a quadratic programming model for farm planning of a region in central macedonia

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A Quadratic Programming Model for Farm Planning of a Region in Central Macedonia Author(s): Basil D. Manos, George I. Kitsopanidis and Elias Meletiadis Source: Interfaces, Vol. 16, No. 4 (Jul. - Aug., 1986), pp. 2-12Published by: INFORMSStable URL: http://www.jstor.org/stable/25060843Accessed: 06-06-2015 13:28 UTC

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A Quadratic Programming Model for Farm

Planning of a Region in Central Macedonia

Basil D. Manos

George I. Kitsopanidis

Department of Agricultural Economics Research

University of Thessaloniki

540 06 Thessaloniki

Greece

Department of Agricultural Economics Research

University of Thessaloniki

Quadratic programming models are used in farm planning because risk and uncertainty are involved in the technical and economic coefficients used and the quantities and prices of resources. A special quadratic programming model (the E-V

model) was used to plan a Greek farm region, the former Lake of Giannitsa. The resulting plan is preferred by farmers to those resulting from the linear and mixed-integer program

ming models and to the previously used plan because it in cludes crops expected to give the highest minimum total

gross margin with the same total fixed costs. The farmers want plans that achieve not only the highest but also the most stable economic results.

Farmers face the problem of allocating and using the resources available at a

given time rationally and efficiently. These resources (land, labor, and capital)

may be allocated to various crop and live

stock enterprises through production

plans, each of which achieves different

economic results. Production plans de

signed to allocate resources efficiently un

der perfect knowledge as well as under

risk and uncertainty are based on various

mathematical models, chiefly mathemati

cal programming models.

A quadratic programming model is

used to program the agricultural produc tion of a region in Greece. The main re

sources considered by the plan are

considered stochastic in time. The results

of the model are useful both to the farm

ers and to the authorities responsible for

Copyright ? 1986, The Institute of Management Sciences INDUSTRIES ? AGRICULTURE/FOOD 0091-2102/86/1604/0002$01.25 PROGRAMMING ?

QUADRATIC

INTERFACES 16: 4 July-August 1986 (pp. 2-12)



FARM PLANNING

planning in this region, especially com

pared with those obtained from linear

and mixed integer programming models.

Risk and Uncertainty in Farm Planning In agriculture, both prices and quan

tities are uncertain. The assumption, common to other fields, that the prices and quantities of resources and the tech

nical and economic coefficients are known

with certainty is not true for farming. The

selection of an optimum production plan is always based on parameters obtained

in the past, although parameter variations

take place during the time the plan is

being used. Such variations include

changes in the yields of crops, in product and resource prices, in the resource re

quirements of the different enterprises, and in the quantities of resources avail

able; all significantly affect the economic

results that can be expected. In the past, research workers attempted

to incorporate these uncertainties into

mathematical programming models in or

der to produce plans that were robust

(relatively insensitive) with respect to var

iations in crop yields, resource prices, and the like.

The standard linear programming

model, which assumes fixed quantities and prices, does not incorporate the ran

dom variations of these parameters. The

same is true of the parametric linear pro

gramming model that investigates the sta

bility of a selected production plan at

fixed intervals of gross margins of the

farm enterprises and the available re

sources. However, it treats each parame ter separately (while the others remain

fixed), and it does not consider each pa rameter as random.

The problem of random variation in the

linear programming model is partially solved by using the mean absolute devia

tion criterion [Hazell 1971], but more

completely by using quadratic program

ming. Quadratic programming can also

be used in the dual form for variations in

the quantities of available resources when

these are treated as random variables

with known probability distributions

[Tinter and Sengupta 1972].

The E-V Model in Quadratic

Programming

Attempts of research workers to incor

porate risk and uncertainty into mathe

matical programming models have led to

the E-V model in quadratic programming. In this model, risk and uncertainty refer

only to the gross margin of farm enter

prises, while constraints are considered

deterministic.

The E-V model, is the quadratic pro

gramming approach to incorporating a

mean-variance criterion in the objective function. The earliest work in capital

budgeting using the quadratic program

ming/mean-variance criterion was done

by Farrer [1962]. The E-V model is so

named because the optimum production

plan is selected on the basis of the ex

pected gross margin E and its variance V.

