*2.2.2. Graphical example*

If it is assumed that convex combinations of players are allowed, then the line segment connecting players A and C shows the possibilities of virtual outputs that can be formed from these two players. Similar segments can be drawn between A and B along with B and C. Since the segment AC lies beyond the segments AB and BC, this means that a convex combination of A and C will create the most outputs for a given set of inputs.

This line is called the efficiency frontier. The efficiency frontier defines the maximum combi‐ nations of outputs that can be produced for a given set of inputs. The segment connecting point C to the HR axis is drawn because of disposability of output. It is assumed that if player C can hit 20 home runs and 10 singles, he could also hit 20 home runs without any singles. We have no knowledge though of whether avoiding singles altogether would allow him to raise his home run total so we must assume that it remains constant.

Since player B lies below the efficiency frontier, he is inefficient. His efficiency can be deter‐ mined by comparing him to a virtual player formed from player A and player C. The virtual player, called V, is approximately 64% of player C and 36% of player A. (This can be determined

**Figure 2.** Graphical example of DEA for player B

Now, as a DEA analyst, we combine parts of different players. First let us analyze player A. Clearly no combination of players B and C can produce 40 singles with the constraint of only 100 at-bats. Therefore player A is efficient at hitting singles and receives an efficiency of 1.0.

Now we move on to analyze player B. Suppose we try a 50-50 mixture of players A and C. This

Note that X = 100 = X(0) where X(0) is the input(s) for the DMU being analyzed. Since lambdaY > Y(0) = (20, 5), then there is room to scale down the inputs, X and produce a virtual output vector at least equal to or greater than the original output. This scaling down factor would allow us to put an upper bound on the efficiency of that player's efficiency. The 50-50 ratio of A and C may not necessarily be the optimal virtual producer. The efficiency, theta, can then

It can be seen by inspection that player C is efficient because no combination of players A and B can produce his total of 20 home runs in only 100 at bats. Player C is fulfilling the role of hitting home runs more efficiently than any other player just as player A is hitting singles more efficiently than anyone else. Player C is probably taking a big swing while player A is slapping out singles. Player B would have been more productive if he had spent half his time swinging for the fences like player C and half his time slapping out singles like player A. Since player B was not that productive, he must not be as skilled as either player A or player C and his

This example can be made more complicated by looking at unequal values of inputs instead of the constant 100 at-bats, by making it a multiple input problem, or by adding more data

If it is assumed that convex combinations of players are allowed, then the line segment connecting players A and C shows the possibilities of virtual outputs that can be formed from these two players. Similar segments can be drawn between A and B along with B and C. Since the segment AC lies beyond the segments AB and BC, this means that a convex combination

This line is called the efficiency frontier. The efficiency frontier defines the maximum combi‐ nations of outputs that can be produced for a given set of inputs. The segment connecting point C to the HR axis is drawn because of disposability of output. It is assumed that if player C can hit 20 home runs and 10 singles, he could also hit 20 home runs without any singles. We have no knowledge though of whether avoiding singles altogether would allow him to raise his

Since player B lies below the efficiency frontier, he is inefficient. His efficiency can be deter‐ mined by comparing him to a virtual player formed from player A and player C. The virtual player, called V, is approximately 64% of player C and 36% of player A. (This can be determined

means that lambda=(0.5, 0.5). The virtual output vector is now,

lambda Y = (0.5 \* 40 + 0.5 \* 10, 0.5 \* 0 + 0.5 \* 20) = (25, 10)

456 Computational and Numerical Simulations

be found by solving the corresponding linear program.

efficiency score would be below 1.0 to reflect this.

points but the basic principles still hold. (*Source:* [16]).

of A and C will create the most outputs for a given set of inputs.

home run total so we must assume that it remains constant.

*2.2.2. Graphical example*

by an application of the lever law. Pull out a ruler and measure the lengths of AV, CV, and AC. The percentage of player C is then AV/AC and the percentage of player A is CV/AC.)

The efficiency of player B is then calculated by finding the fraction of inputs that player V would need to produce as many outputs as player B. This is easily calculated by looking at the line from the origin, O, to V. The efficiency of player B is OB/OV which is approximately 68%. This figure also shows that players A and C are efficient since they lie on the efficiency frontier. In other words, any virtual player formed for analyzing players A and C will lie on players A and C respectively. Therefore since the efficiency is calculated as the ratio of OA/OV or OC/OV, players A and C will have efficiency scores equal to 1.0.

