Use the solver in Excel to find the assignment of persons to tasks that minimizes the total cost. What are the decisions to be made?

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For example, if we assign Person 1 to Task 1, cell C10 equals 1. If not, cell C10 equals 0. What are the constraints on these decisions?

## Assignment problem

What is the overall measure of performance for these decisions? The overall measure of performance is the total cost of the assignment, so the objective is to minimize this quantity.

To make the model easier to understand, create the following named ranges. Explanation: The SUM functions calculate the number of tasks assigned to a person and the number of persons assigned to a task. Total Cost equals the sumproduct of Cost and Assignment. This solution has a total cost of It is not necessary to use trial and error.

We shall describe next how the Excel Solver can be used to quickly find the optimal solution. Note: can't find the Solver button? Click here to load the Solver add-in. This solution gives the minimum cost of All constraints are satisfied.

Assignment Problem. Formulate the Model Trial and Error Solve the Model Use the solver in Excel to find the assignment of persons to tasks that minimizes the total cost. Formulate the Model The model we are going to solve looks as follows in Excel. To formulate this assignment problemanswer the following three questions. Insert the following functions. Trial and Error With this formulation, it becomes easy to analyze any trial solution. Chapter Solver.

### Solving an Assignment Problem

Download Excel File assignment-problem. Follow Excel Easy. Become an Excel Pro Examples.Let there be n agents and n tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.

It is required to perform all tasks by assigning exactly one agent to each task and exactly one task to each agent in such a way that the total cost of the assignment is minimized. Example: You work as a manager for a chip manufacturer, and you currently have 3 people on the road meeting clients. Your salespeople are in Jaipur, Pune and Bangalore, and you want them to fly to three other cities: Delhi, Mumbai and Kerala. The table below shows the cost of airline tickets in INR between the cities:.

The Hungarian algorithm, aka Munkres assignment algorithmutilizes the following theorem for polynomial runtime complexity worst case O n 3 and guaranteed optimality: If a number is added to or subtracted from all of the entries of any one row or column of a cost matrix, then an optimal assignment for the resulting cost matrix is also an optimal assignment for the original cost matrix.

We reduce our original weight matrix to contain zeros, by using the above theorem. We try to assign tasks to agents such that each agent is doing only one task and the penalty incurred in each case is zero.

Skip to content. Related Articles. The table below shows the cost of airline tickets in INR between the cities: The question: where would you send each of your salespeople in order to minimize fair? Recommended Articles. Article Contributed By :.

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Most visited in Mathematical. Load Comments. We use cookies to ensure you have the best browsing experience on our website.Assignment problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum.

Though there problems can be solved by simplex method or by transportation method but assignment model gives a simpler approach for these problems. In a factory, a supervisor may have six workers available and six jobs to fire. He will have to take decision regarding which job should be given to which worker. Problem forms one to one basis. This is an assignment problem. Suppose there are n facilitates and n jobs it is clear that in this case, there will be n assignments.

Each facility or say worker can perform each job, one at a time. But there should be certain procedure by which assignment should be made so that the profit is maximized or the cost or time is minimized. In the table, Co ij is defined as the cost when j th job is assigned to i th worker.

It maybe noted here that this is a special case of transportation problem when the number of rows is equal to number of columns. Any basic feasible solution of an Assignment problem consists 2n — 1 variables of which the n — 1 variables are zero, n is number of jobs or number of facilities. Due to this high degeneracy, if we solve the problem by usual transportation method, it will be a complex and time consuming work.

Thus a separate technique is derived for it. Before going to the absolute method it is very important to formulate the problem. Now as the problem forms one to one basis or one job is to be assigned to one facility or machine. Consider the objective function of minimization type. Following steps are involved in solving this Assignment problem. Locate the smallest cost element in each row of the given cost table starting with the first row. Now, this smallest element is subtracted form each element of that row.

So, we will be getting at least one zero in each row of this new table. Having constructed the table as by step-1 take the columns of the table. Starting from first column locate the smallest cost element in each column. Now subtract this smallest element from each element of that column. Having performed the step 1 and step 2, we will be getting at least one zero in each column in the reduced cost table.

Step is conducted for each row. Nowassignment is made to this single zero by putting the square around it and at the same time, all other zeros in the corresponding rows are crossed out x step is conducted for each column.

