J Syst Sci Syst Eng(Dec 2006) 15(4): 399-418 DOI: 10.1007/s11518-006-5018-2

ISSN: 1004-3756 (Paper) 1861-9576 (Online) CN11-2983/N

LIVESTOCK PRODUCTION PLANNING UNDER ENVIRONMENTAL RISKS AND UNCERTAINTIES Günther FISCHER1

Tatiana ERMOLIEVA2 Harrij van VELTHUIZEN4

Yuri ERMOLIEV3

International Institute for Applied Systems Analysis, Schlossplatz 1, A-2361 Laxenburg, Austria 1

[email protected], 3

2

[email protected]( )

[email protected],

4

[email protected]

Abstract In this paper we demonstrate the need for risk-adjusted approaches to planning expansion of livestock production. In particular, we illustrate that under exposure to risk, a portfolio of producers is needed where more efficient producers co-exist and cooperate with less efficient ones given that the latter are associated with lower, uncorrelated or even negatively correlated contingencies. This raises important issues of cooperation and risk sharing among diverse producers. For large-scale practical allocation problems when information on the contingencies may be disperse, not analytically tractable, or be available on aggregate levels, we propose a downscaling procedure based on behavioral principles utilizing spatial risk preference structure. It allows for estimation of production allocation at required resolutions accounting for location specific risks and suitability constraints. The approach provides a tool for harmonization of data from various spatial levels. We applied the method in a case study of livestock production allocation in China to 2030. Keywords: Spatial production allocation, sequential downscaling, cross-entropy, maximum likelihood, risks and uncertainties.

1. Introduction

1994, USDA Economic Research Service 1998).

The world food economy is increasingly

Increasing

incomes

and

changing

being driven by a shift of diets towards livestock

consumption preferences stimulate production

products.

countries,

intensification with a growing share of livestock

consumption of meat has been growing at 5-6

products coming from industrial and specialized

percent p.a., and that of milk and dairy products

enterprises.

at 3.3-3.5 percent p.a. in the last few decades.

represented a natural farming cycle; livestock

Much of the growth is taking place in China

was kept on grass areas or in confined places

(Huang and Zhang et al. 2003a, Huang and Liu

close to farmland. Primary sources of feed were

2003b, Keyzer and van Veen 2004, Ma and

grass, feed from fodder crops and other crops,

Huang et al. 2004, Simpson and Cheng et al.

household wastes and crop residues. In these

In

the

developing

© Systems Engineering Society of China & Springer-Verlag 2006

Traditional

livestock

systems

Livestock Production Planning under Environmental Risks and Uncertainties

systems, livestock waste and manure were

in fact, will gain by specialization in goods of

considered valuable sources of nutrients for crop

comparative advantage. Accordingly, we may

production or for fuel. The manure was recycled

expect that production should be undertaken by

efficiently,

environmental

the most efficient agent, intensified production

degradation and pollution. With the introduction

on large farms. This is true only if profitability is

of large-scale industrial livestock production,

taken as the only determinant. In reality,

especially of pigs and poultry, this closed cycle

negative impacts of production intensification

is collapsing. Intensive livestock production

are causing great concerns regarding the

enterprises are located close to meat markets,

pollution of natural resources and ecosystems,

near urban areas, and in these locations there is

their potential irreversible changes, risks, and

much more livestock concentrated than land can

sustainability.

causing

minimal

support for proper manure recycling. The

livestock

production

In this paper we demonstrate the need for represents

risk adjusted approaches to planning expansion

important causes of contamination of the

of livestock production. In particular, we

environment through point-source and non-point

illustrate that under exposure to risk, the more

1

pollution . When coinciding with intensive crop

efficient producer has to co-exist and cooperate

cultivation, the problem of pollution through

with less efficient given that the latter is

excess nutrients from livestock operations is

associated with lower, uncorrelated or even

further exacerbated by imbalanced fertilizer

anti-correlated

application. Over-supply of nutrients may lead

important issues of cooperation and risk sharing

to toxic nitrate pollution in the water supply and

among

may cause eutrophication of surface water.

production capacities in order to satisfy a

Hazardous pollution of the atmosphere, water

growing demand, cannot just follow historical

and soil resources by residues of intensive

intensification trends and common proportionate

livestock production is becoming a critical

distribution rules. Of particular interest is a

environmental issue in China (Fischer and

network of large and small producers, i.e.,

Ermolieva et al. 2006, Ma and Huang et al.

backyard farms, that can stabilize the aggregate

2004). The trend is alarming and in some

production,

locations, without appropriate measures, it may

environmental risks and uncertainties, ensure

turn irreversible.

sustainable supply of agricultural products to

Ricardo (1822) stipulated that trading nations,

contingencies.

diverse

producers.

hedge

against

This

raises

Expansion

economic

of

and

markets, etc. Explicitly accounting for risks even in

1

As opposed to point-source pollution, such as from precisely located waste disposal sites or a direct pipe, agricultural non-point pollution occurs as a result of secondary dispersion or land run-off pollution, e.g., when rain falls to the earth or snow melts - and the water runs across fields taking with it topsoil, bacteria, fertilizers, pesticides, and other toxic or harmful materials.