The graphical method is useful in this simple two dimensional example but gets much harder in higher dimensions. The normal method of evaluating the efficiency of player B is by using an LP formulation of DEA (*Source:* [16]).

To conclude this section, DEA models are linear programming methods that calculate the efficiency frontier of a set of DMUs and evaluate the relative efficiency of each unit, therby allowing a distinction to be made between efficient and inefficient DMUs. Those identified as "best practice units" (i.e., those determining the frontier) are given a rating of one, whereas the degree of inefficiency of the rest is calculated on the basis of the Euclidian distance of their input-output ratio from the frontier [17].

Compared to regression or stochastic frontier analysis methods, DEA shows several advan‐ tages. First, DEA allows handling multiple inputs and outputs (with different units) in a noncomplex way. Second, DEA does not require any initial assumption about a specific functional form linking inputs and outputs. While a typical statistical approach (regression analysis) is based on average values, DEA is an extreme point method and compares each producer with only the «best» producers. Efficiency is determined relatively with respect to other production units in the observed group.

( )

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(2)

459

*uj n*

Nonparametric Model for Business Performance Evaluation in Forestry

*s s m m*

*vv v u u*

, , ... , 0 , , ... , 0

2 1

m

2

*Max u y u y*

= ++ +

...

q

used.

incurred in.

activity of forest management.

*vx v x*

Subject to : ... 1

1

from models described in (1) and (2), see [19].

*2.3.1. Sample selection and data description*

1 10 0 1 10 0

> *m s*

³ ³

*u*


0

*j s sj j m mj*

1 1 1 1 0

... – – ... – 0 1, 2, ... ,

+ + -£ =

Where *u0* is the variable allowing identification of the nature of the returns to scale. This model does not predetermine if the value of this variable is positive (increasing returns) or is negative (decreasing returns). The formulation of the output oriented models can be derived directly

In this study, two measures of efficiency are applied – technical and scale efficiency (SE). Measurement of allocative efficiency requires data on production costs which were not available in our data set. For computing the applied models, DEA Excel Solver software was

State forests in the Republic of Croatia (RC) are mostly managed by the company Croatian forests Ltd – they account for approximately 80% of the total forest-covered area or 1,991,537 ha. The company Croatian forests consists of: headquarters in Zagreb, 16 regional forest administrations (FA) and a total of 169 forest offices (FO). In the current three-layer organisa‐ tion of the Croatian forestry, forest office is the organisational unit in which the basic tasks of forestry activities are carried out and most income and direct costs of forest management are

The efficiency analysis of selected forest offices is carried out based on the information adopted from the Croatian forests' ltd yearly reports. Additional applications and more robust data may provide additional insights for the evaluation of forest management.

The research includes 48 forest offices. The selected forest offices are the representatives of four main regions in the Croatian forestry: lowland flood-prone forests (I), hilly forests of the central part (II), mountainous forests (III) and karst/Mediterranean forests (IV). Each region is represented by two forest administrations i.e. by six forest offices from each forest administration. The sample of organisational units (Figure 3) and data involved in this

Inputs and outputs were selected so as to reflect business activities of the investigated decision making units – forest offices as the basic organisational units of the Croatian forestry, which perform the basic professional and technical operations in forest management (regeneration and silviculture of forests, wood harvesting) in a certain part of the forest economic area of RC, and where most income is achieved and direct costs incurred from the core business

research (yearly values of selected inputs and outputs) are shown in Table 1.

*uy uy vx v x*

### **2.3. DEA approach in evaluation of forestry units' performance**

Since DEA was introduced by Charnes, Cooper and Rhodes [14] several analytical models have been developed depending on the assumptions underlying the approach. For instance, the orientation of the analysis toward inputs or outputs, the existance of constant or variable (increasing or decreasing) returns to scale and the possibility of controlling inputs. According to Farrell [18], technical efficiency represents the ability of a DMU to produce maximum output given a set of inputs and technology (output oriented) or, alternatively, to achieve maximum feasible reductions in input quantities while maintaining its current levels of outputs (input oriented). In this study, output oriented DEA seems more appropri‐ ate, given it is more reasonable to argue that forest area, growing stock and other inputs should not be decresed. Instead, the goal of forest sector should be increased outputs of forest management, and improved general state of forests.

