formulation of quadratic assignment problem

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Quadratic Assignment Problem (QAP)

The Quadratic Assignment Problem (QAP) is an optimization problem that deals with assigning a set of facilities to a set of locations, considering the pairwise distances and flows between them.

The problem is to find the assignment that minimizes the total cost or distance, taking into account both the distances and the flows.

The distance matrix and flow matrix, as well as restrictions to ensure each facility is assigned to exactly one location and each location is assigned to exactly one facility, can be used to formulate the QAP as a quadratic objective function.

The QAP is a well-known example of an NP-hard problem , which means that for larger cases, computing the best solution might be difficult. As a result, many algorithms and heuristics have been created to quickly identify approximations of answers.

There are various types of algorithms for different problem structures, such as:

Precise algorithms
Approximation algorithms
Metaheuristics like genetic algorithms and simulated annealing
Specialized algorithms

Example: Given four facilities (F1, F2, F3, F4) and four locations (L1, L2, L3, L4). We have a cost matrix that represents the pairwise distances or costs between facilities. Additionally, we have a flow matrix that represents the interaction or flow between locations. Find the assignment that minimizes the total cost based on the interactions between facilities and locations. Each facility must be assigned to exactly one location, and each location can only accommodate one facility.

Facilities cost matrix:

	L1	L2	L3	L4
F1	0	2	3	1
F2	2	0	1	4
F3	3	1	0	2
F4	1	4	2	0

Flow matrix:

	F1	F2	F3	F4
L1	0	1	2	3
L2	1	0	4	2
L3	2	4	0	1
L4	3	2	1	0

To solve the QAP, various optimization techniques can be used, such as mathematical programming, heuristics, or metaheuristics. These techniques aim to explore the search space and find the optimal or near-optimal solution.

The solution to the QAP will provide an assignment of facilities to locations that minimizes the overall cost.

The solution generates all possible permutations of the assignment and calculates the total cost for each assignment. The optimal assignment is the one that results in the minimum total cost.

To calculate the total cost, we look at each pair of facilities in (i, j) and their respective locations (location1, location2). We then multiply the cost of assigning facility1 to facility2 (facilities[facility1][facility2]) with the flow from location1 to location2 (locations[location1][location2]). This process is done for all pairs of facilities in the assignment, and the costs are summed up.

Overall, the output tells us that assigning facilities to locations as F1->L1, F3->L2, F2->L3, and F4->L4 results in the minimum total cost of 44. This means that Facility 1 is assigned to Location 1, Facility 3 is assigned to Location 2, Facility 2 is assigned to Location 3, and Facility 4 is assigned to Location 4, yielding the lowest cost based on the given cost and flow matrices.This example demonstrates the process of finding the optimal assignment by considering the costs and flows associated with each facility and location. The objective is to find the assignment that minimizes the total cost, taking into account the interactions between facilities and locations.

Applications of the QAP include facility location, logistics, scheduling, and network architecture, all of which require effective resource allocation and arrangement.

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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

The Quadratic Assignment Problem (QAP), discovered by Koopmans and Beckmann in 1957 [1] , is a mathematical optimization module created to describe the location of invisible economic activities. An NP-Complete problem, this model can be applied to many other optimization problems outside of the field of economics. It has been used to optimize backboards, inter-plant transportation, hospital transportation, exam scheduling, along with many other applications not described within this page.

Theory, Methodology, and/or Algorithmic Discussions

Koopmans-beckman mathematical formulation.

Economists Koopmans and Beckman began their investigation of the QAP to ascertain the optimal method of locating important economic resources in a given area. The Koopmans-Beckman formulation of the QAP aims to achieve the objective of assigning facilities to locations in order to minimize the overall cost. Below is the Koopmans-Beckman formulation of the QAP as described by neos-guide.org.

Quadratic Assignment Problem Formulation

$F=(F_{ij})$

Inner Product

$A,B$

Note: The true objective cost function only requires summing entries above the diagonal in the matrix comprised of elements

$F_{i,j}(X_{\phi }DX_{\phi }^{T})_{i,j}$

Since this matrix is symmetric with zeroes on the diagonal, dividing by 2 removes the double count of each element to give the correct cost value. See the Numerical Example section for an example of this note.

