## Quadratic assignment problem

Author: Thomas Kueny, Eric Miller, Natasha Rice, Joseph Szczerba, David Wittmann (SysEn 5800 Fall 2020)

- 1 Introduction
- 2.1 Koopmans-Beckman Mathematical Formulation
- 2.2.1 Parameters
- 2.3.1 Optimization Problem
- 2.4 Computational Complexity
- 2.5 Algorithmic Discussions
- 2.6 Branch and Bound Procedures
- 2.7 Linearizations
- 3.1 QAP with 3 Facilities
- 4.1 Inter-plant Transportation Problem
- 4.2 The Backboard Wiring Problem
- 4.3 Hospital Layout
- 4.4 Exam Scheduling System
- 5 Conclusion
- 6 References

## Introduction

## Theory, Methodology, and/or Algorithmic Discussions

Koopmans-beckman mathematical formulation.

## Quadratic Assignment Problem Formulation

## Inner Product

## Optimization Problem

With all of this information, the QAP can be summarized as:

## Computational Complexity

## Algorithmic Discussions

## Branch and Bound Procedures

## Linearizations

The first attempts to solve the QAP eliminated the quadratic term in the objective function of

## Numerical Example

## Applications

Inter-plant transportation problem.

## The Backboard Wiring Problem

When defining the problem Steinberg states that we have a set of n elements

In his paper he derives the below formula:

The initial placement of components is shown below:

## Hospital Layout

Elshafei identified the following QAP to determine where clinics should be placed:

## Exam Scheduling System

- ↑ 1.0 1.1 1.2 Koopmans, T., & Beckmann, M. (1957). Assignment Problems and the Location of Economic Activities. Econometrica, 25(1), 53-76. doi:10.2307/1907742
- ↑ 2.0 2.1 Quadratic Assignment Problem. (2020). Retrieved December 14, 2020, from https://neos-guide.org/content/quadratic-assignment-problem
- ↑ 3.0 3.1 3.2 Burkard, R. E., Çela, E., Pardalos, P. M., & Pitsoulis, L. S. (2013). The Quadratic Assignment Problem. https://www.opt.math.tugraz.at/~cela/papers/qap_bericht.pdf .
- ↑ 4.0 4.1 Leon Steinberg. The Backboard Wiring Problem: A Placement Algorithm. SIAM Review . 1961;3(1):37.
- ↑ 5.0 5.1 Alwalid N. Elshafei. Hospital Layout as a Quadratic Assignment Problem. Operational Research Quarterly (1970-1977) . 1977;28(1):167. doi:10.2307/300878
- ↑ 6.0 6.1 Muktar, D., & Ahmad, Z.M. (2014). Examination Scheduling System Based On Quadratic Assignment.

## Navigation menu

## Solving an Assignment Problem

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

## 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.

## 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.

## IMAGES

## VIDEO

## COMMENTS

The

assignment problemis a fundamental combinatorial optimizationproblem. In its most general form, theproblemis as follows: Theprobleminstance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-taskassignment.optimalto assign Machine 1 to Task 2, Machine 2 to Task 4, Machine 3 to Task 3, and Machine 4 to Task 1. The total setup time associated with this solution is 11 hours. This completes the solution of theproblem. As noted earlier, every basic feasible solution inan assignmentproblemis degenerate.The Quadratic

Assignment Problem(QAP), discovered by Koopmans and Beckmann in 1957, is a mathematical optimization module created to describe the location of invisible economic activities. An NP-Completeproblem, this model can be applied to many other optimizationproblemsoutside of the field of economics.solution to the LP, so the

optimalvalue of the LP must be at most theoptimalvalue of theassignmentproblem. We consider the dual of the LP: max X i2I u i + X j2J v j s.t. u i + v j c ij for all i2I;j2J Now, we know that xis anoptimalsolution to the primal LP and u;vis anoptimalsolution to the dualThis

problemis a generalization of theassignmentproblemin which both tasks and agents have a size. Moreover, the size of each task might vary from one agent to the other. Thisproblemin its most general form is as follows: There are a number of agents and a number of tasks.MIP solution. Import the libraries. Create the data. Declare the MIP solver. Create the variables. Create the constraints. Create the objective function. This section presents an example that shows how to solve

an assignmentproblemusing both the MIP solver and the CP-SAT solver.An assignmentproblemis a particular case of transportationproblemwhere the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.work only on one job. The

problemis to assign the jobs to the machines, which will minimize the total cost of machining. SESSION 3.2: SOLUTION OF MINIMIZATIONASSIGNMENTPROBLEMThe basic principle is that theoptimalassignmentis not affected if a constant is added or subtracted from any row or column of the cost matrix.