Quadratic assignment problem

Key takeaways

Quadratic assignment problem belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Quadratic assignment problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Quadratic assignment problem from memory before moving on to harder problems.

Reference excerpt

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. It was originally proposed by Tjalling Koopmans and Martin J. Beckmann. The problem models the following real-life problem:

There are a set of n facilities and a set of n locations. For each pair of locations, a distance is specified and for each pair of facilities a weight or flow is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the goal of minimizing the sum of the distances multiplied by the corresponding flows. The problem statement resembles that of the assignment problem, except that the cost function is expressed in terms of quadratic inequalities, hence the name.

Formal mathematical definition The formal definition of the quadratic assignment problem is as follows.

Given a positive integer n {\displaystyle n} and n 4 {\displaystyle n^{4}} cost coefficients { c i , j , p , q } {\displaystyle \{c_{i,j,p,q}\}} , find the n × n {\displaystyle n\times n} matrix [ x i j ] {\displaystyle [x_{ij}]} that minimizes the objective function

∑ i , j ∑ p , q c i , j , p , q x i , j x p , q {\displaystyle \sum _{i,j}\sum _{p,q}c_{i,j,p,q}x_{i,j}x_{p,q}}

subject to the constraints

∑ j x i , j = ∑ i x i , j = 1 , {\displaystyle \sum _{j}x_{i,j}=\sum _{i}x_{i,j}=1,}

x i j ∈ { 0 , 1 } {\displaystyle x_{ij}\in \{0,1\}}

Koopmans-Beckmann formulation The quadratic assignment problem was originally posed by Tjalling Koopmans and Martin J. Beckmann in the following form.

Given square matrices D and T, find the permutation matrix X that minimizes the double-dot product of T with A = X D X ⊺ {\displaystyle A=XDX^{\intercal }} . In other words, given D and T, find { x i j } 1 ≤ i , j ≤ n {\displaystyle \{x_{ij}\}_{1\leq i,j\leq n}} so as to

minimize ∑ 1 ≤ i , j ≤ n a i j t i j subject to ∑ 1 ≤ k ≤ n x i k = ∑ 1 ≤ k ≤ n x j k = 1 , x i j ∈ { 0 , 1 } . {\begin{aligned}{\text{minimize}}\quad &\sum _{1\leq i,j\leq n}a_{ij}t_{ij}\\{\text{subject to}}\quad &\sum _{1\leq k\leq n}x_{ik}=\sum _{1\leq k\leq n}x_{jk}=1,\\&x_{ij}\in \{0,1\}.\end{aligned}}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Quadratic assignment problem

Start with the simplest possible case. Write down what Quadratic assignment problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Quadratic assignment problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Quadratic assignment problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Quadratic assignment problem

In research: Quadratic assignment problem appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Quadratic assignment problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Quadratic assignment problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorial optimization, NP-hard problems, Operations research, so understanding it makes those chapters shorter.
In everyday life: Look for Quadratic assignment problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Quadratic assignment problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quadratic assignment problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Quadratic assignment problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quadratic assignment problem in simple terms?

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. It was originally proposed by Tjalling Koopmans and Martin J.

Why does Quadratic assignment problem matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Quadratic assignment problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Quadratic assignment problem.