Given the selected orientation and the diversity of units characterizing our example, we first applied *CCR model* proposed by Charnes et al. [14]. This model assumes constant returns to scale. Following Cooper et al. [19], we begin by the commonly used measure of efficiency (output/input ratio) and we try to find out the correponding weights by using linear programming in order to maximize the ratio. To determine the efficiency of *n* units (forest offices) *n* linear programming problems must be solved to obtain the value of weights (*vi* ) associated with inputs (*xi* ), as well as the value of weights (*ur*) associated with the outputs (*yr*). Assuming *m* inputs and *s* outputs and transforming the fractional program‐ ming model into a linear programming model, the CCR (Charnes–Cooper–Rhodes) model can be formulated as Cooper et al. [19]:

$$\begin{aligned} \text{Max } \theta &= u\_1 y\_{10} + \dots + u\_s y\_{s0} \\ \text{Subject to:} & \begin{aligned} &u\_1 \mathbf{x}\_{10} + \dots + v\_m \mathbf{x}\_{m0} = 1 \\ &u\_1 y\_{1j} + \dots + u\_s y\_{sj} - v\_1 \mathbf{x}\_{1j} - \dots - v\_m \mathbf{x}\_{mj} \le 0 \\ &v\_{1i}, v\_{2i}, \dots, v\_m \le 0 \\ &u\_{1i}, u\_{2i}, \dots, m\_s \le 0 \end{aligned} \end{aligned} \tag{1}$$

Due to lack of information concerning the form of the production frontier, an extension of CCR model, Banker–Charnes–Cooper (BCC) model was also used. This model incorporates the property of variable returns to scale. The basic formulation of the model, best known as the BCC model is as follows:

$$\begin{aligned} \text{Max } \theta &= u\_1 y\_{10} + \dots + u\_s y\_{.s0} - u\_0 \\ \text{Subject to:} & v\_1 x\_{10} + \dots + v\_m x\_{m0} = 1 \\ & u\_1 y\_{1j} + \dots + u\_s y\_{sj} - v\_1 x\_{1j} - \dots - v\_m x\_{mj} - u\_0 \le 0 \quad \text{( $j = 1, 2, \dots, n$ )} \\ & v\_1, v\_2, \dots, v\_m \ge 0 \\ & u\_1, u\_2, \dots, u\_s \ge 0 \end{aligned} \tag{2}$$

Where *u0* is the variable allowing identification of the nature of the returns to scale. This model does not predetermine if the value of this variable is positive (increasing returns) or is negative (decreasing returns). The formulation of the output oriented models can be derived directly from models described in (1) and (2), see [19].

In this study, two measures of efficiency are applied – technical and scale efficiency (SE). Measurement of allocative efficiency requires data on production costs which were not available in our data set. For computing the applied models, DEA Excel Solver software was used.

### *2.3.1. Sample selection and data description*

producer with only the «best» producers. Efficiency is determined relatively with respect to

Since DEA was introduced by Charnes, Cooper and Rhodes [14] several analytical models have been developed depending on the assumptions underlying the approach. For instance, the orientation of the analysis toward inputs or outputs, the existance of constant or variable (increasing or decreasing) returns to scale and the possibility of controlling inputs. According to Farrell [18], technical efficiency represents the ability of a DMU to produce maximum output given a set of inputs and technology (output oriented) or, alternatively, to achieve maximum feasible reductions in input quantities while maintaining its current levels of outputs (input oriented). In this study, output oriented DEA seems more appropri‐ ate, given it is more reasonable to argue that forest area, growing stock and other inputs should not be decresed. Instead, the goal of forest sector should be increased outputs of

Given the selected orientation and the diversity of units characterizing our example, we first applied *CCR model* proposed by Charnes et al. [14]. This model assumes constant returns to scale. Following Cooper et al. [19], we begin by the commonly used measure of efficiency (output/input ratio) and we try to find out the correponding weights by using linear programming in order to maximize the ratio. To determine the efficiency of *n* units (forest offices) *n* linear programming problems must be solved to obtain the value of weights

outputs (*yr*). Assuming *m* inputs and *s* outputs and transforming the fractional program‐ ming model into a linear programming model, the CCR (Charnes–Cooper–Rhodes) model

