Dem and 12 20 10 10
(27) The net evaluation index for occupied cells in the transportation simplex method is 0.
(44) The optimal solution is found in an assignment matrix when the minimum number of straight lines needed to cover all the zeros equals
(27) Use the Hungarian method to obtain the optimal solution to the following assignment problem in which total cost is to be minimized. All tasks must be assigned and no agent can be assigned to more than one task. Task Agent A B C D 1 10 12 15 25 2 11 14 19 32 3 18 21 23 29 4 15 20 26 28
(42) Explain how the Hungarian method can be used to solve an assignment problem that has a maximization objective.
(35) In assignment problems, dummy agents or tasks are created when the number of agents and tasks is not equal.
(41) Canning Transport is to move goods from three factories (origins) to three distribution centers (destinations). Information about the move is given below. Solve the problem using the transportation simplex method and compute the total shipping cost. Origin Sapply Destination Denand A 200 50 100 125 150 125 Shipping costs are: Destination Origin 3 2 5 9 10 - 5 6 4 (Source B cannot ship to destination Z)
(35) Four jobs must be assigned to four work centers. Only one job can be assigned to each work center, and all jobs must be processed. The cost of processing each job at each work center is shown below. Determine which jobs should be assigned to which work center to minimize total processing cost. Compute the total processing cost. Work Center Job 1 2 3 4 A \ 50 \ 45 \ 50 \ 65 B 25 40 35 20 C 65 60 55 65 E 55 65 75 85
(40) To use the transportation simplex method, a transportation problem that is unbalanced requires the use of
(29) The difference between the transportation and assignment problems is that
(27) A manufacturer of electrical consumer products, with its headquarters in Burlington, Iowa, produces electric irons at Manufacturing Plants 1, 2, and 3. The irons are shipped to Warehouses A, B, C, and D. The shipping cost per iron, the monthly warehouse requirements, and the monthly plant production levels are: Thatehouse Monthly Plant Froduction (units) Plant 1 \ .20 \ .25 \ .15 \ .20 10,000 Plant 2 .15 .30 .20 .15 20,000 Plant 3 .15 .20 .20 .25 10,000 Monthly Warehouse 12,000 8,000 15,000 5,000 Requirements (units) How many electric irons should be shipped per month from each plant to each warehouse to minimize monthly shipping costs?
a.Use the minimum cost method to find an initial feasible solution.
b.Can the initial solution be improved?
c.Compute the optimal total shipping cost per month.
(33) For an assignment problem with 3 agents and 4 tasks, the assignment matrix will have 3 rows and 4 columns.
(36) The transportation simplex method is more efficient than general-purpose linear programming for solving large-sized transportation problems.
(33) A dummy origin in a transportation problem is used when supply exceeds demand.
(34) Solve the following assignment problem using the Hungarian method. No agent can be assigned to more than one task. Total cost is to be minimized. Task Agent A B C D 1 9 5 4 2 2 12 6 3 5 3 11 6 5 7
(23) Explain how the transportation simplex method can be used to solve a transportation problem that has a maximization objective.
(33) If a transportation problem has four origins and five destinations, one of the destinations will not be fully supplied.
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