Now, if the number of marked zeros or the assignments made are equal to number of rows or columns, optimum solution has been achieved. There will be exactly single assignment in each or columns without any assignment. In this case, we will go to step 4. At this stage, draw the minimum number of lines horizontal and vertical necessary to cover all zeros in the matrix obtained in step 3, Following procedure is adopted:.

In step 4, if the number of lines drawn are equal to n or the number of rows, then it is the optimum solution if not, then go to step 6. Select the smallest element among all the uncovered elements. Now, this element is subtracted from all the uncovered elements and added to the element which lies at the intersection of two lines.

This is the matrix for fresh assignments. Repeat the procedure from step 3 until the number of assignments becomes equal to the number of rows or number of columns.

You must be logged in to post a comment.Let there be N workers and N jobs. Any worker can be assigned to perform any job, incurring some cost that may vary depending on the work-job assignment. It is required to perform all jobs by assigning exactly one worker to each job and exactly one job to each agent in such a way that the total cost of the assignment is minimized.

Solution 1: Brute Force We generate n! Since the solution is a permutation of the n jobs, its complexity is O n! Solution 2: Hungarian Algorithm The optimal assignment can be found using the Hungarian algorithm. We can perform depth-first search on state space tree and but successive moves can take us away from the goal rather than bringing closer.

The search of state space tree follows leftmost path from the root regardless of initial state. An answer node may never be found in this approach. We can also perform a Breadth-first search on state space tree. But no matter what the initial state is, the algorithm attempts the same sequence of moves like DFS. It is similar to BFS-like search but with one major optimization. Instead of following FIFO order, we choose a live node with least cost.

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We may not get optimal solution by following node with least promising cost, but it will provide very good chance of getting the search to an answer node quickly. Since Job 2 is assigned to worker A marked in greencost becomes 2 and Job 2 and worker A becomes unavailable marked in red. Now we assign job 3 to worker B as it has minimum cost from list of unassigned jobs.

Finally, job 1 gets assigned to worker C as it has minimum cost among unassigned jobs and job 4 gets assigned to worker C as it is only Job left. Below diagram shows complete search space diagram showing optimal solution path in green. Reference : www. This article is contributed by Aditya Goel. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute geeksforgeeks. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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Download Free PDF. Trisna Darmawansyah. Download PDF. A short summary of this paper. Kotwal, Tanuja S. It is an important problem in mathematics and is also discuss in real physical world.

It is a combinatorial optimization problem in the field of operational research. In a normal case of transportation problem where the objective is to assign the available resources to the activity going on so as to get the minimum cost or maximize total benefits of allocation.

In this paper we proposed modified assignment model for the solution of assignment problem. Here in this paper with the help of numerical examples or problem is solved to show its efficiency and also its comparison with Hungarian method is shown.

A new cost is achieved by using unbalanced assignment problem. Keywords— Unbalanced Assignment problem, Optimization, Hungarian method, proposed method.

### HUNGARIAN METHOD FOR SOLVING ASSIGNMENT PROBLEM - Quantitative Techniques for management

Assignment problem may be any type of problem like person to jobs, teacher to classroom, operators to lathe machine, driver to bus, bus to delivery routes etc. There are various optimization method to solve the assignment problem like genetic algorithm, simulated annealing etc.

Over the 5 decades many variations of assignment problem are proposed e. So to solve an unbalanced assignment problem we propose a new method with space complexity O nm. But it does not always provide a minimal total cost. This paper mainly focus to solve an unbalanced assignment problem by proposing a new method to improve the existing assignment cost.

M: number of worker. Cij :Be the cost of assigning the ith task to jth worker. T: total assigment cost. Yij : 1 if task i is assigned to worker j. Yij: 0 if task I is not assigned to task j. Its worker-task assignment cost matrix is a shown in table 1[3]. Step 2: Obtain table 2 by adding duplicate or one dummy task with all 1 assignment cost to table 1.

Reduced cost matrix obtained from table 2 is shown in table 3. Assign worker to task according to position of the chosen 1 and mark entire row to avoid later redundant assignment as shown in table 4. Assign worker to task according to position of the chosen 1 and mark entire column to avoid later redundant assignment[4].

