assignment problem for maximization

Assignment Problem: Maximization

There are problems where certain facilities have to be assigned to a number of jobs, so as to maximize the overall performance of the assignment.

The Hungarian Method can also solve such assignment problems , as it is easy to obtain an equivalent minimization problem by converting every number in the matrix to an opportunity loss.

The conversion is accomplished by subtracting all the elements of the given matrix from the highest element. It turns out that minimizing opportunity loss produces the same assignment solution as the original maximization problem.

• Unbalanced Assignment Problem
• Multiple Optimal Solutions

Example: Maximization In An Assignment Problem

At the head office of www.universalteacherpublications.com there are five registration counters. Five persons are available for service.

How should the counters be assigned to persons so as to maximize the profit ?

Here, the highest value is 62. So we subtract each value from 62. The conversion is shown in the following table.

On small screens, scroll horizontally to view full calculation

Now the above problem can be easily solved by Hungarian method . After applying steps 1 to 3 of the Hungarian method, we get the following matrix.

Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix.

Select the smallest element from all the uncovered elements, i.e., 4. Subtract this element from all the uncovered elements and add it to the elements, which lie at the intersection of two lines. Thus, we obtain another reduced matrix for fresh assignment. Repeating step 3, we obtain a solution which is shown in the following table.

Final Table: Maximization Problem

Use Horizontal Scrollbar to View Full Table Calculation

The total cost of assignment = 1C + 2E + 3A + 4D + 5B

Substituting values from original table: 40 + 36 + 40 + 36 + 62 = 214.

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Maximization Assignment Problem

Last updated on February 12th, 2020 at 01:52 pm

There are problems where certain facilities have to be assigned to a number of jobs so as to maximize the overall performance of the assignment. The problem can be converted into a minimization problem in the following ways and then Hungarian method can be used for its solution.

• Change the signs of all values given in the table.
• Select the highest element in the entire assignment table and subtract all the elements of the table from the highest element.

Example :  A marketing manager has five salesmen and sales districts. Considering the capabilities of the salesmen and the nature of districts, the marketing manager estimates that sales per month (in hundred rupees) for each salesman in each district would be as follows. Find the assignment of salesmen to districts that will result in maximum sales.

Maximization Problem

Maximization assignment problem is transformed into minimization problem by

Solution:  The given maximization problem is converted into minimization problem by subtracting from the highest sales value (i.e., 41) with all elements of the given table.

Conversion to Minimization Problem

Reduce the matrix row-wise

Matrix Reduced Row-wise

Reduce the matrix column-wise and draw minimum number of lines to cover all the zeros in the matrix, as shown in Table.

Matrix Reduced Column-wise and Zeros Covered

Number of lines drawn ≠ Order of matrix. Hence not optimal.

Select the least uncovered element, i.e., 4 and subtract it from other uncovered elements, add it to the elements at intersection of line and leave the elements that are covered with single line unchanged, Table.

Added & Subtracted the least Uncovered Element

Now, number of lines drawn = Order of matrix, hence optimality is reached. There are two alternative assignments due to presence of zero elements in cells (4, C), (4, D), (5, C) and (5, D).

Two Alternative Assignments

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Variations in assignment problem- case 1: maximization models - assignment problems.

Even though the assignment algorithm is primarily a minimization model, it can be suitably modified and used for maximization models also; whenever the objective of the firm / decision maker is to maximize the return of the allocation, you have to modify the problem in the initial table and use the same Hungarian Algorithm to solve. We will solve one example here.

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Converting job assignment problem into a 0-1 linear programming problem

There are $n$ people who need to be assigned to execute $n$ jobs, one person per job. (That is, each person is assigned to exactly one job and each job is assigned to exactly one person.) The cost that would accrue if the $i$ -th person is assigned to the $j$ -th job is a known quantity $C[i, j]$ for each pair $i, j = 1, \dots , n$ . The problem is to assign the people to the jobs to minimize the total cost of the assignment. Express the assignment problem as a $0$ - $1$ linear programming problem.

Let $M$ be an $n \times n$ matrix where the value at $M_{ij}$ indicates the cost of assigning person $i$ to job $j$ , or $C[i, j]$ .

Let $x$ be an $n \times n$ matrix where the value at $x_{ij}$ denotes whether person $i$ was assigned to job $j$ . If the assignment occurred, $x_{ij}$ is 1; otherwise, it is 0.

I've managed to come up with these definitions, but unfortunately I'm not sure how to proceed with the problem. The canonical form of a linear programming problem is as follows:

maximize $c^Tx$ , subject to $Ax\leq b$ , and $x\geq 0$

However, it seems the goal of this problem is to minimize, not to maximize. How do I convert this minimization problem into a maximization problem? Furthermore, how would the constraints be defined by this sort of problem? Are my definitions adequate for the problem?

