What it means
Kantorovich lived from 1912 to 1986 and was a gifted mathematician from a young age. In 1939, while advising a plywood factory on how to get the most output from its machines, he formulated a method for choosing the best mix of activities when resources are limited.
He published it that year in a work on mathematical methods of organising and planning production. The method is now called linear programming.
It asks a simple question: given a goal, such as maximum profit or minimum cost, and a set of limits, such as available machine hours, materials and labour, what is the best plan? The word "linear" means that the goal and the limits are all expressed with simple proportional relationships, which makes the problem solvable even with many variables.
Kantorovich also introduced the idea of shadow prices, which he called objectively determined valuations. A shadow price tells you how much your best result would improve if you had one more unit of a scarce resource.
This gives managers a rational way to decide whether an extra machine, an extra hour of labour or extra funding is worth paying for. The American mathematician George Dantzig independently developed the simplex method in the 1940s, which became the standard way to solve these problems on a computer.
Kantorovich's early work was not widely known in the West until later, and the 1975 Nobel Prize recognised his contribution alongside Koopmans, who had arrived at similar ideas separately. For a finance professional, the legacy is everywhere.
Budget allocation across projects, production planning, supply chain routing, bond portfolio matching and capital rationing all use optimisation methods that descend from his work. Even a spreadsheet solver add-in is applying Kantorovich-style thinking to a small business problem.
In practice
Real-world examples.
Example
A food manufacturer has limited oven time and two product lines. Its planner uses linear programming to choose how many batches of each product to bake, and the result raises weekly profit by 6% compared with the planner's earlier rule of thumb.
Example
A logistics company needs to assign 30 delivery trucks to routes to minimise fuel cost while meeting every delivery deadline. The operations team feeds the costs and constraints into a solver, which finds a plan that saves about $12,000 a month.
Example
A corporate treasurer has $20,000,000 to spread across five short-term investments, each with its own return and limit on how much can be placed. She uses an optimiser to maximise total yield while staying within credit risk rules, and the result is checked against her own judgement.
Formula
Calculation
A linear programme has this general form: maximise profit = (profit per unit of A x units of A) + (profit per unit of B x units of B), subject to resource limits.
Worked example: a workshop makes products A and B. Product A earns $40 profit per unit and uses 2 machine hours. Product B earns $30 profit per unit and uses 1 machine hour. There are 100 machine hours available, and the market will take no more than 60 units of B.
Constraints: 2A + B <= 100 and B <= 60, with A and B at zero or above.
Test the corner points of the feasible region. At A = 0, B = 60, profit = $0 + $1,800 = $1,800. At B = 60, the machine limit gives 2A = 40, so A = 20, profit = (20 x $40) + (60 x $30) = $800 + $1,800 = $2,600. At B = 0, A = 50, profit = 50 x $40 = $2,000.
The best plan is 20 units of A and 60 units of B for a profit of $2,600. The shadow price of a machine hour is $20, since adding one hour lets A rise by half a unit, which adds half of $40.Case study
Seen in the real world.
Pinecrest Furniture is a fictional manufacturer of tables and chairs. Its owner assumed that the more profitable product, tables, should be made first every week, with chairs filling any spare capacity.
An analyst built a simple linear programme showing that chairs earned more profit per hour of scarce saw time, even though a table earned more per unit. Switching to the optimised mix raised weekly profit from $26,000 to $29,500.
The model also showed a shadow price on saw time of $35 per hour, so renting a second saw for $20 an hour was clearly worthwhile. This is an illustrative story, but it demonstrates the kind of insight that Kantorovich's method provides.
Watch out
Common mistakes.
- Thinking linear programming is a computer technique only. It is a way of framing a decision with a goal and limits, and the computer simply does the arithmetic.
- Believing the highest profit per unit gives the best plan. As the worked example shows, the scarce resource determines which product is most valuable to produce.
- Treating the answer as exact reality. The result is only as good as the numbers and assumptions, and inputs such as prices and demand often change.
Questions
People also ask.
What did Kantorovich win the Nobel Prize for?
He shared the 1975 Nobel Memorial Prize in Economic Sciences with Tjalling Koopmans for contributions to the theory of the optimal allocation of resources.
Did Kantorovich invent the simplex method?
No. The simplex method was developed by George Dantzig in the 1940s, while Kantorovich formulated the linear programming problem and its solution ideas in 1939.
What is a shadow price?
It is the extra profit you would gain from one more unit of a scarce resource. It helps decide whether buying more of that resource is worthwhile.
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