400

simple linear models may considerably alter the composition of production units and their intensification levels. Often, in applications with environmental risks, distribution functions of contingencies are not analytically tractable as they

depend

on

complex

spatio-temporal

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FISCHER, ERMOLIEVA, ERMOLIEV and van VELTHUIZEN

interactions among economic, demographic and

maximum likelihood solutions. We applied the

environmental factors. Analysis of contingencies

method in a case study of livestock production

may be further complicated by the lack or

allocation in China to 2030. We briefly discuss

paucity of information for estimation of these

the main challenges related to the choice of prior

factors. Information is often available only on

information

aggregate levels, for example, information on

procedure and, therefore, to a large extent

the availability of cultivated land and water,

determines the success of the results.

that

guides

the

downscaling

economic indicators, livestock management

Section 2 introduces a prototype production

characteristics, population density. The data may

planning model under risks. We demonstrate

be incomplete, uncertain, or measured indirectly.

with simple examples the need for cooperation

It may also come from various sources and thus

and risk spreading among various kinds of

be inconsistent or given only within plausible

producers. Section 3 presents a practical

intervals. On the other hand, the sources of risks

large-scale

can be rather disperse, intractable, and thus

problem under risk and suitability constraints

require a model-based assessment (Fischer and

and discusses its main challenges. Section 4

Ermolieva et al. 2006).

proposes a sequential downscaling method for

livestock

production

planning

This raises a number of methodological

solution of the livestock allocation problem. In

issues related to spatial livestock facility

Section 5 a connection is made between the

allocation

proposed

planning

under

risks

when

method

and

the

cross-entropy

information on the contingencies may be

maximization principle. Relation between the

disperse, not analytically tractable, or be

cross-entropy and minimax likelihood estimates

available on aggregate levels. For the analysis of

is summarized in Section 6. Section 7 discusses

plausible geographically explicit scenarios of

the relevant numerical experiments for the case

livestock production in China, a downscaling

study, and Section 8 concludes.

procedure

based

on

behavioral

principles

utilizing spatial risk preference structure has been developed and implemented. It allows for

2. Production Intensification, Risks and Cooperation

estimation of production allocation at required

Over the last 20 years, China's demand and

resolutions accounting for location specific risks

production of livestock products has increased

and

approach

remarkably due to rapid development of the

provides a tool for harmonization of data from

suitability

constraints.

The

national economy, urbanization, rising living

various spatial levels.

standards, and population growth (Huang and

The convergence of the proposed algorithm to

solutions

maximizing

a

Zhang et al. 2003a, Huang and Liu 2003b, Ma

cross-entropy

and Huang et al. 2004). Increasing incomes and

function can be considered as an analog of the

changing consumption preferences have boosted

asymptotic consistency analysis in traditional

production and have shifted the composition of

statistical estimation theory, as there is a direct

producers towards specialized enterprises with a

connection of the cross entropy and the

number of advantages: they are more feed

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Livestock Production Planning under Environmental Risks and Uncertainties

efficient and profitable, flexible in terms of management, may better adjust and comply to legislation,

and,

in

general

benefit

from

economies of scale. In a sense, these trends follow the Ricardo’s assertion that trading nations gain from production specialization and intensification, which is true only if risks and negative impacts are not taken into account. In reality, intensive production facilities may be exposed to different contingencies depending on their geographical locations, e.g., occurrences of livestock diseases, market risks. In this case, of

the minimization of the total cost function: c1 x1 + c 2 x 2

(1)

subject to x1 + x 2 = d , x1 ≥ 0 , x 2 ≥ 0 ,

(2)

i.e., the model assumes possible cooperation between producers, which also can be viewed as the free trade. The optimal solution to the problem is x1* = d , x 2* = 0 , i.e., the production

is undertaken by the most efficient producer that accords with Ricardo’s views.

particular interest is a properly organized

2.2 Risk Exposure

network of producers allowing to diversify risks

Consider a simple but more realistic problem of planning production under risks which may reduce the production x1 , x 2 . In this case, the endogenous supply (2) is transformed to a linear function a1 x1 + a 2 x 2 = d , (3)

and provide mutual insurance. Contrary to Ricardo, in this case the less efficient and intensive producer may provide the supply of production and enhance market stability, say, if the producer’s risks are different and weakly or even negatively correlated with others. Let us illustrate this with a stylized model. For the sake of clarity suppose that there are only two producers, i = 1, 2, producing the same good. Let x i denote the production level of

i -th producer; c i is the cost for unit of produce. The production can also be imported, or, so to say, “borrowed” from an external source with price b for unit produce. Assume c1 < c 2 < b , i.e., the cheapest source is the first producer. The goal of the production is to satisfy the exogenous inelastic demand d

of a given

region.

2.1 Absence of Risks In the absence of risks we assume there is no distortion of production and no additional regulations on the size of the production capacities x1 , x 2 . The model is formulated as

402

where a1 , a 2 are contingencies or shocks to x1 , x2 , e.g., due to outbreaks of potential livestock diseases or other hazardous events. We assume that a1 , a 2 are random variables 0 ≤ ai ≤ 1 , i = 1,2 . If endogenous supply a1 x1 + a 2 x 2 falls short of demand d, the residual amount d − a1 x1 − a 2 x 2 must be obtained from external sources at unit import cost b . The planning of production capacities x1 , x 2 can be evaluated from the minimization of total production costs and potential import cost, i.e., the minimization of the function. F ( x) = c1 x1 + c 2 x 2 + bE max{0, d − a1 x1 − a 2 x 2 } x2 ≥ 0, and bE max{0, d where x1 ≥ 0, − a1 x1 − a2 x2 } is the expected import cost when d exceeds the supply the demand a1 x1 + a 2 x 2 . Let

us

consider

the

important

case

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FISCHER, ERMOLIEVA, ERMOLIEV and van VELTHUIZEN

illustrating the role of less efficient producer to

borrowing cost b . It also depends on x1* and

stabilize the supply as an insurer.

all conditions ensuring a positive share x1* of producer 1. Although not at risk ( a 2 = 1 ), the

2.3 Only Efficient Producer Is at Risk

optimal production level of producer 2 is

Assume that only the efficient producer is at

defined

by

equation

(3)

through

risk, that is a 2 = 1 . Let function F (x ) have continuous derivatives, i.e., the cumulative

interdependencies

distribution function of a1 has a continuous

This example also illustrates that the production

density function. It is easy to see that the optimal

level of the producer 1 at risk is implicitly

positive decision

x1*

>0,

x 2*

> 0 exists in the

case when Fx1 (0,0) < 0 , Fx2 (0,0) < 0 . We have

Fx1 (0, 0) = c1 − bEa1 ,

Fx2 (0,0) = c2 − b.

among

producers

participating in the same market with demand d.

constrained

by

interdependencies

among

contingencies, production cost and the import costs.

