Artykuł wyjaśnia, jak porównywać efektywność techniczną i organizacyjną piekarni wtedy, gdy trzeba jednocześnie uwzględnić wiele nakładów i wiele efektów. Autor pokazuje ograniczenia klasycznych wskaźników i modeli ekonometrycznych, a następnie omawia metodę DEA w wersji CCR, opracowaną w 1978 roku przez Charnesa, Coopera i Rhodesa. Na przykładzie pięciu piekarni oferowanych grupie inwestorów przedstawiono, jak zbudować układ nierówności programowania liniowego i rozwiązać go w Pythonie za pomocą bibliotek PuLP i Fractions. Analiza wskazała piekarnie graniczne — Łopatę i Lipniki — oraz zakład najsłabszy, ustępujący im o około 30%.
Assessing Bakery Efficiency Using the CCR-DEA Model
The article explains how to compare the technical and organizational efficiency of bakeries when many inputs and many outputs have to be taken into account at the same time. The author shows the limitations of classic indicators and econometric models, and then discusses the DEA method in its CCR variant, developed in 1978 by Charnes, Cooper and Rhodes. Using the example of five bakeries offered to a group of investors, the text shows how to build a system of linear-programming inequalities and solve it in Python with the PuLP and Fractions libraries. The analysis identified the frontier bakeries — Łopata and Lipniki — and the weakest plant, some 30% behind them.
The DEA CCR model is used to determine the efficiency level of a particular organization in comparison with other, similar organizations. It is therefore a benchmarking tool—that is, a tool for making comparisons, creating rankings and finding differences between similar objects. Benchmarking is easy when objects are compared in terms of one characteristic, such as revenue or profit.
Unfortunately, in economic life we encounter objects of different sizes, where we are often forced to compare several dozen different characteristics simultaneously. Usually, it is not two objects that are compared, but an entire series of them. Comparing such a set of multidimensional objects without the use of specialist tools drastically exceeds the capabilities of human perception.
To make the task of comparing many objects with similar characteristics and functions easier, analysts invented indicators. One such indicator is, for example, revenue per employee or the ratio of profit to revenue.
Indicators effectively reduce differences in the sizes of objects, but they do not solve the problem of the large number of characteristics to be compared. The comparison process becomes laborious when a large number of indicators must be compared. A problem arises concerning the objectivity of analyses in which the analyst decides at their own discretion how to construct the indicators and indicates which indicators are more important and which are less important.
A modern method for comparing the efficiency of economic entities is the DEA (Data Envelopment Analysis) linear-programming algorithm proposed by Charnes, Cooper and Rhodes in 1978. In this article, we will discuss the first version of this algorithm, called CCR after the initial letters of its creators’ surnames.
In particular, the CCR model solves the problem of comparing the efficiency of target technologies, the structure of optimal technologies, and the analysis of reports from the simplex linear-programming methodology.
The DEA method is used in comparative analyses of the efficiency of entities in which a process is described by more than one input and more than one output.
In multidimensional comparisons, the use of both indicators and econometric models is ineffective. These methods fail because, with limited information, we are unable to determine how much of a particular type of input was spent directly on obtaining the individual outputs.
An example of applying the DEA CCR model
A group of investors intends to buy a bakery. Five plants have been offered to them. The analysts working for the investors must indicate which bakeries are the best in terms of technological and organizational efficiency. This information is crucial to deciding which bakery to buy. To avoid a subjective assessment, it was decided to use the DEA CCR algorithm.
The analysts have the following data:
| Bakery | Bobry | Łopata | Lipniki | Nowa | Bartniki |
|---|---|---|---|---|---|
| Infrastructure value, thousands | 939.0 | 466 | 88.83 | 314.6 | 1503.7 |
| Machinery value, thousands | 3493.5 | 976 | 619.2 | 564.2 | 5871.8 |
| Production employment | 57.0 | 23 | 11.7 | 79.3 | 44.0 |
| Costs, thousands | 796.5 | 283 | 67.95 | 278.2 | 2312.2 |
| Production volume, tonnes | 9471 | 5859 | 2239.2 | 2478.0 | 10205.4 |
| Gross profit, thousands | 28 | 12 | 44.4 | 12.6 | 22.8 |
In the DEA method, it is assumed that the technology of object o that is optimal from the point of view of efficiency (1 ≤ o ≤ J) is a linear combination of the empirical technologies of the individual objects.
The description of the technology of every object is its input vector and its output vector.
On the basis of the table, we can assume that the input vector for the Bobry bakery is:
Inputs = [939; 3493.5; 57; 796.5]
Outputs = [9471; 28]To compare the technological efficiency of the Bobry bakery, the following system of linear-programming inequalities must be solved:
939x₁ + 466x₂ + 88.83x₃ + 314.6x₄ + 1503.7x₅ ≤ 939θ₁ (1)
3493.5x₁ + 976x₂ + 619.2x₃ + 564.2x₄ + 5871x₅ ≤ 3493.5θ₁ (2)
57x₁ + 23x₂ + 11.7x₃ + 79.3x₄ + 44.0x₅ ≤ 57θ₁ (3)
796.5x₁ + 283x₂ + 67.95x₃ + 278.2x₄ + 2312.2x₅ ≤ 796.5θ₁ (4)
9471x₁ + 5859x₂ + 2239.2x₃ + 2478.0x₄ + 10205.4x₅ ≥ 9471 (5)
28x₁ + 12x₂ + 44.4x₃ + 12.6x₄ + 22.8x₅ ≥ 28 (6)
θ₂ ≤ 0It is easy to see that the values in the system of inequalities above come from the data table received by the analysts.