There is no column with one 1 in table 4. Therefore, step 5 gives the same matrix as table 4. Step 6:Choose one of remaining 1 in the reduced cost matrix as a position to assign worker to task.An assignment problem can be easily solved by applying Hungarian method which consists of two phases. In the first phase, row reductions and column reductions are carried out. In the second phase, the solution is optimized on iterative basis. Step 0: Consider the given matrix. Step 1: In a given problem, if the number of rows is not equal to the number of columns and vice versa, then add a dummy row or a dummy column.

The assignment costs for dummy cells are always assigned as zero. Step 2: Reduce the matrix by selecting the smallest element in each row and subtract with other elements in that row. Step 3 : Reduce the new matrix column-wise using the same method as given in step 2.

Step 4 : Draw minimum number of lines to cover all zeros. If optimally is not reached, then go to step 6.

Leave the elements covered by single line as it is. Now go to step 4. Step 7: Take any row or column which has a single zero and assign by squaring it. Strike off the remaining zeros, if any, in that row and column X.

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Repeat the process until all the assignments have been made. Note: While assigning, if there is no single zero exists in the row or column, choose any one zero and assign it. Strike off the remaining zeros in that column or row, and repeat the same for other assignments also. If there is no single zero allocation, it means multiple numbers of solutions exist. But the cost will remain the same for different sets of allocations. Example : Assign the four tasks to four operators.

The assigning costs are given in Table. In row A, the smallest value is 13, row B is 15, row C is 17 and row D is The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows:. If the numbers of agents and tasks are equal, then the problem is called balanced assignment.

Otherwise, it is called unbalanced assignment. Commonly, when speaking of the assignment problem without any additional qualification, then the linear balanced assignment problem is meant. Suppose that a taxi firm has three taxis the agents available, and three customers the tasks wishing to be picked up as soon as possible.

The firm prides itself on speedy pickups, so for each taxi the "cost" of picking up a particular customer will depend on the time taken for the taxi to reach the pickup point. This is a balanced assignment problem. Its solution is whichever combination of taxis and customers results in the least total cost.

Now, suppose that there are four taxis available, but still only three customers. This is an unbalanced assignment problem. One way to solve it is to invent a fourth dummy task, perhaps called "sitting still doing nothing", with a cost of 0 for the taxi assigned to it.

This reduces the problem to a balanced assignment problem, which can then be solved in the usual way and still give the best solution to the problem. Similar adjustments can be done in order to allow more tasks than agents, tasks to which multiple agents must be assigned for instance, a group of more customers than will fit in one taxior maximizing profit rather than minimizing cost. Usually the weight function is viewed as a square real-valued matrix Cso that the cost function is written down as:.

The problem is "linear" because the cost function to be optimized as well as all the constraints contain only linear terms. A naive solution for the assignment problem is to check all the assignments and calculate the cost of each one. This may be very inefficient since, with n agents and n tasks, there are n!

Fortunately, there are many algorithms for solving the problem in time polynomial in n. The assignment problem is a special case of the transportation problemwhich is a special case of the minimum cost flow problemwhich in turn is a special case of a linear program.

While it is possible to solve any of these problems using the simplex algorithmeach specialization has more efficient algorithms designed to take advantage of its special structure. In the balanced assignment problem, both parts of the bipartite graph have the same number of vertices, denoted by n.

One of the first polynomial-time algorithms for balanced assignment was the Hungarian algorithm. This is currently the fastest run-time of a strongly polynomial algorithm for this problem. In addition to the global methods, there are local methods which are based on finding local updates rather than full augmenting paths. These methods have worse asymptotic runtime guarantees, but they often work better in practice.

These algorithms are called auction algorithmspush-relabel algorithms, or preflow-push algorithms. Some of these algorithms were shown to be equivalent. Some of the local methods assume that the graph admits a perfect matching ; if this is not the case, then some of these methods might run forever.

These weights should exceed the weights of all existing matchings, to prevent appearance of artificial edges in the possible solution. As shown by Mulmuley, Vazirani and Vazirani, [8] the problem of minimum weight perfect matching is converted to finding minors in the adjacency matrix of a graph.

There is also a constant s which is at most the maximum cardinality of a matching in the graph. The goal is to find a minimum-cost matching of size exactly s.

The most common case is the case in which the graph admits a one-sided-perfect matching i. Unbalanced assignment can be reduced to a balanced assignment.