Therefore, somewhat surprisingly, the less efficient

producer

2

must

be

active

2.4 Both Producers Are at Risks

unconditionally (since c 2 − b < 0 ). The cost

In the case when both producers are exposed

efficient producer 1 is inactive in the case

to risks, i.e., a1 ≠ 1 , a 2 ≠ 1 , the existence of

c1 − bEa1 ≥ 0 , leaving production entirely to the

optimal positive production of both producers

higher-costs producer 2 ( c 2 > c1 ). Only in the

follows from similar equations

case c1 − bEa1 < 0 both producers are active.

Fx1 (0,0) = c1 − bEa1 < 0 ,

The “less cost-efficient” producer 2 is able to

Fx2 (0,0) = c 2 − bEa2 < 0 .

stabilize the aggregate production in the presence of contingencies affecting the “more

The structure of optimal solution is similar as

cost-effective” producer 1. It is important to

in case 2.3. In particular, there may be a

derive the market share of the producer 2. The

situation where c 2 − bEa 2 ≥ 0 , when producer

derivative Fx ( x , x 2 ) = c 2 − bP[d > a 1 x + x2 ]

2 is inactive, but the cost effective producer 1 is

can be found by using formulas for optimality

active now with the insurance provided by the

conditions of stochastic minimax problems (see,

external source (import or borrowing).

2

e.g., Dantzig and Madansky 1961, Ermoliev and

Apart from exogenous risks, the production

Wets 1988, and references therein). This means

and the market are subject to endogenous risks

that the optimal production level

x 2*

> 0 of

dependent on the level of x1 , x 2 . Negative

producer 2 is a quantile defined by the equation

impacts

P[d > a1 x1* + x 2* ] = c 2 / b , assuming x1* > 0 (otherwise x 2* = d ). Thus, the market share of

intensification cause contamination of water, soil,

the

risk-free

higher-cost

producer

2

of

production

increase

and

air in the densely populated areas, which may

is

incur uncertain, possibly highly non-linear costs,

determined by the quantile of the distribution

increasing with increasing x1 , x 2 . In this case,

function describing contingencies a1 of the

the cooperation and market sharing may be

risk-exposed producer 1 and by the ratio of

unconditionally advantageous, as the following

c 2 / b , i.e., of production cost c2 and external

case illustrates.

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Livestock Production Planning under Environmental Risks and Uncertainties

2.5 Quadratic Costs

and model-derived results. Risk exposures are

Assume that costs are increasing non-linear

often

characterized

by

certain

standards

functions, for the sake of simplicity, quadratic,

commonly imposed as additional “safety”

c1 x12 + c2 x22 , c1 < c2 , and there are no production

constraints on admissible values of some

distortions, i.e., a1 = a 2 = 1 . The problem is to

indicators, e.g., constraints on ambient standards

minimize

in the pollution control.

c1 x12 + c 2 x 22 subject to the demand-supply constraints x1 + x 2 = d , x1 ≥ 0 , x 2 ≥ 0 . At least one producer must be active, say producer 1. It is easy to see from the standard optimality condition that the optimal level is x1* =

c1 d . Therefore, the optimal level for c1 + c2

producer 2 is x 2* =

c2 c1 + c 2

d . In other words,

both x1* > 0 , x 2* > 0 , i.e., unconditional on the cost effectiveness of the producer 1, the increasing non-linear production costs require co-existence and cooperation of both producers. These examples, in particular 2.4. and 2.5.,

In the following Section we formulate a large-scale problem of livestock production planning in China under aggregate, incomplete, uncertain

information

environmental

risks

taking and

into

account

location-specific

feasibility constraints, and propose an approach for spatial production allocation.

3. Spatial Planning of Livestock Production Facilities Environmental and health risks associated with livestock production intensification, in particular, such as livestock manure and waste processing, livestock disease outbreaks, as well

emphasize that market shares are to a larger

as deterioration of water quality (Simpson and

extent determined by the contingencies of

Cheng 1994, USDA Economic Research Service

producers, in which case the less efficient but

1998)

with lower risk, producer will likely have a

intensification and allocation cannot just follow

higher share than a more efficient, but with

the historical trends. This raises the question

higher risk exposure, producer. For the sake of

how to adjust the production facilities in

simplicity, we characterized contingencies by a

response to increasing demand but without

probability

exacerbating the environmental and health

contingencies,

distribution.

In

reality,

e.g.,

livestock

the

diseases,

environmental pollution, demand fluctuations, economic geographical

instabilities,

temporal

complex

patterns

that

future

production

problems. The

main

components

of

production

increases are the creation of new and/or the

of

expansion of existing facilities, which, however,

occurrences, are subject to spatial interactions.

need to be restricted with respect to location-

Their mutual probability distribution functions

specific

may not be analytically tractable and thus

constraints

require specific simulation and downscaling

information on current environmental condition,

procedures allowing for estimation of required

livestock

values based on all available auxiliary statistics

characteristics, availability of cultivated and

404

and

have

indicate

economic-environmental and

indicators

composition,

suitability aggregating management

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FISCHER, ERMOLIEVA, ERMOLIEV and van VELTHUIZEN

grazing land for livestock feeding and manure

livestock production systems with different

recycling, manure and chemical fertilizers in

management characteristics, productivity,

excess of crop uptake, density of population,

technologies, and farm size. Commercial

water supply, etc.

farms use higher quality production inputs

The data to derive the indicators on required

such as grain feeds, breeding and housing

resolutions is often missing and can only be

techniques, credit and market information,

estimated indirectly, e.g., via a simulation model

veterinary services (livestock health control),

and a set of equations utilizing available

while subsistence farmers often rely on crop

information on various scales (see, for example,

residues and household wastes for livestock

Fischer and Ermolieva el al. 2006, Fischer and

feeds, low level of health services and

Ermolieva et al. 2005). The information that

control. High input systems will have higher

guides spatial allocation of livestock production

productivity and are likely to be found near

may come in various forms and from diverse

markets.

sources (Cao 2000, CCAP 2002, Fang 2001, FAO 2004, FAOSTAT 2002, Huang and Liu 2003b, SSB 2001, SSB 2002): 1.