*j s sj j m mj*

Due to lack of information concerning the form of the production frontier, an extension of CCR model, Banker–Charnes–Cooper (BCC) model was also used. This model incorporates the property of variable returns to scale. The basic formulation of the model, best known as the

*uy uy vx v x j n*

+ + £ =

... – – ... – 0 1, 2, ... ,

), as well as the value of weights (*ur*) associated with the

( )

(1)

other production units in the observed group.

458 Computational and Numerical Simulations

**2.3. DEA approach in evaluation of forestry units' performance**

forest management, and improved general state of forests.

(*vi*

) associated with inputs (*xi*

*Max u y u y*

= ++

...

BCC model is as follows:

q

can be formulated as Cooper et al. [19]:

*vx v x*

Subject to : ... 1

1 10 0 1 10 0

*s s m m*

++ =

*vv v uu m*

, , ... , 0 , , ... , 0

*m s* £ £

1 1 1 1

State forests in the Republic of Croatia (RC) are mostly managed by the company Croatian forests Ltd – they account for approximately 80% of the total forest-covered area or 1,991,537 ha. The company Croatian forests consists of: headquarters in Zagreb, 16 regional forest administrations (FA) and a total of 169 forest offices (FO). In the current three-layer organisa‐ tion of the Croatian forestry, forest office is the organisational unit in which the basic tasks of forestry activities are carried out and most income and direct costs of forest management are incurred in.

The efficiency analysis of selected forest offices is carried out based on the information adopted from the Croatian forests' ltd yearly reports. Additional applications and more robust data may provide additional insights for the evaluation of forest management.

The research includes 48 forest offices. The selected forest offices are the representatives of four main regions in the Croatian forestry: lowland flood-prone forests (I), hilly forests of the central part (II), mountainous forests (III) and karst/Mediterranean forests (IV). Each region is represented by two forest administrations i.e. by six forest offices from each forest administration. The sample of organisational units (Figure 3) and data involved in this research (yearly values of selected inputs and outputs) are shown in Table 1.

Inputs and outputs were selected so as to reflect business activities of the investigated decision making units – forest offices as the basic organisational units of the Croatian forestry, which perform the basic professional and technical operations in forest management (regeneration and silviculture of forests, wood harvesting) in a certain part of the forest economic area of RC, and where most income is achieved and direct costs incurred from the core business activity of forest management.

According to the *Forest Act,* along with conventional production of wood, forest management must also provide additional outputs. They are related to silviculture, protection and use of forests and forest land for construction and maintenance of forest infrastructure, all in accordance with general European criteria for ensuring sustainable forest management. Also, the goal of Croatian forests ltd. and its administrations and offices is business profitability. Most income comes from sold wood and hence the segment related to maintaining and enhancing the production function of forests (increment of growing stock) becomes increas‐ ingly important. Accordingly, the inputs and outputs considered in this example are:

**DMU**

Area I1, 103 ha

G. stock I2, m3/ha

Costs I3, 105 kn

**Inputs Outputs**

Income O1, 105kn

Harvest O2, m3/ha

Nonparametric Model for Business Performance Evaluation in Forestry

Investments O3, km

http://dx.doi.org/10.5772/57042

B. renewal O4, ha

461

Employees I4, N

9. Novska 11.73 263.02 289.54 64 320.66 4.04 0.70 649.10 10. Okučani 6.56 276.00 124.73 26 144.76 3.91 0.98 91.69 11. S. Brod 6.23 210.00 217.88 61 229.30 5.21 0.00 461.13 12. Trnjani 5.77 265.00 145.37 55 128.97 3.40 1.00 237.00 Hilly forests of the central part (II) Forest Administration Zagreb (C) 13. D. Stubica 2.60 239.04 23.24 12 21.12 2.47 0.00 51.00 14. Krapina 4.47 248.00 115.63 37 100.67 5.01 2.27 457.00 15. Novoselec 10.50 211.03 243.74 63 289.85 4.89 2.00 991.28 16. Popovača 7.64 201.00 158.69 52 168.71 3.24 3.00 829.00 17. Samobor 6.46 232.00 85.62 21 75.59 3.61 1.72 179.55 18. Zagreb 6.59 270.00 166.56 35 140.90 4.09 0.00 269.12 Forest Administration Koprivnica (D) 19. Čakovec 3.36 139.00 59.97 23 45.62 2.41 0.00 679.60 20. Ivanec 2.86 235.00 79.96 22 61.49 4.54 0.00 41.93 21. Koprivnica 6.53 331.00 219.34 65 215.91 5.05 0.00 556.06 22. Križevci 9.78 298.68 235.18 67 255.43 5.24 5.50 679.87 23. Ludbreg 5.00 271.00 129.40 35 123.20 4.30 0.50 380.00 24. Varaždin 5.12 187.00 108.90 37 85.73 1.71 0.00 119.82 Mountainous forests (III) Forest Administration Delnice (E) 25. Gerovo 7.04 316.13 181.84 53 202.73 6.21 0.00 118.17 26. Gomirje 5.43 297.30 119.95 39 118.33 4.58 0.00 55.92 27. Klana 6.81 251.12 96.88 38 79.79 2.82 3.50 59.08 28. Mrkopalj 9.25 314.00 179.28 50 190.23 4.92 0.00 894.00 29. Prezid 5.57 336.45 127.39 44 128.10 5.10 0.00 91.00 30. R. Gora 6.20 361.00 167.88 44 177.89 5.34 0.26 48.00 Forest Administration Gospić (F) 31. Brinje 17.25 208.00 215.07 43 212.38 2.55 7.10 390.85 32. D. Lapac 20.07 193.57 172.41 41 213.71 1.89 9.24 40.35

Inputs


### Outputs




According to the *Forest Act,* along with conventional production of wood, forest management must also provide additional outputs. They are related to silviculture, protection and use of forests and forest land for construction and maintenance of forest infrastructure, all in accordance with general European criteria for ensuring sustainable forest management. Also, the goal of Croatian forests ltd. and its administrations and offices is business profitability. Most income comes from sold wood and hence the segment related to maintaining and enhancing the production function of forests (increment of growing stock) becomes increas‐

ingly important. Accordingly, the inputs and outputs considered in this example are:

**3.** Expenditures, I3 – money spent in hundred-thousand croatian kunas (7,5 kn ≈ 1 EUR)

**1.** Revenues, O1 –yearly income in hundred-thousand croatian kunas (7,5 kn ≈ 1 EUR)

**4.** Biological renewal of forests, O4 – area of conducted silvicultural and protection works

Employees I4, N

Lowland flood-prone forests (I) Forest administration Vinkovci (A) 1. Gunja 5.84 234.00 300.10 68 315.51 4.30 1.80 547.34 2. Otok 10.72 418.00 470.31 100 538.41 7.13 0.00 3846.34 3. Strizivojna 4.31 294.00 149.90 40 160.61 4.42 0.00 510.00 4. Strošinci 4.84 394.00 141.83 40 141.04 4.28 1.21 493.87 5. Vinkovci 5.70 234.11 219.23 77 226.77 4.98 0.00 1748.59 6. Županja 6.54 364.00 177.64 61 393.10 8.78 0.00 583.70 Forest Administration Nova Gradiška (B) 7. N. Gradiška 12.39 242.95 320.47 85 288.71 5.12 3.10 1221.10 8. N. Kapela 8.40 218.75 151.21 71 130.73 3.61 0.00 229.98

**Inputs Outputs**

Income O1, 105kn

Harvest O2, m3/ha Investments O3, km

B. renewal O4, ha

**2.** Growing stock, I2 – volume of forest stock in cubic meters per hectare

**2.** Timber production, O2 – timber harvested in cubic meters per hectare

Costs I3, 105 kn

**3.** Investments in infrastructure, O3 – forest roads built in kilometres

**1.** Land, I1 – forest area in thousand hectares

460 Computational and Numerical Simulations

**4.** Labour, I4 – number of employees in persons

Inputs

Outputs

in hectares

Area I1, 103 ha

G. stock I2, m3/ha

**DMU**


**Table 1.** Input and output data of DMUs selected for efficiency measurement

There are 48 forest offices evaluated in this model. For the basic DEA models, the number of offices (units under consideration) should be a minimum of between 3 to 5 times the total number of input and output factors. Thus, we have limited the total number of inputs and outputs to eight factors.