This model gives a solution to the prob lem of risk and uncertainty adequate for

planning agricultural production. In fact,

the most important parameters are yields and prices of farm products. These pa rameters vary randomly through time and

strongly affect the production plan and

the level of gross margin expected. Varia

tions in the quantities of the resources

available and their prices, on the other

July-August 1986 3



MANOS, KITSOPANIDIS

hand, are usually small and have little in

fluence on the production plan suggested or the level of gross margin expected

[Kitsopanidis 1966]. Fluctuation in gross margin, which is

the main criterion for comparing farm en

terprises for their contributions to the op timum production plan, is measured by its variance, and it can be incorporated into the E-V model. In this case, the

gross margin of each farm enterprise is

considered a random variable, and the

variance of the expected total gross mar

gin for the whole region is a function of

the variances and covariances of the gross

margins of the farm enterprises studied

[Kitsopanidis 1967].

According to the E-V criterion, each

farmer will prefer a production plan with

higher V only if E is also greater

dE ( _> 0) , and this ratio should increase

oV

at an increasing rate with increases in

d2E V(? >0) [Hazell 1971].

The E-V model is expressed by min V = x'Dx

Ax><lb 0 < X < ?max,

ex = X U

x >0

where x is the n-vector of levels x} of ac

tivities; =

1,2,...,n,

x' is the transpose of x,

b is the m-vector of levels b? of avail

able resources / = l,2,...,m,

A is the m x n matrix of technical and

economic coefficients a{) of resource /

and activity /, c is the n-vector of the mean gross

margin cj of activities / =

1,2,...,n,

D is the n x n matrix of covariances o^ of gross margin c, and c;,

E ̂ is the maximum total gross margin of the linear programming approach,

X is a parameter that takes values in

the interval (0, ?max). Both the means of the gross margins

and a?; are unknown parameters (popula tion parameters) and estimates of them

must be found (appendix). The objective of the above model is the

minimization of the variance in gross

margins achieved by production plans when X is varied from zero to Emax.

This parametric process corresponds to

a sequence of efficient pairs (E^V^),

(E2,V2)/-- in relation to the sequence of

critical values X1,X2/..., where ?, and V,

are estimates of E, and V, respectively. For each critical value X^

= ?^, the mini

mum variance V^ corresponds to that

point of the feasible region where the iso

gross margin line is tangent to the iso

variance curve [How and Hazell 1968].

This point gives the u, optimum plan (ap

pendix) which includes farm enterprises with small correlation coefficients rif (posi tive or negative), because of the term

XXSjjXiXj of variance and the relation ri] ?

si}/V Sifijj which joins the correlation co

efficient with the covariance of c? and c;. The selection of the most efficient plan

from the sequence of (EifV), i = 1,2,... is

achieved in different ways. The most

common way is to construct confidence

intervals for the gross margin of the pro duction plans (appendix).

Description of the Region and the

Construction of the Model

The E-V model is used to plan the agri

INTERFACES 16:4 4



FARM PLANNING

cultural production of a region in Central

Macedonia that was formerly the Lake of

Giannitsa. The region consists of a culti

vated area of 161,672 irrigated stremmas

(one stremma equals 1,000 square me

ters). In this area, 3,519 man years are

available for farm labor along with 1,928

tractors and 33 machines for harvesting

grain, 22 for harvesting sugar beets, and

27 for cotton. Casual labor is also used

(111,724 man days during the year), and

17 sugar beet and two cotton harvesting machines are hired from other regions

during the peak periods. The total capital invested (except for land) amounts to 2.12

billion drachmas ($30.3 million). Data for the period 1976-81 were col

lected from the various local services of

the Ministry of Agriculture and the Agri cultural Bank, from the National Statisti

cal Service, from the secretaries of 34

villages, from the records and accounts of

73 farms, from 968 farm enterprises (us

ing special questionnaires), and from 19

farmer-owners of various machines.

The investigation in this region was un

dertaken by the Department of Agricul tural Economics Research at the Univer

sity of Thessaloniki in conjunction with

the local services of the Ministry of Agri culture. The purpose of the investigation

was to prepare the most efficient produc tion plans with the highest and most sta

ble economic return for the local area.