2.

3. 4. 5.

6.

3.1 Data Harmonization and Downscaling Problems

Available survey statistics on livestock at

Compiling the available data on land use and

county level comes from China 1997 year

agricultural activities consistently across spatial

census. It provides data on the number of

and temporal resolutions, e.g., provincial and

ruminants,

cows,

county level, is a considerable and challenging

buffaloes, goats, sheep, and monogastric

data harmonization and rescaling exercise. Data

livestock – pigs and poultry.

aggregated from county or province level may

On the level of provinces, the data is

be inconsistent with the national estimates, may

available from statistical year books and

contain errors, or be available only in terms of

livestock

cover

plausible ranges, etc. Conversely, the available

livestock produce by livestock type, e.g.,

national statistics do not give any clues as to the

beef, pork, mutton, poultry, milk and eggs

heterogeneity of processes and values at

production;

locations. The type of problems, when we have

Data on crop production at province and

only limited, partial, aggregate or incomplete

county level for a number of years;

statistics,

Information on fertilizer use and application

estimation and data inference methods based on

at county level;

the ability to obtain an infinite number of

Information on cultivated and grazing areas

observations from an unknown true probability

at county level, available from county

distribution. The new estimation problems can

surveys. However, the values are not always

be

consistent

generalizing the definitions in Bierkens and

e.g.,

census.

with

cattle,

The

milking

statistics

provincial

information,

termed

challenges

traditional

“downscaling”

statistical

problems

(by

sometimes missing, and

Finke et al. (2000) and require the development

Expert estimates and survey data on

of rescaling procedures. The recursive sequential

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Livestock Production Planning under Environmental Risks and Uncertainties

downscaling procedure proposed in Section 4

∑ aij u il xijl = Vi .

can be used in a variety of practical situations, in particular,

for

data

harmonization

and

j ,l

Another

related

problem

of

data

downscaling. The main idea is to rely on an

harmonization and downscaling is the estimation

appropriate optimization principle using all

of the spatial dimensions of livestock feed

possible constraints connecting observable and

balances. A significant fraction of agricultural

unobservable dependent variables. In Fischer

primary production and many byproducts are

and Ermolieva et al. (2006) we prove based on

consumed by livestock. Available statistics,

the duality theorem that the solution of the

nationally and internationally, cover only part of

downscaling procedure converges to the solution

feed consumption. For instance, national-level

of the related cross-entropy maximization

data available from the FAO provide accounts of

problem.

primary crop production used for feed (e.g., livestock

grain maize or sweet potato), as well as of crop

production and allocation, a typical downscaling

by-products like protein cakes from oil seed

problem is to derive spatially explicit estimates

processing or from sugar refineries. Statistics on

of current livestock activities, i.e., production

availability and use of other important feed

levels, feeding modes, scale of operations. The

sources, e.g., such as crop residues, pastures,

available

and agricultural and household wastes are

For

the

analysis

information

of

on

future

livestock

is

summarized as follows. Total livestock aij of type i, i = 1, m , in location j,

j = 1, n , is

available from agricultural census. There may also be available information on technical characteristics of livestock management system and productivity u il of livestock i in system l, the price π il of livestock product, the national or subnational production Vi of livestock i . The task is to estimate the livestock production

i in system l and location j = 1, n . Let xijl be the unknown estimates of shares of livestock

i in system l and location j. Formally, the problem is to find such unknown xijl which are consistent with the local statistics on farming and livestock as well as with national estimates of livestock production, and satisfying equations

∑l xijl = 1 , xijl ≥ 0 , i = 1, m , j = 1, n . Since the production Vi of livestock i in the country is known, we have to meet equation

406

generally lacking. Also, the available data sources

lack

indications

concerning

the

distribution of feed resources among different livestock

types.

To

project

agricultural

development consistent and complete feed balances are essential for estimating future crop demand related to indirect crop consumption via livestock.

Ample

technical

estimates

and

information are available on livestock herd structure, secondary crop products (e.g., share of crop residue production in relation to primary products),

spatial

grassland

yields,

and

livestock-specific energy demand and nutritional requirements

(e.g.,

in

terms

of

protein

concentration of applicable feed sources). A downscaling method in this case will use the ancillary sources of information to compile a prior reference distribution of feed requirements and then apply an optimization principle (e.g., cross-entropy

maximization)

subject

to

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FISCHER, ERMOLIEVA, ERMOLIEV and van VELTHUIZEN

accounting constraints by livestock system (of

transport and environmental impacts.

feed energy and nutritional requirements) and

Denote the expected national supply increase

feed categories (of feed supply from different

in the livestock product i by d i , i = 1 : m . Let

sources).

xijl be the unknown portion of the supply increase i related to location j and management system l . The problem is to find xijl satisfying the following system of equations:

In its simplest form, the estimation problem of the livestock feed balance is to find the unknown estimates of feed source j consumed by livestock type i . Total feed requirements of particular livestock systems, denoted by vectors ai , i = 1, m, are estimated from livestock

l, j

∑ xijl = d i ,

(4)

statistics

xijl ≥ 0 ,

(5)

∑ xijl ≤ b jl ,

(6)

(number

and

type

of

animals;

management characteristics; livestock products) and technical information. Availability of feed resources b j of different types j, j = 1, n is known from statistics or, for some types, inferred from data on crop production and agronomic technical parameters. Formally, the problem can be described by a system of n

equations

∑ xij = b j , i =1

j = 1, n,

n

∑ d ij x ij = a i

i =1

i = 1, m , where the matrix {d ij } refers to nutritional contents of different feed sources (e.g., digestible energy, crude protein, etc.). Note, that there is great flexibility in formulating the dietary constraints, for instance to include seasonal nutrition balances or to bring in geographical characteristics and immobility of certain feed sources.