Optimization Problem

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

$\min _{X\in P}\langle F,XDX^{T}\rangle$

Computational Complexity

QAP belongs to the classification of problems known as NP-complete, thus being a computationally complex problem. QAP’s NP-completeness was proven by Sahni and Gonzalez in 1976, who states that of all combinatorial optimization problems, QAP is the “hardest of the hard”. [2]

Algorithmic Discussions

While an algorithm that can solve QAP in polynomial time is unlikely to exist, there are three primary methods for acquiring the optimal solution to a QAP problem:

Dynamic Program
Cutting Plane

Branch and Bound Procedures

The third method has been proven to be the most effective in solving QAP, although when n > 15, QAP begins to become virtually unsolvable.

The Branch and Bound method was first proposed by Ailsa Land and Alison Doig in 1960 and is the most commonly used tool for solving NP-hard optimization problems.

A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores branches of this tree, which represent subsets of the solution set. Before one lists all of the candidate solutions of a branch, the branch is checked against upper and lower estimated bounds on the optimal solution, and the branch is eliminated if it cannot produce a better solution than the best one found so far by the algorithm.

Linearizations

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

$min\sum _{i=1}^{n}\sum _{j=1}^{n}c{_{\phi (i)\phi (j)}}+\sum _{i=1}^{n}b{_{\phi (i)}}$

in order to transform the problem into a (mixed) 0-1 linear program. The objective function is usually linearized by introducing new variables and new linear (and binary) constraints. Then existing methods for (mixed) linear integer programming (MILP) can be applied. The very large number of new variables and constraints, however, usually poses an obstacle for efficiently solving the resulting linear integer programs. MILP formulations provide LP relaxations of the problem which can be used to compute lower bounds.

Numerical Example

Qap with 3 facilities.

$D={\begin{bmatrix}0&5&6\\5&0&3.6\\6&3.6&0\end{bmatrix}}$

Cost for Each Permutation in
Permutation	Cost
(123)	91.4
	99.8
	98.4
	86.5
	103.3
	90

$\phi _{4}=(13)$

Applications

Inter-plant transportation problem.

The QAP was first introduced by Koopmans and Beckmann to address how economic decisions could be made to optimize the transportation costs of goods between both manufacturing plants and locations. [1] Factoring in the location of each of the manufacturing plants as well as the volume of goods between locations to maximize revenue is what distinguishes this from other linear programming assignment problems like the Knapsack Problem.

The Backboard Wiring Problem

As the QAP is focused on minimizing the cost of traveling from one location to another, it is an ideal approach to determining the placement of components in many modern electronics. Leon Steinberg proposed a QAP solution to optimize the layout of elements on a blackboard by minimizing the total amount of wiring required. [4]

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

$E=\left\{E_{1},E_{2},...,E_{n}\right\}$

as well as a set of r points

$P_{1},P_{2},...,P_{r}$

In his paper he derives the below formula:

$min\sum _{1\leq i\leq j\leq n}^{}C_{ij}(d_{s(i)s(j))})$

In his paper Steinberg a backboard with a 9 by 4 array, allowing for 36 potential positions for the 34 components that needed to be placed on the backboard. For the calculation, he selected a random initial placement of s1 and chose a random family of 25 unconnected sets.

The initial placement of components is shown below:

After the initial placement of elements, it took an additional 35 iterations to get us to our final optimized backboard layout. Leading to a total of 59 iterations and a final wire length of 4,969.440.

Hospital Layout

Building new hospitals was a common event in 1977 when Alealid N Elshafei wrote his paper on "Hospital Layouts as a Quadratic Assignment Problem". [5] With the high initial cost to construct the hospital and to staff it, it is important to ensure that it is operating as efficiently as possible. Elshafei's paper was commissioned to create an optimization formula to locate clinics within a building in such a way that minimizes the total distance that a patient travels within the hospital throughout the year. When doing a study of a major hospital in Cairo he determined that the Outpatient ward was acting as a bottleneck in the hospital and focused his efforts on optimizing the 17 departments there.

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

$min\sum _{i,j}\sum _{k,q}f_{ik}d_{jq}y_{ij}y_{kq}$

For the Cairo hospital with 17 clinics, and one receiving and recording room bringing us to a total of 18 facilities. By running the above optimization Elshafei was able to get the total distance per year down to 11,281,887 from a distance of 13,973,298 based on the original hospital layout.