The application of these plans on each

family farm and in the whole region was

undertaken by the local agriculturists re

sponsible for regional agricultural

development. The E-V model used in agricultural

Activities

Farm Enterprises Resources

A. Annual Crops B.

1. Wheat 11

2. Barley 12

3. Corn 13

4. Tobacco (Burley) 14

5. Cotton (hand picked) 15

6. Cotton (machine 16

picked) 17 7. Sugar beet 18

8. Tomatoes 19

9. Beans 20

10. Various irrigated

crops

Perennial Crops Alfalfa (1) Peaches (table) (1) Peaches (processing) (1) Apples (1) Pears (1) Alfalfa (2) Peaches (table) (2) Peaches (processing) (2)

Apples (2) Pears (2)

C. Labor

21. Family Casual

22. April 23. May 24. June 25. July 26. August 27. September 28. October

29. November

D. Machinery Cotton-harvesters

30. Owned

31. Hired

Sugar-beet harvesters

32. Owned

33. Hired

Combine 1

34. Owned

35. Hired

Combine 2

36. Owned

37. Hired

Tractors

38. Owned

Hired tractors

39. April 40. May 41. June 42. July 43. August 44. September 45. October

E. Variable capital 46. Borrowed

Table 1: The 46 activities of the E-V model are divided into farm enterprises and farm resources. The perennial crops indexed (1) refer to the average for the existing plan irrespective of the age of trees, whereas those indexed (2) refer to the average for their age and compete with annual

crops. Combine (1) harvests wheat and barley; combine (2) harvests corn. Irrigated crops in clude potatoes, melons, and some other garden crops.

July-August 1986 5



MANOS, KITSOPANIDIS

planning in this region includes 46 activi

ties and 76 constraints [Manos 1984]. The

activities are divided into farm enterprises

(Xj) and farm resources (u?) (Table 1). The

constraints refer to land, labor, machin

ery, and variable capital (Table 2). One

more constraint sets the level of gross

margin for the production plan at the de

sired level (E{ =

X,; the left side corre

sponds to the objective function of the

linear programming model cx-qu, where

q? is the variable cost of the resource

activity Uj). The reasons for using resource activi

ties, as well as those of farm enterprises, are economic and technical. Most impor

tant, the E-V model can be immediately transformed to linear, parametric linear,

and mixed-integer programming models

(integer units of labor and machinery); and the machinery (owned and hired) re

quired by the plan can be immediately

estimated, as can be surplus available

capital and its interest and the total capi tal required for hired labor and machin

ery. The estimate for hired labor and

tractors is figured in monthly units re

quired and their cost.

The various constraints, especially those for labor and machinery, are in

cluded in the model in such a way that

each farm enterprise can be entered into

the production plan at the desired level.

Also the available labor, machinery, and

capital are allocated to the individual

farm enterprises according to their month

ly requirements. This allocation gives

priority to the resources belonging to the

region over hired ones from other regions. The objective function of this model ex

presses the variance V in the total gross

A. Land (constraints 1-20) 1. Total (

= )

2. Wheat (max 1976-81 period)

20. Pears 2 (max 50% 1981 period)

B. Labor (constraints 21-41) 21. Family total (max available)

41. November (max casual used)

C. Machinery (constraints 42-73) 42. Cotton machinery (max required) 43. Cotton machinery (max owned available)

73. Tractors hired October

D. Capital (constraints 74-75) 74. Own capital (max available) 75. Borrowed capital (max zero to infinity)

E. Gross Margin (constraint 76) 76. Expected gross margin (

= zero to Emax)

Table 2: A sample of the constraints for the E-V model.

margin resulting from the variances and

covariances of gross margins c; for the

farm enterprises during the period 1976

81. Each time, its value is the product of

vectors x' and x by the gross margins covariance matrix D.