3.2 Livestock Production Allocation In the case of livestock production allocation, the objective is to allocate the increase of national demand for livestock products among the locations and the main production systems in the best possible way while accounting for various risks associated with production and suitability

criteria

regarding

profitability,

i

b jl is aggregate risk constraint restricting the expansion of production in system l and location j , l = 1 : L , j = 1 : n , i = 1 : m . Apart from b jl , there may be additional limits imposed on xijl , xijl ≤ rijl , which can be associated with legislation, for example, to restrict production i within a production “belt”, or to exclude from urban or protected areas, etc. Thresholds b jl and rijl may either indicate that livestock in excess of these values is strictly prohibited or it incurs measures such as taxes or premiums, for eradication of the risks, say, livestock diseases outbreaks or environmental pollution. In this sense, they are analogous to the risk constraints from the catastrophe and insurance theory (Ermolieva and Ermoliev 2005). Values b jl and rijl may be reasonably treated in priors. Formulation (4)-(6) belongs to the transportation type of problems, however there may be more general constraints, for example, of the type ∑ aijl xijl = b jl or i ∑ aijl xijl = b j , which require extension of il approaches proposed in this paper (see Fischer and Ermolieva et al. 2006, Fischer and Ermolieva et al. 2005). There may be infinitely where

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Livestock Production Planning under Environmental Risks and Uncertainties

many solutions of equations (4)-(6). The

the expected initial allocation of d i to k is

important determinant in the allocation problem

y ik0 = qik d i , i = 1, m . However, allocation y 0ik

is a trade-off between the efficiency and the

may not satisfy constraint ∑ y 0ik ≤ bk , j = 1, n .

risks. The information on the current production facilities, threshold values b jl , rijl , and costs

i

can be used to derive a prior probability qijl as

It is easy to derive the relative imbalances β k0 = bk / ∑i yik0 and update z ik0 = y ik0 β k0 ,

our belief or confidence that a unit of demand

i = 1, m , and the constraint

d i should be allocated to management system l in location j . For example, the likelihood qijl can be in essence inversely proportional to costs and values rijl (see, e.g., Wilson 1970).

∑ yik ≤ bk i

is

satisfied, k = 1,2,... , but the estimate z ik0 may cause imbalance for (7), i.e.,

0 ∑ k z ik ≠ d i

Calculate α i0 = d i / ∑k z ik0 , i = 1, m , and update 1 yik = z ik0 α i0 , an so on. The estimate yiks can be

4. Rebalancing Algorithms

represented

Solution xijl can be derived from a certain behavioral principle (see, for example, Ermoliev

qiks = ( qik β ks −1 ) / ( ∑ j qik β ks −1 ) ,

and Leonardi 1981) or from a cross-entropy

k = 1,2,... .

optimization principle. Let us consider the first

calculated.

approach. We can renumerate all pairs (l , j ) , l = 1, L, j = 1, n by k = 1, 2,..., K . In new

qiks +1

notation, the problem is formulated as finding yik satisfying constraints:

as

(

{ }

y s = y iks

Assume

β ks

Find s j

= qik β / ∑i qik β

yiks = qikk di ,

s j

),

i = 1, m, has

= b k / ∑i y iks

been and

i = 1, m , k = 1,2,... ,

and so on. In this form the procedure can be viewed as a redistribution of required supply increase d i

∑ y ik = d i ,

(7)

by applying sequentially adjusted q iks +1 , e.g., by

y ik ≥ 0 ,

(8)

using a Bayesian type of rule for updating the

(9)

qik0 = qik . The update is done on an observation

k

∑ yik ≤ bk , i = 1 : m , k = 1,2,... , i

prior

distribution:

qiks +1 = qik β ks / ∑ i qik β ks ,

with a prior qik aggregating information on

of imbalances of basic constraints rather than

suitability.

observations of random variables. A rebalancing

It is reasonable to allocate more livestock to

procedure, similar to the one described above,

locations with higher demand, productivity, or

was proposed by G.V. Sheleikovskii (see

feed access, with the preference structure according to prior qik . In this case, initial

references in Bregman 1967) for estimation of

portion of production i allocated to k can be

its

derived as qik d i , ∑ k qik = 1 for all i . But it

maximizing

may lead to violation of the constraint (9). The sequential rebalancing (Bregman 1967, Fischer

is given in Fischer and ∑ yij ln yij / qij Ermolieva et al. (2006). Constraints (7), (9)

and Ermolieva et al. 2006) proceeds as follows. Assume that relying on prior probability qik ,

of a Hitchcock-Koopmans transportation model.

408

passenger flows between the regions. A proof of convergence the

to

the

optimal

cross-entropy

solution function

belong to very specific transportation constraints

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING

FISCHER, ERMOLIEVA, ERMOLIEV and van VELTHUIZEN

For more general type of constraints, the algorithm

was

described

in

Fischer

given by Shannon (1948)

and

S ( p1 , p 2 ,..., p n ) = −∑ p i ln p i ,

Ermolieva et al. (2006) and the proof of its

i

convergence to a cross-entropy solution was

and this is termed the entropy of the probability

based on the application of duality theorem.

distribution. The probability that maximizes this measure is said to be the least uncertain or the

5. Cross-entropy Formulation By taking into account the convergence of described in Section 4 sequential algorithm to the cross-entropy maximizing solution, we can formulate the following equivalent approach (see discussion in Fischer and Ermolieva et al. 2006). Let us normalize equations (7)-(9) by using variables z ik = yik / d i . The allocation problem defined by (7)-(9) can be equivalently reformulated as one to find such shares z ik of additional production i allocated to k, k = 1, n , satisfying constraints

k = 1,2,... under the constraints (10)-(12) we have to maximize the entropy − ∑ z ik ln z ik i,k

subject to equations (10)-(12) which represent the known relationships among variables. The choice of z ik can be guided by available prior information q ik on current distribution of livestock, land availability, and feed abundance, proximity to markets, etc. In fact, as we mentioned before, the prior can aggregate all information on risks and suitability thresholds

∑ z ik = 1 ,

(10)

0 ≤ z ik ≤ 1

(11)

∑ zik d i ≤ bk , i = 1 : m , k = 1,2,....