Exam Scheduling System

The scheduling system uses matrices for Exams, Time Slots, and Rooms with the goal of reducing the rate of schedule conflicts. To accomplish this goal, the “examination with the highest cross faculty student is been prioritized in the schedule after which the examination with the highest number of cross-program is considered and finally with the highest number of repeating student, at each stage group with the highest number of student are prioritized.” [6]

$n!$

↑ 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.

Quadratic Assignment Problem (QAP)

Andrey Karenskih

Jul 2, 2024

Introduction

The Quadratic Assignment Problem (QAP) is one of the fundamental optimization problems in operations research and has long been challenging to solve, particularly for dynamic, fast-changing scenarios. In this blog post, we demonstrate how we formulated the QAP as a QUBO problem, ran it on the LightSolver platform and benchmarked it against two solvers from the Google OR-Tools suite.

The Challenge: Real-time Assignments

Imaginee you are the logistics manager of a bustling e-commerce warehouse, where hundreds of orders flood in every hour and new shipments of goods arrive continuously. Your challenge is to ensure that each item is stored in an optimal location to minimize the time and effort needed for picking and packing. As customer demands fluctuate and new products are introduced, the complexity of assigning storage locations dynamically increases. This is where the QAP comes into play, helping you determine the best arrangement of products to enhance efficiency, reduce operational costs, and accelerate order fulfillment in real-time.

Considered one of the most challenging NP-hard problems , the QAP has been used in fields like facility layout design, electronic design, and logistics, traditionally. It involves assigning a set of facilities to a set of locations in such a way that the total cost, influenced by both the distance between locations and the interaction between facilities, is minimized. Imagine trying to determine the best positions for machines in a factory or servers in a data center, where the objective is to reduce the total cost of transporting materials or data between them. The QAP captures this real-world problem, seeking an optimal solution among the many possible layouts. Despite its practical importance, solving QAP efficiently is notoriously difficult due to the combinatorial explosion of possible assignments as the number of facilities and locations increases.

This challenge becomes intractable with conventional computing platforms for dynamic use cases such as the above-mentioned e-commerce warehouse optimization, where solutions must be obtained within seconds or minutes to deliver optimal value. Additional dynamic QAP applications include network design in telecommunications (real-time allocation of transmitters to frequencies to adapt to traffic patterns), dynamic hospital layout (assigning rooms and facilities in function of needs and emergencies), path planning and task assignment for robot arms in manufacturing settings, dynamic electronic component placement for PCB assembly, emergency response resource allocation, and traffic flow management in smart cities. Delivering optimized solutions for such fast-paced scenarios demands advanced solving algorithms and highly efficient computing platforms.

Putting QUBO to work

Let’s start by putting down the mathematical definition of the QAP: MATHEMATICAL DEFINITION

𝑛 facilities to be assigned to 𝑛 locations.
d i , j is the distance between location i and location j.
f i , j is the flow between facility i and facility j
A solution is a bijective function σ:{1,…,n}→{1,…,n} which assigns facility i to location σ(i)
For a given solution σ, the objective value of QAP is: ∑ i , j f i , j d σ i , σ j

Our goal is to find an assignment σ that minimizes the cost defined above. QUBO FORMULATION Here is how we created the QUBO:

We introduce the binary variables x ij that represent the assignment of the ith facility to the j th location.
For σ to be bijective we must have each facility assigned to exactly one factory. To achieve this, we add a penalty term if these conditions are violated: P e n a l t y x = λ ∑ i ∑ j x i j – 1 2 + λ ∑ j ∑ i x i j – 1 2
The objective value introduced before can be expressed in terms of the variables x i j as follows: Obj ( x ) = ∑ i , j , k , l f ij d kl x ik x jl
Finally, the QUBO formulation for a QAP problem is: m i n χ i , j ∈ { 0 , 1 } O b j x + Penalty x

Benchmark against Google OR-Tools Solvers

To evaluate the feasibility of our platform for dynamic QAP use cases, we decided to limit the run time for this benchmark to 60 seconds. We chose data sets from the COR@L library , starting with instances of 12 locations and increasing the problem size until the solvers were unable to provide solutions. Our evaluation criterium was the solution quality, expressed by the size of the gap from the best-known objective value, with the smallest gap being the best result. Here is how the LightSolver platform performed against CP-SAT and GSCIP from the Google-OR package 9.9.3963:

LightSolver constantly delivered higher solution quality (smaller gaps from the optimal known solution) than the two Google OR-Tools solvers, regardless of the problem size. Within the set runtime, CP-SAT could not generate solutions for instances with 50+ facilities, nor could GSCIP for problems of 60+ facilities.