The Suggested Plans and the

Expected Results

The farm plans and the corresponding economic results suggested by the E-V

model are shown in Table 3. As the gross

margin increases from 1.55 to 1.98 billion

drachmas, the farm plan changes in order

to yield the smallest possible variance for

each level of gross margin. The changes in farm plans from 1-6 mainly involve a

decrease in land devoted to barley, toma

toes, beans and wheat, and an increase

for tobacco, while the land area devoted

INTERFACES 16:4 6



FARM PLANNING

to the other crops remains unchanged. The decrease of some crops and the cor

responding increase of others depends on

the variance of each crop and the covari

ance of some of them. In other words,

the substitution of some crops by others

according to the total gross margin adds

the smallest possible variance to the total

variance of the gross margin: substitu

tions take place between crops that have a

low degree of correlation or the weigKted

average of correlation coefficients is abso

lutely low, irrespective of whether the

partial correlation coefficients are high or

low, positive or negative. Table 3 also shows the rapid increase of

the variance (from 7.4 to 13.0 percent) as

the gross margin moves continuously to

higher levels. In fact, when the gross

margin increases from 1.55 to 1.98 billion

drachmas (27.5 percent), its standard de

viation increases from 114.8 to 257.2 mil

lion drachmas (124.0 percent). The rate of

increase of the total gross margin com

Farm Enterprises and

Economic Results Optimum Farm Plans in Stremmas and Economic

Results in 1,000 Drachmas 12 3 4 5

A. Farm Enterprises 1. Wheat

2. Barley 3. Corn

4. Alfalfa 1

5. Alfalfa 2

6. Tobacco

7. Cotton (hand) 8. Cotton (machine) 9. Sugar beet

10. Tomatoes

11. Beans

12. Various irrigated crops 13. Peaches (table) 1

14. Peaches (table) 2

15. Peaches (processing) 1

16. Peaches (processing) 2

17. Apples 1

18. Apples 2

19. Pears 1

20. Pears 2

Total

35,342

2,640

38,068

1,458

2,391

2,415

28,249

2,833

25,777

11,185

2,164 504

2,437

1,219

2,501 817

781

391

333

_167 161,672

21,677

38,068

1,458

2,391

4,655

12,967

36,499

21,024

11,185

2,164 504

2,437

1,219

2,501

1,251 781

391

333

167

161,672

35,342

2,640

38,068

1,458

2,391

12,847

24,015

9,319

14,809

10,848

2,164 504

2,437

2,501

1,048 781

333

167

161,672

35,342

1,929

38,068

1,458

2,391

13,539

12,631

7,604

25,777

11,185

2,164 504

2,437

1,219

2,501

1,251 781

391

333

167

161,672

30,921

38,068

1,458

2,391

18,040

25,122

13,161

16,976

8,421

504

2,437

2,501

781

391

333

167

161,672

29,667

38,068

1,458

2,391

18,983

26,256

5,258

25,213

6,620

504

2,437 89

2,501 555

781

391

333 167

161,672

B. Gross Margin (E) 1. E

2. Standard deviation (ct?) 3. Variance coefficient 4. Pr(E?2(TE)

= 95.45%

C. Fixed Costs D. Net Profits

E. Return on Labor

F. Return on Capital per cent

G. Farm Income

1,555,001

114,873 7.4

1,325,255

1,784,747

1,643,469

-88,468

746,218 4.2

1,443,059

1,642,050

132,987 8.1

1,376,076

1,908,024

1,643,469

-1,419

833,267 4.9

1,531,199

1,701,061

144,898 8.5

1,411,265 1,990,856

1,643,469

57,592

892,278 5.4

1,590,165

1,829,007

184,730 10.1

1,459,547 2,198,467

1,643,469

185,538

1,020,224 6.2

1,698,903

1,902,012

219,492 11.5

1,463,028 2,340,996

1,643,469

258,543

1,093,228 6.8

1,742,387

1,978,546

257,211 13.0

1,464,124 2,492,968

1,643,469

335,077

1,169,725 7.1

1,769,440

Table 3: Optimum farm plans showing the land area devoted to different crops and their eco

nomic results according to the level of gross margin and its variance. Return on labor includes the wages of labor used and net profit. Return on capital includes interest on capital invested and net profit. Farm income includes land rent, labor wages, interest on capital, and net profit.

July-August 1986 7



MANOS, KITSOPANIDIS

pared with its standard deviation is

greater at low levels and smaller at high levels of the gross margin. Inversely, this

means that the rate of increase of the

standard deviation is higher as the total

gross margin becomes higher. The rate of

increase of both the gross margin and its

standard deviation is better shown in Fig ure 1. The behavior of the gross margin and the standard deviation is due to the

fact that the probability of failure or suc

cess increases rapidly as the expected in

come increases. So, taking as a base the

95.45 percent confidence intervals (two

standard deviations), it is obvious that

the vertical distance between the two

curves EG and AB (which represent the

minimum and maximum gross margins

Minimum Gross Margin Expected

3 l-,-,-1-1-1-1-1-1-r? 100 120 140 160 180 200 220 240 260 280

Figure 1: Standard deviation of gross margin (million drachmas).

expected) increases as the medium gross

margin expected (curve CD) is increased.