(12)

k

i

In this case, shares

most compact. Thus, to find the shares zik , i = 1, m ,

for bk and z ik . In this case, the cross-entropy maximization principle derives the estimates z ik from maximization of the function − ∑ z ik ln

z ik can be interpreted as

probabilities that a portion of demand increase will be assigned to location j. These probabilities are not known. All we know are the constraints

i,k

z ik q ik

(13)

under constraints (4)-(6), where ∑ z ik ln i,k

z ik qik

(10)-(12). The value on the right hand side in the

defines the so-called Kullback-Leibler distance (Kullback 1959) between distributions z ik and

constraint (12) can be interpreted as an

qik .

expectation value. Given this information, what is

the

best

estimate

of

the

probability

6. Entropy and Minimax Likelihood

distribution? The problem is to find a probability

In many applications the uncertainty can be

assignment which avoids bias while agreeing

characterized or interpreted in probabilistic

with whatever information is given. The

terms either as a frequency of underlying

information theory provides us with a unique

random variables or a degree of our believe. For

criterion allowing to define a measure of the

example, shares z ik in problem (10)-(12) can

uncertainty represented by a discrete probability

be interpreted as probabilities that a portion of

distribution. This measure of the uncertainty was

the supply increase will be assigned to location j.

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING

409

Livestock Production Planning under Environmental Risks and Uncertainties

This

interpretation

is

used

in

sequential

(14) are probabilities p iN = µ i / N , i = 1, r . A

downscaling method of Section 4. A key

key question

problem

concerned with large sample properties of the

in

the

probabilistic

models

of

for

statistical

estimation

is

piN , i = 1, r , i.e., with asymptotic

uncertainty is the estimation of the true

estimate

probability distribution. The standard statistical

analysis of p iN for N → ∞ . In particular, the

estimation theory deals with the situation when

consistency of piN requires the convergence of

the information on this distribution can be derived by random observations of underlying random variables. In this case, the most natural principle for selecting an estimate from a given sample of observations is the

piN

to

is observed. A downscaling problem deals with

true

probability

r

Epξ = ∑ pi* ln pi ,

(15)

i =1

likelihood proposed by Fisher (1922). This maximize the probability that the given sample

underlying

distribution pi , i = 1, r for N → ∞ . The log likelihood function (14) is the sample mean approximation of the expectation

maximum

principle requires that the estimate has to

the

where unknown pi* is approximated by its piN = µ i / N .

frequency problems

the

In

available

downscaling

information

about

the estimation of practically unobservable

unknown probability distribution

variables. Let us show that the maximum

is given not by a sample of observations, but by

entropy principle which is extensively used for

a number of constraints of type (10)-(12)

these new type of estimation problems (see, e.g.,

connecting this distribution with characteristics

Bierkens and Finke et al. 2000, Fischer and

of observable variables. Let us denote by Ρ the

Ermolieva 2006, Wilson 1970), can be viewed

set

as an extension of the maximum likelihood

constraints. If x = ( x1 ,..., x r ) ∈ Ρ , then we can

principle.

of

all

distributions

pi*

, i = 1, r

satisfying

these

consider r

Let xi , i = 1,2,..., N be independent observations of a random variable x with unknown parameter θ . The empirical estimate

as an approximation of the expectation function

µ i of the probability P( x = xi ) is calculated

(15) analogous to the log likelihood function

as the number of times the value xi has been

(14). We can derive now a conclusion analogous

∑ xi ln pi

(16)

i =1

observed in the sample, ∑ µ i = N . The true

to maximization of the sample mean approximation (1 / N )∑ir=1 µ i ln pi . Let us show

probability that P( x = xi ) = pi is unknown and can be estimated by maximizing function

that if an approximate probability distribution x = ( x1 ,..., x r ) ∈ P , then

N

i =1

r

Π i piµi under constraints ∑ pi = 1 , pi > 0 , i =1

i = 1, r , which is traditionally substituted by the log likelihood function L( x, xi ) = ∑i ln piµi = ∑i µ i ln pi .

(14)

The solution to the maximization problem

410

r

r

max ∑ xi ln pi = ∑ xi ln xi . p∈Ρ i =1

(17)

i =1

Indeed, the log likelihood function (16) is defined for any feasible probability distribution x ∈ P. The

worst-case

principle

leads

to

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING

FISCHER, ERMOLIEVA, ERMOLIEV and van VELTHUIZEN

minimization of the maximum log likelihood

demand for livestock products coherent with

function defined by (17):

urbanization processes (Huang and Zhang et al.

r

2003a), demographic change (Cao 2000) and

r

min max ∑ xi ln p j = min ∑ xi ln xi , x∈Ρ

expected growth of incomes (Huang and Liu

x∈Ρ i =1

p∈Ρ i =1

i.e., to the principle of maximizing entropy − ∑ir=1 xi ln xi . In the case of a given prior distribution

qij ,

we

may

require

the

minimization of the difference between the log likelihood function (16) for p ∈ Ρ and the log likelihood function ∑ir=1 q i ln p i for the given prior qi , i = 1, r from Ρ : r r ⎡ ⎤ min ⎢ max ∑ xi ln pi − ∑ xi ln qi ⎥ x∈Ρ ⎣ p∈Ρ i =1 i =1 ⎦

x = min ∑ xi ln i . x∈Ρ qi

2003b, Keyzer and van Veen 2004). Production allocation

and

intensification

levels

are

projected from the base year data for the main livestock types (pigs, poultry, sheep, goat, meat cattle, milk cows) and management systems (grazing, industrial, specialized, traditional) on the level of China counties (2434 China counties). The livestock production by location and management systems is described by the system of equations (7)-(9) where d i is the demand for product i , i = 1 : n , and k = 1,2,... enumerates locations and management systems. Production and intensification level is found by