**Variable Mean St. deviation Min Max Total**

Nonparametric Model for Business Performance Evaluation in Forestry

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463

Area, 103 ha 11.42 10.36 2.60 49.87 547.96 G. stock, m3/ha 214.98 91.94 51.85 418.00 - Costs, 105 kn 152.35 93.61 23.24 470.31 7312.99 Employees, N 42 21 8 100 2007

**Figure 3.** Sample of the organisational units (Forest offices) included in the research

Income, 105 kn 157.20 106.40 21.12 538.41 7545.68 Harvest, m3/ha 3.06 2.19 0.00 8.78 - Investments, km 2.24 4.29 0.00 22.59 107.48 B. renewal, ha 422.26 606.34 30.21 3846.34 20268.47

**Table 2.** Descriptive statistics of the variables used in the DEA model

Inputs

Outputs

Table 2 presents the descriptive statistics of the variables used in the analysis. A wide variation in both inputs and outputs is noticable. The input use is in some cases twenty times larger than that used by other offices, while variation in output variables is even higher. Such variation in the level of input and output implies that there are big differences between conditions under which individual forest offices operate. These differences are not unexpected, since the sample involves all representative areas managed by Croatian forests. However, it may also be a sign of poor management of resources in individual forest offices.

**Figure 3.** Sample of the organisational units (Forest offices) included in the research

**DMU**

Area I1, 103 ha

462 Computational and Numerical Simulations

G. stock I2, m3/ha

**Table 1.** Input and output data of DMUs selected for efficiency measurement

of poor management of resources in individual forest offices.

outputs to eight factors.

There are 48 forest offices evaluated in this model. For the basic DEA models, the number of offices (units under consideration) should be a minimum of between 3 to 5 times the total number of input and output factors. Thus, we have limited the total number of inputs and

Table 2 presents the descriptive statistics of the variables used in the analysis. A wide variation in both inputs and outputs is noticable. The input use is in some cases twenty times larger than that used by other offices, while variation in output variables is even higher. Such variation in the level of input and output implies that there are big differences between conditions under which individual forest offices operate. These differences are not unexpected, since the sample involves all representative areas managed by Croatian forests. However, it may also be a sign

Costs I3, 105 kn

**Inputs Outputs**

Income O1, 105kn

Harvest O2, m3/ha Investments O3, km

B. renewal O4, ha

Employees I4, N

33. Gospić 34.95 142.00 268.40 59 225.58 0.99 6.00 389.00 34. Gračac 49.87 140.66 204.33 45 167.67 0.77 4.15 329.64 35. Korenica 25.05 171.92 299.38 50 289.70 1.60 15.73 190.59 36. Udbina 20.99 144.62 268.05 61 246.95 1.67 22.59 139.23 Karst/Mediterranean forests (IV) Forest Administration Buzet (G) 37. Buje 7.55 75.34 58.56 33 61.52 0.12 0.00 307.81 38. Buzet 2.63 129.98 49.26 15 47.09 1.02 0.00 97.70 39. C-Lošinj 9.36 82.91 40.04 13 39.74 0.21 0.00 205.00 40. Opatija 9.04 154.00 76.53 24 77.33 1.23 0.00 99.00 41. Poreč 7.05 77.17 42.50 19 45.32 0.10 0.00 118.59 42. Rovinj 6.55 76.53 39.70 16 44.00 0.03 0.00 120.00 Forest Administration Split (H) 43. Brač 9.61 81.54 27.67 8 27.72 0.00 0.00 30.21 44. Dubrovnik 19.51 108.16 44.09 9 45.45 0.00 0.00 115.94 45. Makarska 7.24 115.00 40.81 13 50.85 0.00 1.35 198.15 46. Sinj 44.14 51.85 67.19 27 71.95 0.01 7.30 113.86 47. Šibenik 28.14 91.57 53.01 19 60.69 0.00 0.00 105.00 48. Zadar 28.72 121.63 138.33 27 118.17 0.02 6.48 157.31


**Table 2.** Descriptive statistics of the variables used in the DEA model