This can be seen also in Table 3, where,

for farm plan 1, the difference between

the maximum and minimum gross mar

gin expected is 459.5 million drachmas;

for farm plan 4, this difference increases

to 738.9 million drachmas; and for farm

plan 6, this difference becomes 1,028.8

million drachmas, which is more than

double that for farm plan 1.

The choice of the most efficient plan is

based on the degree of certainty with

which each level of gross margin can be

achieved in relation to fixed costs. From

this standpoint, the first two plans are

rejected, because their fixed costs are

higher than the gross margin expected. Given the fact that all the other plans

present a very small probability

(Pr(E^O)^O.OOOl) of achieving negative

gross margins (because of the positive limits of the 95.45 percent confidence in

tervals), we conclude that the most profit able plan must be chosen on the basis of

the irdnimum total gross margin expected. The highest minimum gross margin ex

pected, 1,464 billion drachmas for plan 6,

corresponds to point H on curve EG in

Figure 1. Farm plan 6 is expected to give a level of gross margin equal to 1.979 bil

lion drachmas with a degree of certainty 95.45 percent, which means that the

gross margin will not be less than 1.464

billion drachmas.

Considering the other economic results,

plan 6 is the most profitable and achieves

a higher return on labor and capital in

vested and a higher-farm income than the

other five plans even though it is based

on the same fixed costs. More specifically,

INTERFACES 16:4 8



FARM PLANNING

the net profit for plan 6 is 335,077 million

drachmas, equivalent to 2,072 drachmas

per stremma or 95,197 drachmas per fam

ily member working on each farm of the

region. These figures are significant be

cause the majority of the farms in this re

gion are family farms. On the other

hand, the return on labor increases as

much as 1,170 billion drachmas, which

corresponds to 1,141 drachmas per eight

hours, while the farm income increases as

much as 1,769 billion drachmas, or to as

much as 502,825 drachmas per family member working on the farm. Farm in

come is the farm family's living; this liv

ing then is 502,825 drachmas for a family with one member and 1,005,750 drachmas

for a family with two members working on the farm. Finally, the return on capital increases to as much as 7.1 percent,

which shows a more productive use of

the capital invested.

A Comparison of the Existing Plan, the

E-V Plan, and the Linear and

Mixed-Integer Program Plans

Table 4 shows the economic results

achieved or expected with the existing

plan and the plans suggested by the E-V

model, the linear, and the mixed-integer

programs.

The farm plan suggested by the E-V

model differs from the existing plan. It in

creases or decreases the land area covered

by some crops; specifically, the area cov

ered by corn, cotton to be picked by

hand, sugar beet, and alfalfa increases. In

contrast, the area covered by cotton to be

picked by machine, barley, tomatoes for

processing, and beans decreases. These

changes lead to a better use of the family labor available and the capital invested. In

fact, the degree of family employment in

creases about 6.1 percent annually, while

the utilization of machinery increases 0.6

percent for tractors, 30 percent for sugar beet harvesters, and 20 percent for corn

harvesters. Available resources are better

utilized, with the expected gross margin about three percent greater even though it is based on the same fixed costs. The

same is true for the profits (about 20 per cent greater), the return on labor (about five percent greater), the return on capital

(increases from 6.5 to 7.1 percent), and

farm income (about four percent greater). The farm plan suggested by the E-V

model also differs from the linear pro

gramming plan and the mixed-integer

programming plan in the land area cov

ered by certain crops. The E-V model

plan includes more wheat compared with

the LP and MIP plans and more corn and

tomatoes. However, the E-V model as

signs less land to tobacco and to cotton to

be picked by machine compared with the

LP and the MIP models. Family labor,

farm owned machinery, and the capital invested are all utilized a bit more effi

ciently by the LP and MIP plans through the more productive crops of tobacco and

cotton. The gross margin expected for the

LP plan is 2,000.845 million drachmas, for

the MIP plan it is 2,000.573 million drach

mas and for the E-V model it is 1,978.546

million drachmas all based on the same

fixed costs. As a result of this difference,

a higher profit is expected with the LP

and MIP plans than with the E-V plan. This profit affects return on labor and

farm income slightly, while the return on

capital remains unchanged. The farm plan resulting from the LP

July-August 1986 9



MANOS, KITSOPANIDIS

Farm Enterprises and

Economic Results Farm Plans in Stremmas and Economic Results in 1,000 Drachmas

E-V_Existing_LP _MIP A. Farm Enterprises

1. Wheat 2. Barley 3. Corn

4. Alfalfa 1 5. Alfalfa 2 6. Tobacco

7. Cotton (hand picked) 8. Cotton (machine picked) 9. Sugar beet

10. Tomatoes

11. Beans

12. Various irrigated crops 13. Peaches (table) 1 14. Peaches (table) 2 15. Peaches (processing) 1 16. Peaches (processing) 2 17. Apples 1 18. Apples 2 19. Pears 1

20. Pears 2

Total

B. Gross Margin (E) 1. E

2. Standard deviation (ct?) 3. Variance coefficient

4. Pr(E?2a?) = 95.45%

C. Fixed Costs

D. Net Profits E. Return on Labor

F. Return on Capital per cent

G. Farm Income

Table 4: A comparison of the E-V plan mixed-integer Programs.

29,667

38,068 1,458 2,391

18,983 26,256

5,258 25,213

6,620

504

2,437 89

2,501 555 781 391 333

_167

161,672

1,978,546.0 257,211.0

13.0

1,464,124.0 2,492,968.0 1,643,469.0

335,077.0 1,169,763.0

7.1

1,769,440.0

29,005 1,441

26,385 1,458

18,888 17,182 31,910 20,540

7,207 1,100

504

2,437

2,501

781

333

161,672

1,923,803.0 242,244.0

12.6

1,439,316.0 2,408,290.0 1,643,469.0

280,334.0 1,115,020.0

6.5

1,705,447.0

2,400

20,658 1,458 2,391

20,309 26,813 33,385 25,544

504

2,437

2,501

781 391 333

_167

161,672

2,000,845.0 278,306.0

13.9

1,444,233.0 2,557,457.0

1,643,469.0

357,376.0 1,192,062.0

7.1

1,783,380.0

23,790

21,600 1,458 2,391

20,284 26,795 32,610 25,638

504 2,437

2,501

781 383 333

_167

161,672

2,000,573.0 278,019.0

13.9

1,444,535.0 2,556,611.0

1,643,469.0

357,104.0 1,191,790.0

7.1

1,783,108.0

with existing one and those suggested by the linear and

model is the most efficient compared with

the other plans because it better utilizes

the resources (land, labor, machinery,

capital) available. This is true under con

ditions of certainty for prices and quan

tities of all parameters used in the

planning. But these conditions are not

certain in actual practice; planning agri cultural production in this region is based

on parameters that are averages for the

period 1976-81. So the LP plan cannot

guarantee that the economic results ex

pected are achieved. On the other hand,

the E-V model plan is more stable from

both the technical and the economic point of view, because it takes into account the

variances and covariances of gross mar

gins for the various crops in the period

INTERFACES 16:4 10



FARM PLANNING

1976-81. In fact, the variance of the total

gross margin in the E-V model plan is

13.0 percent instead of 13.9 percent as it

is in the LP plan. This is true for the min

imum expected gross margins (with 95.45

percent probability) of the two plans: 1.464 billion drachmas for the E-V plan and 1.444 billion drachmas for the LP

plan. The expected minimum gross mar

gin for the E-V model plan will thus be

19,589 million drachmas higher than the

corresponding LP model plan. The ex

pected minimum gross margin of the E-V

plan is higher than that of the LP plan; therefore, under conditions of risk and

uncertainty, the former must be chosen.

Conclusions

The E-V model was used in farm plan

ning for a region in Central Macedonia

(formerly the Lake of Giannitsa), a low

lying area with good quality soil. The

plan specifies crops that yield the highest

expected minimum gross margin com

pared to the existing plan and those pro duced by the LP and MIP models. The

model is used by the Greek agricultural authorities in collaboration with the Agri cultural Economics Research Department to produce regional farm plans for the

farmers to give them not just the highest but also the most stable economic results.

APPENDIX (1) Assuming that c; represents random

variables, distributed normally with mean

|x; and variance a2y, the total gross margin (E

= ex) of the production plan, as a linear

combination of them, will follow a normal distribution with mean

n n

[i= E(cx) = S XjE(c})

= 2 Xjfy, y = l y

= l

and variance

nn

V= Var(cx) = 22 cr^Xy,

where

a = ( covic^Cj) i+] 9 \ var(c}) i=j

From n independent and mutually ex

clusive samples of gross margins of size r

(one sample of r years for each activity ;) we take estimations |x(

= E) and V

n

|X = 2 CjXj

= ex,

nn

where

1 r s.y= -?\ S (Cu- ?,)

. (Ck -

c?)

are the estimates of o-?,.