7. Numerical Experiments In the numerical experiments, for the

the sequential procedure relying on prior qik . Two

scenarios

of

future

production

are

analysis of current and plausible future livestock

compared:

production allocation and intensification in

representing the case in which the increase of

China we derived a solution by applying the

production is allocated proportionally to demand

sequential downscaling procedure described in

increase, which is concentrated in the proximity

Section 4. The solution of the procedure

to densely populated urban areas, and (ii) a

converges to the cross-entropy maximizing

scenario that combines the demand driven

solution. The case study also required a spatially

preference structure of the first scenario with

explicit simulation model2 (Fischer and Ermoli-

information on population densities and its

eva et al. 2006) that in each time period

vulnerability to environmental risks caused by

generates levels and geographic distribution of

livestock production. In the first scenario, the

2

The model is spatially detailed and runs for a time horizon of 30 years, with a 5-year time step. It performs analyses at county (2434 China counties) and province (31 provinces) levels, for which it employs specific down- and up- scaling procedures. For each allocation scenario, the simulation model evaluates the environmental pressure stemming from non-point and point pollution due to nutrients losses in livestock housing and manure handling facilities, from non-effective crop fertilization, and from application of nutrients from fertilizer and livestock manure that are in excess of crop uptake capacities.

(i)

an

intensification

scenario,

prior is calculated to reflect the spatial profile of the demand for livestock products. In the second scenario, the prior is adjusted in such a way that it assigns less “preference” weight to locations with

higher

environmental

population pressure

density

(where

and

livestock

densities are higher and nutrients fertilizers are in excess of crop uptake capacities). The intensification scenario implicitly minimizes the

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING

411

Livestock Production Planning under Environmental Risks and Uncertainties

transportation

costs

as

the

production

production but with little confined livestock; C:

concentrates in the proximity of large markets in

Slight environmental pressure counties with

urban areas with high demand. In the alternative

low environmental pressure from confined

scenario, which is a compromise between the

livestock

demand driven production allocation and the

environmental pressure, i.e., counties with

considerations of health and environmental risks,

moderate environmental pressure from confined

the production is shifted to more distant

livestock

production;

locations

pressure,

i.e.,

characterized

by

availability

of

cultivated land, lower livestock and population

production;

D:

E:

counties

Moderate

Environmental

with

substantial

urbanization and environmental pressure from

density, which increases transportation. However,

confined

the measure of goodness for the scenarios

Environmental pressure, i.e., counties with

livestock

production;

F:

accounts not only for the transportation cost but

substantial urbanization and high environmental

also includes environmental and health risk

pressure from livestock production, and G:

High

proxies. Thus, the two scenarios are compared

Extreme environmental pressure i.e., counties

with respect to number of people in China’s

with high degree of urbanization coinciding with

regions exposed to different categories of

high environmental pressure from confined

environmental risks. Environmental risks are

livestock production.

measured in terms of environmental pressure in

Figure 1 presents the above classification of

relation to the coincidence of three factors:

environmental pressure for the year 2000. Figure

density of confined livestock, human population

1 indicates that currently (i.e., year 2000)

density, and availability of cultivated land. For

hot-spots of environmental pressure are located

this purpose 2434 counties were classified as

mainly in provinces covering the North China

follows: (i) percentage of land under cultivation

Plain, the Sichuan basin, and several locations

(less than 10 percent, 10 to 50 percent, and more

along the coast of South China. Locations of

than 50 percent); (ii) livestock unit cultivated

livestock production concentrate around or in

land ratio i.e., total live weight of confined

the vicinity of areas where the livestock demand

livestock per ha available cultivated (less than

grows fast, e.g. highly populated and urban areas.

300 kg/ha, 300 to 600 kg/ha, and more than

Figure 2 presents diagrams of the distribution of

600kg/ha), and (iii) population density (less than

current population against the mapped classes of

100 persons per square kilometer, 100 to 1000

severity

persons, and more than 1000 persons per square

livestock. The left diagram shows absolute

kilometer). The resulting 27 combined classes

numbers, i.e. million people per class and region.

were reduced and broadly classified into seven

The diagram on the right gives shares of

of

environmental

pressures

from

categories, namely: A: No confined livestock,

population within each region falling into

i.e., counties in scarcely populated areas (desert

respective classes. For year 2000, the estimates

or mountain/plateau) and with very little

suggest that about 20 percent of China’s

confined

population lives in counties characterized as

livestock;

B:

No

environmental

pressure, i.e., counties with substantial crop

412

having

high

or

extreme

severity

of

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING

FISCHER, ERMOLIEVA, ERMOLIEV and van VELTHUIZEN

environmental pressure from intensive livestock

followed by the South region with 27 percent

production. In the “intensification” scenario, by

population in two highest pressure classes in

2030 this population share increases to 36

2000 and with 44 percent in 2030.

percent (Figure 3.a.), i.e., from one-fifth in 2000

Looking at the second allocation scenario,

to about one-third in 2030. Looking only at the

the positive changes are quite visible (see Figure

highest pressure class, the South region appears

3.b.). The estimate of people living in highest

to have the largest number of people and the

pressure class for South region changes from 45

highest population share in such unfavorable

to 42 million. Percentage of population in the

environmental pressure, about 38 million or 22

two highest pressure classes for the same region

percent of population in 2000 increasing to

varies between scenarios as 43 for the bad and

nearly 45 million or 17 percent in 2030.

41 for the good. For North region, 56.5 percent

The region with the highest occurrence of

of total population will live in two highest

people (both absolute and relative) in the two

pressure classes in 2030 in bad scenario and

highest pressure classes is the North region, with

only 52.3 – in good. In North East, the highest

more than 40 percent of the population. In 2030,

pressure classes change percentage from about

the estimated share becomes 57 percent,

18 to less than 10.