The input and output vectors of the Bobry bakery also appear on the right-hand side of the inequalities.
We proceed in the same way with the other bakeries.
Łopata bakery
939x₁ + 466x₂ + 88.83x₃ + 314.6x₄ + 1503.7x₅ ≤ 446θ₂ (1)
3493.5x₁ + 976x₂ + 619.2x₃ + 564.2x₄ + 5871x₅ ≤ 976θ₂ (2)
57x₁ + 23x₂ + 11.7x₃ + 79.3x₄ + 44x₅ ≤ 23θ₂ (3)
796.5x₁ + 283x₂ + 67.95x₃ + 278.2x₄ + 2312.2x₅ ≤ 283θ₂ (4)
9471x₁ + 5859x₂ + 2239.2x₃ + 2478x₄ + 10205.4x₅ ≥ 5859 (5)
28x₁ + 12x₂ + 44.4x₃ + 12.6x₄ + 22.8x₅ ≥ 12 (6)
θ₂ ≤ 0Lipniki bakery
939x₁ + 466x₂ + 88.83x₃ + 314.6x₄ + 1503.7x₅ ≤ 88.83θ₃ (1)
3493.5x₁ + 976x₂ + 619.2x₃ + 564.2x₄ + 5871x₅ ≤ 619.2θ₃ (2)
57x₁ + 23x₂ + 11.7x₃ + 79.3x₄ + 44x₅ ≤ 11.7θ₃ (3)
796.5x₁ + 283x₂ + 67.95x₃ + 278.2x₄ + 2312.2x₅ ≤ 67.95θ₃ (4)
9471x₁ + 5859x₂ + 2239.2x₃ + 2478x₄ + 10205.4x₅ ≥ 2239.2 (5)
28x₁ + 12x₂ + 44.4x₃ + 12.6x₄ + 22.8x₅ ≥ 44.4 (6)
θ₂ ≤ 0Nowa bakery
939x₁ + 466x₂ + 88.83x₃ + 314.6x₄ + 1503.7x₅ ≤ 314.6θ₄ (1)
3493.5x₁ + 976x₂ + 619.2x₃ + 564.2x₄ + 5871x₅ ≤ 564.2θ₄ (2)
57x₁ + 23x₂ + 11.7x₃ + 79.3x₄ + 44x₅ ≤ 79.3θ₄ (3)
796.5x₁ + 283x₂ + 67.95x₃ + 278.2x₄ + 2312.2x₅ ≤ 278.2θ₄ (4)
9471x₁ + 5859x₂ + 2239.2x₃ + 2478.0x₄ + 10205.4x₅ ≥ 2478 (5)
28x₁ + 12x₂ + 44.4x₃ + 12.6x₄ + 22.8x₅ ≥ 12.6 (6)
θ₂ ≤ 0Bartniki bakery
939x₁ + 466x₂ + 88.83x₃ + 314.6x₄ + 1503.7x₅ ≤ 1503.7θ₅ (1)
3493.5x₁ + 976x₂ + 619.2x₃ + 564.2x₄ + 5871x₅ ≤ 5871.8θ₅ (2)
57x₁ + 23x₂ + 11.7x₃ + 79.3x₄ + 44x₅ ≤ 44θ₅ (3)
796.5x₁ + 283x₂ + 67.95x₃ + 278.2x₄ + 2312.2x₅ ≤ 2312.2θ₅ (4)
9471x₁ + 5859x₂ + 2239.2x₃ + 2478x₄ + 10205.4x₅ ≥ 10205.4 (5)
28x₁ + 12x₂ + 44.4x₃ + 12.6x₄ + 22.8x₅ ≥ 22.8 (6)
θ₂ ≤ 0There are many methods of solving the linear-programming inequalities above without using computers. It is worth, however, using computer programs or simplex calculators available online. I used two free Python libraries named PuLP and Fractions.
| Mill | Score |
|---|---|
| Bobry | 0.705 |
| Łopata | 1 |
| Lipniki | 1 |
| Nowa | 0.814 |
| Bartniki | 0.914 |
The Łopata and Lipniki bakeries proved to be the best technologically. These are so-called frontier objects. When the CCR model is applied, it most often does not indicate one best object, but rather a group of objects that are at a relatively similar level. Bobry proved to be the worst bakery, being 30% worse than the frontier bakeries.
Wojciech Moszczyński — graduate of the Department of Econometrics and Statistics of Nicolaus Copernicus University in Toruń; specialist in econometrics, finance, data science, and management accounting. He specializes in the optimization of production and logistics processes. He conducts research in the area of the development and application of artificial intelligence. For years he has been engaged in the popularization of machine learning and data science in business environments.