Since its initial introduction in the 1950s, the QAP continues to challenge the operations research community. Despite the amount of data available in real time today, many solvers cannot generate solutions for dynamic scenarios due to the explosion of combinatorial possibilities. LightSolver’s platform delivers accurate and ultra-fast solutions that enable organizations to optimize their operations, saving valuable time, resources, and in the case of emergency response resource allocation, even lives.

Would you like to see how LightSolver can tackle YOUR optimization challenge? Contact us at [email protected]

About the author

Andrey Karenskih is a simulation and benchmark expert at LightSolver and has a background in algorithm research and machine learning. When not working at the office or studying for his M.Sc. in mathematics at Tel Aviv University, Andrey enjoys doing origami. His most complex project so far is a crocodile comprised of over 300 steps, created over three days with very short nights.

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Quadratic Assignment Problem

The objective of the Quadratic Assignment Problem (QAP) is to assign $n$ facilities to $n$ locations in such a way as to minimize the assignment cost. The assignment cost is the sum, over all pairs, of the flow between a pair of facilities multiplied by the distance between their assigned locations.

Problem Statement

The quadratic assignment problem (QAP) was introduced by Koopmans and Beckman in 1957 in the context of locating “indivisible economic activities”. The objective of the problem is to assign a set of facilities to a set of locations in such a way as to minimize the total assignment cost. The assignment cost for a pair of facilities is a function of the flow between the facilities and the distance between the locations of the facilities.

Consider a facility location problem with four facilities (and four locations). One possible assignment is shown in the figure below: facility 2 is assigned to location 1, facility 1 is assigned to location 2, facility 4 is assigned to location 3, and facility 4 is assigned to location 3. This assignment can be written as the permutation $p = \{2, 1, 4, 3\}$, which means that facility 2 is assigned to location 1, facility 1 is assigned to location 2, facility 4 is assigned to location 3, and facility 4 is assigned to location 3. In the figure, the line between a pair of facilities indicates that there is required flow between the facilities, and the thickness of the line increases with the value of the flow.

To calculate the assignment cost of the permutation, the required flows between facilities and the distances between locations are needed.

\[\begin{array}{|c|c|c|}
\text{facility } i & \text{facility } j & \text{flow}(i,j) \\ \hline \hline
1 & 2 & 3 \\
1 & 4 & 2 \\
2 & 4 & 1 \\
3 & 4 & 4
\end{array}\]

$\begin{array}{|c|c|c|}
\text{location } i & \text{location } j & \text{distance}(i,j) \\ \hline
1 & 3 & 53 \\
2 & 1 & 22 \\
2 & 3 & 40 \\
3 & 4 & 55
\end{array}$

Then, the assignment cost of the permutation can be computed as $f(1,2) \cdot d(2,1) + f(1,4) \cdot d(2,3) + f(2,4) \cdot d(1,3) + f(3,4) \cdot d(3,4)$ = $3 \cdot 22 + 2 \cdot 40 + 1 \cdot 53 + 4 \cdot 55$ = 419. Note that this permutation is not the optimal solution.

Mathematical Formulation

Here we present the Koopmans-Beckmann formulation of the QAP. Given a set of facilities and locations along with the flows between facilities and the distances between locations, the objective of the Quadratic Assignment Problem is to assign each facility to a location in such a way as to minimize the total cost.

Sets $N = \{1, 2, \cdots, n\}$ $S_n = \phi: N \rightarrow N$ is the set of all permutations

Parameters $F = (f_{ij})$ is an $n \times n$ matrix where $f_{ij}$ is the required flow between facilities $i$ and $j$ $D = (d_{ij})$ is an $n \times n$ matrix where $d_{ij}$ is the distance between locations $i$ and $j$

Optimization Problem $\text{min}_{\phi \in S_n} \sum_{i=1}^n \sum_{j=1}^n f_{ij} \cdot d_{\phi(i) \phi(j)}$

The assignment of facilities to locations is represented by a permutation $\phi$, where $\phi(i)$ is the location to which facility $i$ is assigned. Each individual product $f_{ij} \cdot d_{\phi(i) \phi(j)}$ is the cost of assigning facility $i$ to location $\phi(i)$ and facility $j$ to location $\phi(j)$.