• linear-programming
• word-problem
• binary-programming

The required LP is

$\begin{align} \min & \sum_{i=1}^n\sum_{j=1}^n M_{ij} x_{ij} \\ & \sum_{i=1}^n x_{ij} = 1 \quad \forall j \in \{1,\dots, n\} \tag{each job is assigned to exactly one person} \\ & \sum_{j=1}^n x_{ij} = 1 \quad \forall i \in \{1,\dots, n\} \tag{each person is assigned to exactly one job} \\ & x_{ij} \in \{0,1\} \quad \forall i,j \in \{1, \dots, n\} \end{align}$

To understand the objective function, for each job $j$ , there exist unique $i_0 \in \{1, \dots, n\}$ such that $x_{ij} = \begin{cases} 1 &\text{if } i = i_0 \\ 0 &\text{otherwise,} \end{cases}$ so the cost for the $j$ -th job is $M_{i_0j} = \sum \limits_{i=1}^n M_{ij}$ . Sum the previous equality over $j \in \{1,\dots,n\}$ to get the above double sum.

How do I convert this minimization problem into a maximization problem?

To minimize a quantity is equivalent to maximizing its additive inverse.

Furthermore, how would the constraints be defined by this sort of problem?

Use the standard technique of transforming an equality constraint to a set of two inequality constraints:

$$x = a \iff \begin{cases} - x &\le -a \\ x &\le a. \end{cases}$$

Are my definitions adequate for the problem?

They are more than enough. You don't need to define $M$ , since you already have $C$ , which represents the costs.

To write the LP is a more concise matrix form, denote

• $\mathbf{1} \in \{0,1\}^n$ the $n \times 1$ vector filled with $1$ .
• $C = (C[i,j])_{ij} = (c_{ij})_{ij} \in \mathbb{R}^{n \times n}$ : the matrix $C$ is defined by $C[i,j]$ as the $(i,j)$ -th element, so that
• $(C^T)_{ij} = c_{ji}$ : the $(i,j)$ -th element of $C^T$ is the $(j,i)$ -th element of $C$ .
• $X = (x_{ij})_{ij} \in \{0,1\}^{n \times n}$

Remarks: I prefer using capital letters for matrices, and use small letters with a subscript to denote the corresponding matrix elements.

Note that the cost function is \begin{align} \sum_{i=1}^n \sum_{j=1}^n C[i,j] x_{ij} &= \sum_{i=1}^n \sum_{j=1}^n (C^T)_{ji} x_{ij} \tag{take transpose of $C$} \\ &= \sum_{j=1}^n \left( \sum_{i=1}^n (C^T)_{ji} x_{ij} \right) \tag{change order of summation} \\ &= \sum_{j=1}^n \left( C^T X \right)_{jj} \tag{defintion of matrix product} \\ &= \mathop{\mathrm{tr}}(C^T X). \tag{definition of trace} \end{align} $\sum\limits_{i=1}^n x_{ij} = 1$ can be rewritten as $\mathbf{1}^T X = \mathbf{1}^T$ ; $\sum\limits_{j=1}^n x_{ij} = 1$ can be rewritten as $X \mathbf{1} = \mathbf{1}$ . Hence the LP in matrix form is $\min\{\mathop{\mathrm{tr}}(C^T X) \mid \mathbf{1}^T X = \mathbf{1}^T, X \mathbf{1} = \mathbf{1}, X \in \{0,1\}^{n \times n} \}$

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Related Multiple Choice Questions

• The maximization or minimization of a quantity is the
• An alternative optimal solution to a minimization transportation problem exists whenever opportunity cost corresponding to unused route of transportation is:
• An assignment problem is considered as a particular case of a transportation problem because
• An assignment problem is a special case of transportation problem, where
• In applying Vogel's approximation method to a profit maximization problem, row and column penalties are determined by:
• Which of the following is used to come up with a solution to the assignment problem?
• Which method usually gives a very good solution to the assignment problem?
• In the general linear programming model of the assignment problem,
• Which of the following is not true regarding an LP model of the assignment problem? ]
• The assignment problem constraint x31 + x32 + x33 + x34 ≤ 2 means

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Solving an Assignment Problem

This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver.

In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3). Note that there is one more worker than in the example in the Overview .

The costs of assigning workers to tasks are shown in the following table.

The problem is to assign each worker to at most one task, with no two workers performing the same task, while minimizing the total cost. Since there are more workers than tasks, one worker will not be assigned a task.

MIP solution

The following sections describe how to solve the problem using the MPSolver wrapper .

Import the libraries

The following code imports the required libraries.

Create the data

The following code creates the data for the problem.

The costs array corresponds to the table of costs for assigning workers to tasks, shown above.

Declare the MIP solver

The following code declares the MIP solver.

Create the variables

The following code creates binary integer variables for the problem.

Create the constraints

Create the objective function.

The following code creates the objective function for the problem.

The value of the objective function is the total cost over all variables that are assigned the value 1 by the solver.

Invoke the solver

The following code invokes the solver.

Print the solution

The following code prints the solution to the problem.

Here is the output of the program.

Complete programs

Here are the complete programs for the MIP solution.

CP SAT solution

The following sections describe how to solve the problem using the CP-SAT solver.

Declare the model

The following code declares the CP-SAT model.

The following code sets up the data for the problem.

The following code creates the constraints for the problem.

Here are the complete programs for the CP-SAT solution.

Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License , and code samples are licensed under the Apache 2.0 License . For details, see the Google Developers Site Policies . Java is a registered trademark of Oracle and/or its affiliates.

Last updated 2023-01-02 UTC.