. i : c;=

- 2 ct? r

fc=l is the expected gross margin of activity ;.

cK;, K =

l,2,...r is the observed value of

gross margin of activity j on the year k of

sample /. (2) Algebraically, the k plan xK > 0 of

E-V model is optimum if the conditions of Kuhn-Tucker are satisfied, that is, if there is a vector yK

> 0 such that

yKA =

DxK

yK{AxK_- b)_= 0

(DxK -yKA)xK = 0

where yK is the vector of dual values that

correspond to the constraints of the E-V model. This optimum plan is a local opti mum, but if the function x'Dx is convex

(i.e. x'Dx^O) then it is also the global optimum.

(3) The set of efficient pairs-solutions (?j,^), (E2,V2),..., presents increasing

gross margin followed by increasing its variance (Figure 2).

July-August 1986 11



MANOS, KITSOPANIDIS

?-\ >(0

Gross Margin

Figure 2: Standard deviation of gross margin versus the level of the gross margin.

r)rr *

The ratio ??-> , which gives the dE

rate of increase of the standard deviation

of the gross margin (v? =

\fV) in relation

to the increase of gross margin E, is posi tive and greater than one. This means

that the rate of increase of the variance is

higher than the rate of increase of the

gross margin, mainly for high levels of

gross margin. The selection of the most efficient pair

plan is achieved in one of the following

ways:

(a) by selecting from each farmer the

plan suitable for him depending on the

desirable level of gross margin in relation

to its variance.

(b) by constructing confidence intervals

for the gross margin of the production

plans and choosing the one whose gross

margin has the highest probability of

being positive. (c) by finding algebraically or graphi

cally the production plan when the farm

er's utility function is known.

Using the second method, (b), the most

commonly used, the probability Pr(E^O) is calculated for each plan with gross

margin E. Then a confidence interval, which includes the expected gross margin E with a given probability level, is con

structed. The most efficient plan then is

the plan whose gross margin has the

smallest probability of being negative.

References

Fairer, D. F. 1962, The Investment Decision under

Uncertainty, Prentice Hall, Englewood Cliffs, New Jersey.

Hazell, P. 1971, "Linear alternative to quad ratic and semivariance programming for

farm planning under uncertainty," American

Journal of Agricultural Economics, Vol. 53, No.

1, pp. 53-62.

How, R. and Hazell, P. 1968, "Use of quadratic

programming in farm planning under un

certainty," Department of Agricultural Econ

omics, Cornell University, Ithaca, New York,

A. E. Res. 250.

Kitsopanidis, G. 1966, "Programming proce dures for farm planning under variable

input-output coefficients," Agricultural Eco

nomic Review, Vol. 11, No. 2, pp. 257-274,

Thessaloniki, Greece.

Kitsopanidis, G. 1967, "TeTpo^coviKo?

npO7pa|X|X0tTia^xo<;-MLa <pap|uio7T| crrnv

XXt|vlkt| 7 ?)p7ia" (Quadratic program

ming ? An application to Greek farming),

Agricultural Economics, Vol. 49, No. 1, pp. 1

19, Athens, Greece.

Manos, B. 1984, "IIpo7pa|X|xaTicr|xo?

7 ?)p7iKT|<; TTapa7<ji)7T|<; T <o<; Xl(xvt)s Tiav

vlt (1)v" (Planning of agricultural production in Giannitsa Lake region), Ph.D. disserta

tion, Thessaloniki University, Thessaloniki, Greece.

Tinter, G. and Sengupta, J. 1972, Stochastic

Economics; Stochastic Processes, Control, and

Programming, Academic Press, New York.

A letter from Elias Meletiadis, Director

of the Agricultural Service of provincial

Giannitsa, confirms that "we have a close

contact with the Department of Agricul tural Economics Research of the Univer

sity of Thessaloniki in regional farm

planning. This department, using techni

cal and economic data of this region, pre

pares optimum farm plans which we try to apply in practice in collaboration with

local agriculturists and interested

farmers."

INTERFACES 16:4 12



a quadratic programming model for farm planning of a region in central macedonia

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