Solve some QAPs!

Follow the links below to test your skill at finding good solutions to QAPs of various sizes. Notice that as the problem size increases, it becomes much more difficult to find an optimal solution. As $n$ increases beyond a small number, it becomes impossible to enumerate and evaluate all possible assignment vectors. Instead, advanced solution algorithms are required to solve larger instances.

QAP of size 4

Qap of size 5, qap of size 6, qap of size 7, qap of size 8, qap of size 9.

Anstreicher, K.M. 2003. Recent advances in the solution of quadratic assignment problems. Mathematical Programming Series B 97 , 27 - 42.
Çela, E. 1998. The Quadratic Assignment Problem: Theory and Algorithms . Kluwer Academic Publishers, Dordrecht.
Koopmans, T. C. and M. J. Beckmann. 1957. Assignment problems and the location of economic activities. Econometrica 25 , 53 - 76.
QAPLIB Home Page

The Quadratic Assignment Problem

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The Quadratic Assignment Problem

Related Papers

Encyclopedia of Optimization

Panos Pardalos

Panos M Pardalos

Abstract This paper aims at describing the state of the art on quadratic assignment problems (QAPs). It discusses the most important developments in all aspects of the QAP such as linearizations, QAP polyhedra, algorithms to solve the problem to optimality, heuristics, polynomially solvable special cases, and asymptotic behavior.

Güneş Erdoğan

The Quadratic Assignment Problem (QAP) is one of the hardest combinatorial optimization problems known. Exact solution attempts proposed for instances of size larger than 15 have been generally unsuccessful even though successful implementations have been reported on some test problems from the QAPLIB up to size 36. In this dissertation, we analyze the binary structure of the QAP and present new IP formulations. We focus on “flow-based” formulations, strengthen the formulations with valid inequalities, and report computational experience with a branch-and-cut algorithm. Next, we present new classes of instances of the QAP that can be completely or partially reduced to the Linear Assignment Problem and give procedures to check whether or not an instance is an element of one of these classes. We also identify classes of instances of the Koopmans-Beckmann form of the QAP that are solvable in polynomial time. Lastly, we present a strong lower bound based on Bender’s decomposition.

Cesar Beltran-Royo

Pesquisa Operacional

Paulo Boaventura Netto

We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branch-and-bound type algorithm for the quadratic assignment problem.

Computers & Operations Research

Ricardo P M Ferreira

Ilyes Beltaef

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Environment and Planning B: Urban Analytics and City Science

The Quadratic Assignment Problem: An Analysis of Applications and Solution Strategies

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DOI: 10.1090/dimacs/016/01
Corpus ID: 9547090

The Quadratic Assignment Problem: A Survey and Recent Developments

P. Pardalos , F. Rendl , Henry Wolkowicz
Published in Quadratic Assignment and… 1993
Mathematics, Computer Science

347 Citations

Selected topics on assignment problems, new heuristic rounding approaches to the quadratic assignment problem, quadratic assignment problem: a survey and applications, a survey of quadratic assignment problems, a survey of the quadratic assignment problem, equality of complexity classes p and np: linear programming formulation of the quadratic assignment problem, on the equality of complexity classes p and np: linear programming formulation of the quadratic assignment problem, a survey of the quadratic assignment problem, with applications.

Highly Influenced

QAPLIB – A Quadratic Assignment Problem Library

Quadratic and multidimensional assignment problems, 218 references, a parallel branch and bound algorithm for the quadratic assignment problem, a parallel algorithm for the quadratic assignment problem, quadratic assignment problems, a new lower bound for the quadratic assignment problem, applications of parametric programming and eigenvalue maximization to the quadratic assignment problem, special cases of the quadratic assignment problem, on the mapping problem, a heuristic for quadratic boolean programs with applications to quadratic assignment problems, an introduction to parallelism in combinatorial optimization, a graph theoretic analysis of bounds for the quadratic assignment problem, related papers.

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The Quadratic Assignment Problem

pp 1713–1809

Cite this chapter

Rainer E. Burkard 3 ,
Eranda Çela 3 ,
Panos M. Pardalos 4 &
Leonidas S. Pitsoulis 4

4523 Accesses

80 Citations

The quadratic assignment problem (QAP) was introduced by Koopmans and Beckmann in 1957 as a mathematical model for the location of a set of indivisible economical activities [113]. Consider the problem of allocating a set of facilities to a set of locations, with the cost being a function of the distance and flow between the facilities, plus costs associated with a facility being placed at a certain location. The objective is to assign each facility to a location such that the total cost is minimized. Specifically, we are given three n x n input matrices with real elements F = ( f ij ), D = ( d kl ) and B = ( b ik ), where f ij is the flow between the facility i and facility j , d kl is the distance between the location k and location l , and b ik is the cost of placing facility i at location k . The Koopmans-Beckmann version of the QAP can be formulated as follows: Let n be the number of facilities and locations and denote by N the set N = {1, 2,..., n}.

These authors have been supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung.

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Quadratic Assignment Problems

A linear formulation with $$o(n^2)$$ variables for quadratic assignment problems with manhattan distance matrices.

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Burkard, R.E., Çela, E., Pardalos, P.M., Pitsoulis, L.S. (1998). The Quadratic Assignment Problem. In: Du, DZ., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0303-9_27

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Quadratic Assignment Problem (QAP)
The Quadratic Assignment Problem (QAP) is an optimization problem that deals with assigning a set of facilities to a set of locations, considering the pairwise distances and flows between them.. The problem is to find the assignment that minimizes the total cost or distance, taking into account both the distances and the flows. The distance matrix and flow matrix, as well as restrictions to ...
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Quadratic Assignment Problem Formulation Parameters = is an n x n matrix where the required flow between facilities and . = is an n x n matrix where the distance between facilities and and . Intuitively, the product of distance and flow represents cost, and the objective is to minimize this cost. ...
PDF The Quadratic Assignment Problem
The quadratic assignment problem (QAP) was introduced by Koopmans and Beckmann in ... 2.2 Concave Quadratic Formulation In the objective function of (3), let the coeﬃcients cijklbe the entries of an n2 ×n2 matrix S, such that cijkl is on row (i− 1)n+ kand column (j− 1)n+ l. Now let Q:= S− αI,
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Introduction. The Quadratic Assignment Problem (QAP) is one of the fundamental optimization problems in operations research and has long been challenging to solve, particularly for dynamic, fast-changing scenarios. In this blog post, we demonstrate how we formulated the QAP as a QUBO problem, ran it on the LightSolver platform and benchmarked ...
A comprehensive review of quadratic assignment problem: variants
The quadratic assignment problem (QAP) has considered one of the most significant combinatorial optimization problems due to its variant and significant applications in real life such as scheduling, production, computer manufacture, chemistry, facility location, communication, and other fields. QAP is NP-hard problem that is impossible to be solved in polynomial time when the problem size ...
Quadratic assignment problem
The quadratic assignment problem (QAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics, from the category of the facilities location problems first introduced by Koopmans and Beckmann.. The problem models the following real-life problem: There are a set of n facilities and a set of n locations.
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The quadratic assignment problem (QAP) is considered one of the most difficult optimization problems to solve optimally. The QAP is a combinatorial optimization problem stated for the first time by Koopmans and Beckmann ().Early papers on the subject include Gilmore (), Pierce and Crowston (), Lawler (), and Love and Wong ().The problem is defined as follows.
Quadratic Assignment Problems
The quadratic assignment problem (QAP) was introduced by Koopmans and Beckmann in 1957 as a mathematical model for the location of indivisible economical activities [].Consider the problem of allocating n facilities to n locations, with the cost being a function of the distance and flow between the facilities plus costs associated with placing a facility at a certain location.
PDF A Survey of the Quadratic Assignment Problem, with Applications
The quadratic assignment problem (QAP) was originally introduced in 1957 by Tjalling C. Koopmans and Martin Beckman [26] who were trying to 1. ... 2.2.2 A Quadratic 0-1 Formulation What follows is an equivalent formulation of the QAP as a quadratic 0-1 in-teger program. This formulation was originally used by Koopmans-Beckman
quadratic assignment problem
The Quadratic Assignment Problem (QAP) has been the subject of an enormous amount of research efforts; ... We present here a generalized formulation of the assignment problem between N objects and M labels, from which most commonly studied variations are special cases. We express all families of assignment problem in this generalized form for ...
PDF Quadratic Assignment Problem
The Quadratic assignment problem (QAP) is one of the fundamental, interesting and challenging combinatorial optimization problems from the category of the facilities location/allocation problems. QAP considers the problem of allocating a set of n facilities to a set of. n locations, with the cost being a function of the distance dkl between the ...
(PDF) The quadratic assignment problem: A survey and recent
The Quadratic Assignment Problem (QAP) is one of the hardest combinatorial optimization problems known. ... H subgraph of G ; H Gg: 0 0 The graph theoretic formulation of quadratic assignment problem was proposed by Christo des and Gerrards as follows. Let G and G be graphs with edge weights a : E 7! <; b : E 7! <. 0 0 THE QUADRATIC ASSIGNMENT ...
Quadratic Assignment Problem
The quadratic assignment problem (QAP) was introduced by Koopmans and Beckman in 1957 in the context of locating "indivisible economic activities". The objective of the problem is to assign a set of facilities to a set of locations in such a way as to minimize the total assignment cost. The assignment cost for a pair of facilities is a ...
Quadratic Assignment Problem
The quadratic assignment problem (QAP) is a combinatorial optimization problem, that although there is a substantial amount of research devoted to it, it is still, up to this ... The mathematical formulation of this problem leads to an NP-hard anti-Monge-Toeplitz QAP.
A survey for the quadratic assignment problem
The international literature identifies the quadratic assignment problem (QAP) as the problem of finding a minimum cost allocation of facilities into locations, taking the costs as the sum of all possible distance-flow products. The main motivation for this survey is the continuous interest in QAP, shown by a number of researchers worldwide ...
(PDF) The Quadratic Assignment Problem
The GLB can easily pro- The proposed formulation differs from the standard quadratic assignment problem (QAP) (e.g., [51]), in terms of its sets of constraints. In the QAP, facilities need to be ...
(PDF) The Quadratic Assignment Problem
The solution of QAP (F, D) produced by an ǫ-approximation algorithm is called an ǫ-approximate solution. Theorem 3.2 (Sahni and Gonzalez [164], 1976) The quadratic assignment problem is strongly NP-hard. For an arbitrary ǫ > 0, the existence of a polynomial time ǫ-approximation algorithm for the QAP implies P = N P.
The Quadratic Assignment Problem: An Analysis of Applications and
A wide variety of practical problems in design, planning, and management can be formulated as quadratic assignment problems, and this paper discusses this class of problem. Since algorithms for producing optimal solutions to such problems are computationally infeasible for all but small problems of this type, heuristic techniques must usually ...
Quadratic Assignment Problem Example
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meaning of quadratic assignment problem, solving techniques and we will give a survey of some developments and researches. Keywords: 6 2 0 3 4Quadratic Assignment Problem, formulation, Exact Algorithm, NP-complete, Bound, Heuristic. 1. INTRODUCTION Quadratic assignment problem (QAP) was firstly introduced
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The quadratic assignment problem (QAP) is a combinatorial optimization problem, that although there is a substantial amount of research devoted to it, it is still, up to this date, not well solvable in the sense that no exact algorithm can solve problems of size n > 20 in reasonable computational time. The QAP can be viewed as a natural extension of the linear assignment problem (LAP; cf. also ...
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This paper surveys some of the most important techniques, applications, and methods regarding the quadratic assignment problem. . Quadratic Assignment Problems model many applications in diverse areas such as operationsresearch, parallel and distributedcomput-ing, and combinatorial data analysis. In this paper we survey some of the most important techniques, applications, and methods regarding ...
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The quadratic assignment problem (QAP) was introduced by Koopmans and Beckmann in 1957 as a mathematical model for the location of a set of indivisible economical activities [113]. ... M. S. Bazaraa and H. D. Sherali, Benders' partitioning scheme applied to a new formulation of the quadratic assignment problem, Naval Research Logistics ...

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