science

Theory of two-level planning

Theory of two-level planning is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Theory of two-level planning rather than just read about it. In short: The theory of two-level planning (alternatively, Kornai–Liptak decomposition) is a method that decomposes large problems of linear optimization into sub-problems. This decomposition simplifies the solution of the overall problem.

Key takeaways

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

Reference excerpt

The theory of two-level planning (alternatively, Kornai–Liptak decomposition) is a method that decomposes large problems of linear optimization into sub-problems. This decomposition simplifies the solution of the overall problem. The method also models a method of coordinating economic decisions so that decentralized firms behave so as to produce a global optimum. It was introduced by the Hungarian economist János Kornai and the mathematician Tamás Lipták in 1965. It is an alternative to Dantzig–Wolfe decomposition.

Description The LP problem must have a special structure, known as a block angular structure. This is the same structure required for the Dantzig Wolfe decomposition:

There are some constraints on overall resources (D) for which a central planning agency is assumed to be responsible, and n blocks of coefficients (F1 through Fn) that are the concern of individual firms. The central agency starts the process by providing each firm with tentative resource allocations which satisfy the overall constraints D. Each firm optimizes its local decision variables assuming the global resource allocations are as indicated. The solution of the firm LP's yield Lagrange multipliers (prices) for the global resources which the firms transmit back to the planning agency. In the next iteration, the central agency uses the information received from firms to come up with a revised resource allocation; for example if firm i reports a high shadow price for resource j, the agency will grant more of this resource to this firm and less to other firms. The revised tentative allocations are sent back to the individual firms and the process continues. It has been shown that this process will converge (though not necessarily in a finite number of steps) towards the global solution for the overall problem. (In contrast the Dantzig Wolfe method converges in a finite number of steps). The DW and KL methods are dual: in DW the central market establishes prices (based on firm demands for resources) and sends these to the firms who then modify the quantities they demand, while in KL the central agency sends out quantity information to firms and receives bids (i.e. firm specific pricing information) from firms.

See also Dantzig–Wolfe decomposition Benders' decomposition Column generation

References J. Kornai, T. Liptak: Two-level Planning, Econometrica, 1965, Vol. 33, pp. 141–169. [1]

Worked examples

Example 1 — a first encounter with Theory of two-level planning

Start with the simplest possible case. Write down what Theory of two-level planning 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 Theory of two-level planning 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 Theory of two-level planning 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 Theory of two-level planning

In research: Theory of two-level planning 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 Theory of two-level planning 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: Theory of two-level planning is common in secondary-school and first-year university syllabi. It links to neighbouring topics Decomposition methods, Linear programming, so understanding it makes those chapters shorter.
In everyday life: Look for Theory of two-level planning 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.

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How to study Theory of two-level planning in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Theory of two-level planning 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.
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Frequently asked questions

What is Theory of two-level planning in simple terms?

Why does Theory of two-level planning 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 Theory of two-level planning?

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 Theory of two-level planning.