No confined livestock No environmental pressure Slight environmental pressure Moderate environmental pressure Environmental pressure High environmental pressure Extreme environmental pressure

Figure 1 Environmental pressure from confined livestock production, 2000

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING

413

Livestock Production Planning under Environmental Risks and Uncertainties

350

NW

Extreme pressure

300

SW

High pressure

250 200 150

Pressure

S

Moderate pressure

C

Slight pressure

100 50 0 N

NE E

C

E

No pressure

NE

No confined

N

S SW NW

0.0

0.2

0.4

0.6

0.8

1.0

Figure 2 Absolute (million people) and relative (share of total population) distribution of population according to classes of severity of environmental pressure from livestock, 2000. The label on the horizontal axis indicate China regions: N, NE, E, C, S, SW, NW stand for North, North-East, East, Center, South, South-West, North-West, respectively NW

NW

SW

SW

S

S

C

C

E

E

NE

NE

N

N 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Figure 3 Relative (share of total population) distribution of population according to classes of severity of environmental pressure from livestock, 2030: a. “intensification” scenario, b. environmentally friendly scenario

8. Conclusions This paper addresses some important aspects of agricultural production planning under risks,

environmental and health problems and violate threshold constraints. In

many

real-world

applications,

the

uncertainties and incomplete information. With

distribution functions of contingencies are not

simple but realistic examples we illustrate the

tractable analytically due to the complexity of

need for co-existence and cooperation of various

interacting factors and spatial relationships. To

agricultural producers with diversified risks,

derive estimates of contingencies, production

which enhances stability of agricultural markets.

planning relies on modeling of the dependencies

In particular, we show that production expansion

and interactions among the processes as well as

can not merely follow historical intensification

on appropriate downscaling and upscaling

trends which in many cases would lead to

algorithms that derive estimates of variables on

414

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING

FISCHER, ERMOLIEVA, ERMOLIEV and van VELTHUIZEN

required scales using all available auxiliary

China: Prospects to 2045 by place of

information. We analyzed a downscaling method

residence and by level of education. Interim

only

Report IR-00-026, International Institute for

for

situations

when

the

available

information is given in the form of average

Applied

values. For many practical situations this

Austria

Systems

Analysis,

Laxenburg,

assumption may be rather strong, which calls for

[4] CCAP (2002). Estimates of province-level

more rigorous probabilistic treatments. For

meat production and consumption, 1980 –

example, a prior probability qik for a demand

1999: Database prepared for CHINAGRO

d i to be produced in location k generates only

project. Center for Chinese Agricultural

a

random

portion

of

production

ξik ,

∑ k ξ ik = d i . The analysis of only average flows ∑ zik d i = bk for many production location i problems may be rather limited since parameters

Policy, Beijing [5] Dantzig, G. & Madansky, A. (1961). On the solution of two-stage linear programs under uncertainty. In: Proceedings of Fourth

z ik can be associated with additional risks.

Berkeley

Similarly, the average aggregate d i may not

Statistics and Probability: 165-176, 1961,

completely

Univ. California Press, Berkley

reflect

the

heterogeneity

and

Symposium

on

Mathematical

uncertainties of demand. More rigorous risk

[6] Ermolieva, T. & Ermoliev, Y. (2005).

based analysis requires probabilistic treatment of

Catastrophic risk management: flood and

these constraints. Another important issue is to

seismic risks case studies. In: Wallace, S.W.,

use in (10)-(12) not only worst-case distributions

Ziemba,

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Stochastic

with goals of the overall decision making

MPS-SIAM

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Philadelphia, PA, USA

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W.T.

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and World Resource Institute, Washington

risks, in particular of extreme catastrophic

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nature.

Recent

practical

applications

and

scientific publications cover problems of spatial Günther Fischer is leader of the Land Use

estimation

Change and Agriculture (LUC) Program at

agricultural

International Institute for Applied Systems

planning under risks and uncertainties; fast

Analysis (IIASA, Laxenburg, Austria), focusing

Monte Carlo optimization for insurance and

on global climate change impacts and adaptation,

mitigation of catastrophic losses; population

on selected regional analyses in Asia and Europe

aging, globalization and economic growth under

to support sustainable and efficient use of land

demographic

and water resources, and on development of

Ermolieva received the Kjell Gunnarson’s Risk

analytical

Management

tools.

He

earned

degrees

in

and

downscaling

values;

and Prize

of

agricultural

economic of

aggregate production

shocks.

Swedish

Dr.

Insurance

mathematics and in data/information processing

Society and the Dr. Aurelio Peccei Award of the

from the Technical University, Vienna, and joined

International Institute for Applied Systems

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING

417

Livestock Production Planning under Environmental Risks and Uncertainties

Analysis.

discontinuous systems subject to abrupt changes and catastrophic risks.

Yuri Ermoliev graduated from Kiev State

University, Department of Mathematics. He

Harrij van Velthuizen is senior scientist in the

received his Ph.D. in applied mathematics from

Land Use Change and Agriculture Program at

the same university. Dr. Ermoliev holds the State

IIASA. Harrij van Velthuizen is land resources

Award in Science both of the Ukraine and of the

ecologist and specialist in agro-ecological

USSR. He is a Member of the Ukrainian

zoning. He was a member of the working group

Academy of Sciences. Dr. Ermoliev has been

that developed FAO's Agro-Ecological Zones

Head of the Department of Mathematical

(AEZ) methodology. In the capacity of senior

Methods of Operations Research at the Institute

consultant and chief technical advisor of various

of Cybernetics of the Ukrainian Academy of

organizations of the United Nations he has done

Sciences, Kiev. From 1979 to 1984 he was

extensive work on agro-ecological assessments

employed at IIASA undertaking research in

for agricultural development planning covering

non-differentiable and stochastic optimization

over twenty countries in Asia, Africa, South

problems. In 1991 he was a visiting professor at

America and Europe. Since 1995, Dr. van

the University of California at Davis and

Velthuizen has been engaged with the activities

returned to IIASA as co-leader of the Risk,

of

Uncertainty

Dr.

Agriculture Program. In 2001, he joined IIASA

concern

stochastic

as research scholar and worked on enhancement

and

models,

Ermoliev's optimization

and

Complexity

publications methods

Project.

of

the

the

IIASA’s

AEZ

Land

Use

methodologies

Change

for

and

various

path-dependent adaptation processes, pollution

applications including agricultural and forest

control

production potentials and impacts of climate

problems,

energy

and

agriculture

modeling, reliability theory, and optimization of

418

variability and climate change on food security.

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING