Land use analysis using multiple goal linear
programming

¹⁾ Adresse : LUW, B.P. 430, 6700 AK Wageningen, les Pays-Bas
²⁾ Adresse : AB-DLO, B.P. 14, 6700 AA Wageningen, les Pays-Bas
³⁾ Adresse : IER, B.P. 258, Bamako, Mali

IER, Bamako AB-DLO, Wageningen, Haren DAN-UAW, Wageningen

P S S

Rapports PSS Nº 31 Wageningen, 1997

Rapports du projet Production Soudano-Sahélienne (PSS)

Table of contents

• 1. Introduction to the course
  • 1.1. Objectives of the course
  • 1.2. Organization of the manual
• 2. Why explorative land use studies?
  • 2.1. Introduction
  • 2.2. General methodology
  • 2.3. Examples of different applications of the methodology for explorative land use studies
  • 2.4. Background and motives of the case: Competing for limited resources in the 5th region of Mali
    • 2.4.1. General characteristics on the economy of Mali
    • 2.4.2. The PSS project
  • 2.5. Background and motives of the case: Four perspectives for the rural areas in the European Community (Rabbinge et al., 1994a)
  • 2.6. Exercises
• 3. Linear programming and multiple criteria analysis
  • 3.1. Introduction
  • 3.2. Linear programming
    • 3.2.1. An example
    • 3.2.2. Decision variables or activities, constraints and objective function
    • 3.2.3. Formulation
    • 3.2.4. Transformations
    • 3.2.5. Assumptions and restrictions
    • 3.2.6. Solutions
    • 3.2.7. Simplex-algorithm
    • 3.2.8. Post-optimal analysis or sensitivity analysis
    • 3.2.9. Example of a linear programming problem: Communal farming in Tagota Village in Mara Region, Tanzania
    • 3.2.10. Software for solving LP-problems
  • 3.3. Multiple criteria optimization problems
    • 3.3.1. Attributes, objectives, targets and goals
    • 3.3.2. Pareto optimal or efficient solutions
    • 3.3.3. Trade-offs amongst criteria
    • 3.3.4. Multiple criteria decision making techniques
  • 3.4. Optimization software used in this course
    • 3.4.1. LP 88
    • 3.4.2. XPRESS
  • 3.5. Exercises
• 4. Methodology for explorative land use studies
  • 4.1. Defining the system
  • 4.2. Land evaluation
  • 4.3. Quantification of input-output relations: Concepts and definitions
    • 4.3.1. Introduction
    • 4.3.2. Target-oriented approach - complete information
    • 4.3.3. Production level, production situation, production technique and production orientation
    • 4.3.4. Aggregation level and time horizon of input-output relations
    • 4.3.5. Quantification of non-substitutable inputs
    • 4.3.6. Quantification of substitutable inputs
    • 4.3.7. Emissions and immissions
    • 4.3.8. Production functions or technology sets
  • 4.4. Identification and quantification of constraints
  • 4.5. Policy views and objective functions
  • 4.6. Interactive multiple goal programming
  • 4.7. Generating land use scenarios
  • 4.8. Sensitivity analysis
  • 4.9. General discussion
  • 4.10. Exercises
• 5. The European Community case study
  • 5.1. System definition
  • 5.2. Land evaluation
  • 5.3. Input-output combinations
  • 5.4. Constraints
  • 5.5. Policy views and objective functions
  • 5.6. Land use scenarios
  • 5.7. The EC study: a summary
    • 5.7.1. System definition
    • 5.7.2. Land evaluation
    • 5.7.3. Quantification of input-output combinations
    • 5.7.4. Identification and quantification of constraints
    • 5.7.5. Policy views and objective functions
    • 5.7.6. Interactive procedures and land-use scenarios generated
    • 5.7.7. Sensitivity analysis
  • 5.8. Exercises
• 6. The PSS study of the Sudano Sahelian zone of Mali
  • 6.1. The definition of the system
  • 6.2. Resources of the Soudano-Sahelian zone of Mali
    • 6.2.1. Introduction
    • 6.2.2. The sub-zones and the agro-ecological units
    • 6.2.3. The resources per sub-zone
    • 6.2.4. Resources of the Koutiala district
    • 6.2.5. The socio-economic and institutional environment of Koutiala
  • 6.3. Agricultural production activities
    • 6.3.1. Introduction
    • 6.3.2. Crop activities
      • 6.3.2.1. Definition criteria
    • 6.3.3. Livestock activities
      • 6.3.3.1. Livestock in herds
      • 6.3.3.2. Feed quality and quantification of feed input
      • 6.3.3.3. Fattening
    • 6.3.4. Transformation activities
      • 6.3.4.1. Fallow activities
      • 6.3.4.2. Rangeland activity
      • 6.3.4.3. Burning of residues
      • 6.3.4.4. Plowing crop residues into the soil
      • 6.3.4.5. Mulching
      • 6.3.4.6. Compost making
      • 6.3.4.7. Crop residue transport from field to farm
      • 6.3.4.8. Transport of organic matter from farm to field
    • 6.3.5. The technical coefficient generator (GCT) of the PSS project
      • 6.3.5.1. Introduction
      • 6.3.5.2. The type of activities
      • 6.3.5.3. The main calculation files and help files
      • 6.3.5.4. Preparing and starting the GCT
      • 6.3.5.5. Examining the calculation files and the data base
      • 6.3.5.6. Generating coefficients for a large number of activities
  • 6.4. Restrictions
  • 6.5. Development objectives and the economic environment
  • 6.6. An example of an MGLP model for one climatic zone
  • 6.7. Case study
• Literature

1. Introduction to the course

At the beginning of this course Gerrard Winstanly (1650) is quoted:

Do not all strive to enjoy the land? The gentry strive for the land, the clergy strive for the land, and buying and selling is an art whereby people endeavour to cheat one another of the land. (A New Yeers Gift to the parliament and Armie, 1650)

This is a quote from a long time ago, but the message he tried to convey, that land was a basic necessity that all sections of society tried to enjoy, has always been and still is valid. This implies that for the same piece of land, there are often several intended uses. Clearly, this may lead to conflict situations. A very old example of such a conflict situation is the following:

`Abram was very rich in cattle, silver and gold. And he went on his journeys to the south, to Bethel, to the place where his tent had been before, between Bethel and Ai, to the place of the altar that he had made there before: and there Abram called on the name of the Lord. And Lot, who went with Abram, also had flocks, and herds, and tents. And the land was not able to bear them, while they were dwelling together. Their possessions were too great, they could not stay together. And there was strife between the herdmen of Abram's cattle and the herdmen of Lot's cattle. ..

And Abram said to Lot: Please, let there be no strife between me and you, and between my herdmen and your herdmen; for we are brothers. Is not the whole land before you? Please, separate yourself from me; if you go to the left, I will go to the right. If you will go to the right, then I will go to the left.' (the Bible, Genesis 12:2-9)

In our days land has become even more scarce. Almost nowhere conflicts can be resolved by simply separating, like in the case of Abram and Lot, because in most parts of the world we are no longer in a situation that there is still `new' land that can be used. This observation implies that it is to be expected that conflicts will be more and more difficult to resolve. A second implication is that today land use is increasingly intensive, which often leads to a degradation of this precious resource. Degradation of a part of the land leads to an increasing pressure on land that is still of good quality, and the danger exists to end up in a downward spiral. As land becomes more scarce, it also becomes more important that its utilization is sustainable. Not only the present generation but also future generations should be able to enjoy the benefits of the land.

In many places worldwide attention has been given to development of techniques, mainly decision supporting, that accommodate the problems that accompany the above mentioned two points. This course endeavors to play a role in acquainting participants with the latest state of the art techniques. On purpose we refrain from using the term `land use planning' as is so often done by many authors on this topic. The term `land use planning' only too often gives the connotation that results of a study are converted into a set of actions which will lead to the realization of a desired goal. Everybody involved in land use studies knows that usually this is not the case. These studies increase the understanding and insight of present or intended land use and as such contribute to policy making and decision making. Hardly ever are these studies directly followed by a `blue-print' of actions which are to be taken. It is not the researcher who takes any action and indeed it often is not the policy maker who takes any direct action. In the end it is often at the farm household level that decisions are made concerning the utilization of land, especially in developing countries. We have therefore preferred to use the term Land Use Analysis (LUA), with which we follow, among others, Schipper (1996).

A second remark concerning the title is that the analysis is not limited to the use of land only, but, in principle, also includes other natural resources, and, for example, labor. For this reason it might have been preferable to replace the word `land' by `natural resources' or `agricultural resource use'. The main reason why this was not done, is that in the literature the term land use has become so widely used for the type of study with which this course deals, that it would be misleading not to use it. Another justification is that other natural resources and their use are always related to the use of land. Thus, land refers not only to the land itself, but also to everything it carries or contains as other resource, like trees, water, nutrients, and organic matter.

1.1. Objectives of the course

Recent developments in research have provided useful tools for LUA. This course will strictly limit itself to the analysis of land use for agricultural purposes. It will, however, be possible to show the effects for non-agricultural purposes. Land used for agriculture cannot be used for non-agricultural activities. As Fresco (1994, p 5) states:

`During the last decade, the combined use of simulation models, expert systems, and geographic information systems, various types of data bases and multiple goal planning techniques have allowed us to formulate technical options for land use in a much more precise and varied way. Not as one dimensional blue prints, but as scenarios for policy choices.'

The presentation of scenarios is not a new technique; it is already well-known in demography and economic planning. Scenarios are not to be confused with forecasts: they do not predict but allow us to explore technical options based on explicit assumption given a set of goals. More specifically, the cost of attainment of one goal can be expressed in terms of the reduction of the attainment of other goals, thus forcing the policy makers and land users to make explicit choices. In other words land use models may help to make potential conflicts more visible'.

Within the context of the PSS project a model was developed that can be used as an instrument for exploring options for sustainable land use in arable agriculture and livestock husbandry in the sudano-sahelian zone of West Africa. The model can also be used to demonstrate the trade-off between different conflicting objectives.

The general objective of this course is to utilize the experience gained during the PSS project and to acquaint course participants with the techniques and methodology developed in the project. This is consistent with the objective of this project to support and strengthen local research capacity in Mali.

To achieve this general objective the following will be done during the course:

1. acquaint course participants with the latest state of the art techniques of analysis in the field of land use analysis. This will be done by explaining and describing these techniques of analysis.

2. in order to deepen understanding and insight of these land use analysis techniques, course participants will exercise with these techniques.

3. show how these techniques have actually been applied in LUA. Course participants will be introduced to two case studies, one conducted by the Netherlands Scientific Council for Government Policy (WRR, 1992), the other one conducted by the PSS project carried out by the Institut Economie Rurale (IER, Bamako, Mali) and the Institute for Agro-biological and Soil fertility research (AB-DLO, Wageningen, The Netherlands).

1.2. Organization of the manual

The central question that will be addressed in Chapter 2 is: Land use analysis: why? We have already briefly touched on this subject in Chapter 1, however, the question will be answered in more depth in Chapter 2. This will be done by presenting a general discussion and the motivations for the two major case studies mentioned above.

When considering LUA studies one comes across many techniques of analysis. The technique of analysis that one may consider most appropriate will depend on the objectives of a study. In this course, only brief attention will be given to the various techniques. Main focus of the course is to acquaint participants with those techniques that can be used to analyze strategic options for development. Particular attention will be given to linear programming (LP) and to multiple criteria decision making (MCDM). Both techniques of analysis are very often used in LUA studies and an understanding of them is essential. They will be treated in Chapter 3. Problems concerning LUA today have become so complex that solving them without the use of computers and highly sophisticated software packages has become unthinkable. There are two computer packages that will be used during this course, namely LP88 and XPRESS. The former will mainly be used to introduce linear programming and the latter to exercise with more complicated problems. The two computer packages will be introduced at the end of Chapter 3.

In Chapter 4 the general methodology for explorative LUA is presented and discussed. This is done by focusing on the following components of explorative land use analysis studies: (1) system definition, (2) land evaluation, (3) concepts and definitions concerning the quantification of input - output relations, (4) the identification and quantification of constraints, (5) policy views and objective functions, (6) interactive multiple goal linear programming, (7) generation of land use scenarios and finally, (8) sensitivity analysis.

The manner in which the LUA methodology - described in Chapter 4 - was actually used in the European case study and in the PSS case study is described in Chapters 5 and 6 respectively. As this course particularly focuses on the work that has been done in the PSS project, the European case study will be treated in less detail. In Chapter 6, course participants will have the opportunity to exercise with a technical coefficient generator and the MGLP model that were developed by the researchers of the PSS project. These exercises will give more insight to participants concerning the strong and weak points of such studies. The last part of Chapter 6 will introduce the case study that is to be developed by the course participants. Using knowledge that they have accumulated during earlier stages of the course, for a specified region in Mali, the course participants will:
* make a study proposal that will be presented and discussed,
* generate the required data, making use of a technical coefficient generator,
* develop a multiple goal linear programming model;
* generate scenarios,
* conduct sensitivity analysis,
* interpret the results
* and finally, write a brief report, that will be presented and discussed in a plenary session.

With the aid of the TCG and the MGLP model it should be possible to execute this case study in relatively little time, and to concentrate on the policy views, the quantification of objectives and the development of scenarios.

2. Why explorative land use studies?

2.1. Introduction

Facing the future, everybody is in favor of sustainable development and it is likely that people take it as the guiding principle for their view on future development and future land use. However, everyone seems to have an own idea about sustainability. Sustainable development is hard to make operational, because of the various dimensions of sustainability (ecological, economic and social dimensions), the uncertainties about relations between these dimensions, and the difficulty to balance consequent environmental, economic and social risks. No wonder that there are many perceptions of sustainable development, which can be classified according to the perceptions of environmental risks and societal risks, e.g., the possibility to realize societal changes (WRR, 1995).

Explorative land use studies aim at revealing and quantifying the trade-off between the different perceptions of sustainable development and the conflicting objectives involved. They aim at showing policy makers consequences of different policy aims for land use and thus at helping them in choosing aims.

In explorative land use studies the future possibilities are considered with an open mind. In these studies the past is not used as a measure for the future. Existing political, economical and societal structures are not taken for granted or extrapolated to the future, but they are treated in an explicit way. New possibilities are explored by combining technical possibilities with explicit political or societal desires. Explorative studies do not give blue prints how to arrive at new situations; they just present static end views. Realizing new technical possibilities and changing political or social structures is often a matter of political desire and an appropriate policy. Obviously, therefore, these studies need a follow up in order to answer questions of how to realize the end views and the appropriate policy instruments.

Explorative land use studies can be carried out for various reasons. As stated, ultimately the aim is to operationalize options for sustainable development. More down to earth, immediate motives can be various. Reclamation of a new territory can be a reason to carry out an explorative study. Setting of research priorities may be a factor. There may be problems with agricultural production, either shortage of food (in some Third World countries) or overproduction (e.g. in the European Union). Socio-economic problems with agricultural employment and farmers' income may play a role. Environmental issues and nature or landscape conservation may be motives.

In this chapter the general methodology for explorative land use studies is introduced. After this introduction a historical overview of the literature on explorative land use studies is given with special reference to the motives to carry out such studies. Finally, an extensive description is given of the immediate motives to carry out two examples of explorative studies, which are used as case studies in this course.

2.2. General methodology

Figure 2.1 highlights the general building blocks of the methodology for explorative land use studies.

The central technique that is used in the methodology is the Interactive Multiple Goal Linear Programming technique. IMGLP is a linear programming technique, not with one but with several objective functions. In each run of an IMGLP model, one objective function is optimized while the other objectives are used as constraints. Upper or lower bounds can be put on these `goal constraints'. For different policy views land use scenarios can be generated by optimizing and putting bounds on the most relevant combination of objectives. Chapter 3 deals with the principles and backgrounds of linear programming and the IMGLP method.

The IMGLP model is fed with different kinds of information. First of all, input-output relations on agricultural production, based on a land evaluation, are fed into the model. The input-output tables represent the quantification of different production activities on specific land units. They tell us the required inputs to produce certain outputs on a land unit with certain climatic and soil characteristics. Different production ecological concepts are used for a systematic and scientific quantification of the input-output relations.

Secondly, technical or fixed constraints are formulated which represent information about the various resources, such as area, water and available nutrients (manure). They can also be used to quantify more normative constraints like the consumption and trade of agricultural products in the system.

Finally a set of objective functions is distilled from the prevailing policy views that can be distinguished in the society of the system under study. Usually the objectives concern ecological, agricultural, economic and social aspects of sustainability.

With these three blocks of information the IMGLP model can be built. Land use scenarios can be generated with the IMGLP model for each of the policy views in the society of the system, by optimizing and putting upper or lower bounds on the most relevant combination of objectives for each of these policy views. The results for the scenarios include the values for the objective functions and the optimal regional land allocation (maps). Policy makers can now see how their priorities affect land use and how the effects are distributed over the system under study.

Figure 2.1 General methodology for explorative land use studies.

2.3. Examples of different applications of the methodology for explorative land use studies

The methodology for explorative land use studies presented in this course is rather new. Examples of explorative land use studies are not numerous. The examples differ in the time horizon of the study; some are more explorative (distant future) than others (near future).

The methodology builds on the work presented in Veeneklaas (1990). In that work, technical and economic analysis is dovetailed. A simple structured input-output model with technical relations was constructed. Poorly understood behavioral and normative relations are represented as constraints. The methodology uses an operations research technique, interactive multiple goal linear programming, an iterative procedure which makes it possible to deal with various objectives. This procedure was first used by Nijkamp & Spronk (1980) and Spronk & Veeneklaas (1983). The Netherlands Scientific Council for Government Policy used the methodology in its study Scope for growth (WRR, 1987), focusing on the economic future of the Netherlands and on environmental issues.

For typical agricultural or land use problems several studies using the methodology have been carried out for the regional level. De Wit et al. (1988) presented a case study for a Mediterranean agro-pastoral region, the semi-arid zone in the Mediterranean Basin located in the northern Negev of Israel. Agricultural activities in that region consist of sheep husbandry and dryland farming. The approach resulted in options reflecting the various viewpoints or policy views for that region. Three viewpoints were distinguished: traditionalists, settlement agency, settlers' view, each with different opinions on the relevance of various objectives concerning development aid, the area under extensive systems, the import of concentrates, employment, the number of settlers in the region and the desired consumption. Table 2.1 gives an example of scenarios for the three policy views. The different attitudes against the various objectives affect the land use and the use of various sheep breeds.

The first real agricultural application of the method is that in the Mariut project (Ayyad & Van Keulen, 1987). That project aimed at assessing the potentials of different agricultural systems for the purpose of regional planning in the Northwestern coastal zone of Egypt. The main agricultural activities distinguished in the region were fruit production (olive and fig trees), barley production and animal husbandry (sheep and goat meat production). Optional land use systems were formulated and their economic feasibility and impact on the natural resources were investigated. The results aimed at helping planners to introduce or promote those systems.

An extensive study using the methodology, was carried out for the fifth region of Mali, in the framework of the second five-year plan, formulated in cooperation with the World Bank (Van Keulen & Veeneklaas, 1993). The study aimed at increasing insights in the agricultural production systems in the Region, and their major constraints, as a basis for the formulation of a land use plan, that would take into account the potentials of the natural resources and the development objectives of the various actors involved in the development process. This study is used as a case study in this course. This Chapter will discuss the background and motives to carry out this study.

The Netherlands Scientific Council for Government Policy carried out the study `Ground for Choices' (WRR, 1992; Rabbinge & Van Latesteijn, 1992) for future land use in the European Community. Their study shows the conflicts arising from increasing agricultural productivity, market saturation, differences in production and productivity within the EC and increasing concern for regional employment, environment, nature and landscape. This study is also used as case study in the course. This Chapter will discuss the background and motives of this study.

In Costa Rica a more or less explorative study is carried out for the Atlantic Zone, in the USTED study (Alfaro et al., 1994). Problems with the marketing of agricultural products and growing concern about loss of bio-diversity, sustainable exploitation of natural resources and widespread land speculation were arguments for the Wageningen Agricultural University to carry out a study on future land use in the Atlantic Zone.

Table 2.1 Summary of results of the IMGLP model for three policy views, in the case study for a Mediterranean agropastoral region

		traditionalists	settlement	settlers agency	unit
bounds on the goals
development aid	=	0	0	0	$/year
extensive system	>	600	75	75	103 ha
imported concentrates	=	0	free	free	kg
hired labour	<	free	200	free	p. year
results for objectives
employment		13000	15000	10600	p. year
settlers		11740	14800	6000	p. year
linear growth rates settlers		43	74	0	p. year/year
hired labour		1260	200	4600	p. year
consumptive income		121	196	169	106 $
consumptive income/settler		11	14	28	103 $/year
some results on land use
nitrogen fertilizer use		1600	650	608	103 kg
phosphorus fertilizer use		243	866	643	103 kg
area wheat/fallow		0	0	0	ha
area wheat/wheat		8400	3700	100	ha
area wheat/legume		851	15100	16900	ha
total area wheat		9251	18800	17000	ha
area extensive systems		60	12	8	103 ha
number of sheep of various breeds
Awassi systems		0	0	0	ewe eq.
improved Awassi systems		122300	42700	35300	ewe eq.
Merino systems		15100	75300	70500	ewe eq.
Finn cross systems		10	27200	27100	ewe eq.
total number of sheep		137410	145200	132900	ewe eq.

Source: De Wit et al. (1988).

The principles of the methodology are also used in studies at farm level. Schans (1991) carried out a study for the optimization of arable farming systems that integrate economic and ecological goals. Farming systems in the Dutch Flevopolders with ware potatoes as a major cash crop were used as a case study. Similar studies were carried out for dairy farming systems (Van de Ven, 1994) and arable farms with flowerbulb-based rotations in the Netherlands (Rossing et al., 1997). New research projects are started in 1994, to investigate possibilities for mixed farming systems (arable and dairy systems) in the Netherlands, both at the farm and regional level.

Stroosnijder & Van Rheenen (1993) applied the methodology for small mixed farms in East Java, taking into consideration aspects of food security, farmer's welfare and environmental concern. The results of this study are published in Van Rheenen (1995).

2.4. Background and motives of the case: Competing for limited resources in the 5th region of Mali

2.4.1. General characteristics on the economy of Mali

The principal characteristics of the economy of Mali are essentially those of most of the developing countries, that is (Table 2.2) .

Table 2.2 The national product of Mali in 1990, its subdivision and some other characteristics of the malian economy of Mali (in billion F cfa)

		distribution of the GNP	%
gross national product	666	primary sector	45.6
growth rate in volume (%)	1.0	food crops	13.1
per capita GNP (thousands of F cfa)	82.4	cash crops	3.4
		animal production	22.9
imports	168	fisheries	0.9
of which : food products	19	tree plantations	5.2
machines and cars	60
oil products	15	secondary sector	12.8
exports	94	industry	6.5
of which : cotton	44	mines	1.5
livestock	25	art	1.3
gold and diamonds	13	buildings. public works	3.6
exports/imports (%)	56
		tertiary	37.0
trade balance (exp. - imp.)	-74	transport	4.7
goods and services balance	-96	trade	17.9
current balance	-34	services	14.4
balance of payments	20
budget :
- receipts	250	import tax	4.6
- expenditures	261
- deficit	11	total	100
public debt
- current	1046
- services	19
- ratio service/exports (%)	20

Source: DNSI, 1992.

* The per capita Gross National Product (GNP) is low (around US $270).
* The primary sector is the largest contributor to the GNP, almost 50 %, which is for almost 100 % accounted for by crop and animal production. The contribution of the tertiary sector is around 37 % of GNP. The contribution of industrial activities to GNP is still weak, and is mainly constituted by cotton, cloth, sugar, non alcoholic drinks, cigarettes and electricity.
* Imports are essentially machines and cars (36 % in 1990), followed by food products and oil products.
* Main export products are: cotton (47 % in 1990), livestock, gold and diamonds
* The trade balance is negative
* The budget has an important deficit
* Mali has a big public debt, which is greater than the GNP for one year. And although the debt service is about 25 % of the value of exported products, the debt is not becoming smaller.

Like many other African countries, Mali is in a period of transition where its economic environment is concerned. The Structural Adjustment Programs (SAP) that are carried out are characterized by less state intervention at many societal levels and various other economic reforms. The most important measures taken in this program are the progressive liberalization of the cereal market, the reform of the cotton marketing, limitation of public expenditure (especially salaries), and finally the reform of the public sector : liquidation, privatization and rehabilitation. To strengthen the SAP, the Franc CFA has undergone a devaluation of 50 %, which should help Mali and other countries with the same currency to become more competitive.

Crop and livestock husbandry

Only around 4 % of the area of Mali is arable land, 75 % of which is actually occupied by crops (Table 2.3). Food crops occupy 85 % of cropped area in Mali. Cash crops occupy only a relatively small area and also represent a relatively small percentage of total product value (7,6 % of the added value of the primary sector). Cereals represent 90 % of area under food crops. This predominance of cereals is also crucial for livestock, as it implies a big availability of byproducts that can be used as animal feed. Groundnut and cowpea yield considerable quantities of good quality hay.

Prices of a number of important crops are given in Table 2.4.

Table 2.3 Potential land occupation in Mali

	area
total area	124 million ha
arable land	5.2 million ha
cropped area	3.9 million ha

Source: Project PIRL cited by DNSI (1991).

Table 2.4 Producer prices for various crops (F cfa kg^-1 ), 1980-90

	cultures
year	millet, sorghum	maize	rice	groundnut	cotton (1st choice)
1981	35	35	38	40	55
1982	43	45	50	45	55
1983	45	48	55	55	65
1984	50	50	60	na	65
1985	50	50	60	na	75
1986	55	55	70	na	85
1987	55	55	70	na	85
1988	50	36	70	68	85
1989	42	33	70	69	85
1990	63	51	78	na	85
1991	80	56	81	na	93
1992	47	36	58	na	93
1993	51	41	na	101	na
1994	45	37	na	na	na

(na : not available)
Sources: (DNSI, SIM/PAM, DPAER/IER)

Population, education and health

In 1987 Mali had a population de 7,7 million habitants (DNSI, 1991d), which comes down to an average population density of 6 inhabitants per km². The growth rate is around 2.7 % per year (Banque Mondiale, 1994). The population is not evenly distributed between the 7 administrative regions, because 75 % of the population is concentrated on 25 % of the territory in the regions of Ségou, Sikasso, Mopti, and Koulikoro and in the town of Bamako (see Table 2.5). The regions of the North, Gao and Tombouctou, to which is recently added the region of Kidal, have less than 900,000 inhabitants in 1987, while they occupy more than 60 % of the national territory. The demographic data show that the population is essentially rural (78 %).

The sedentary population represents 96 %, the nomadic population 4 %. There are many ethnic groups in Mali, of which the most important (numerically) are : Bambara, Malinké, Sarakolé, Sonrhaï, Dogon, Sénoufo, Bobo, Minianka, Fulani, Maure and Tamachecks. The nomadic ethnies are mainly pastoralists: the Fulani, the Maures and the Tamacheck.

Table 2.5 Structure of the population of the Mali according to various ways of classification

socio-demographic category	%	administrative region	%	density
< 8 year	24	Kayes	13.9	8.9
8-14 year	22	Koulikoro	15.6	12.5
15-59 year	48	Sikasso	17.0	18.7
> 60 year	6	Ségou	17.4	20.7
urban	22	Mopti	16.6	16.2
rural	78	Tombouctou	6.0	0.9
nomad	4	Gao	5.0	1.2
sedentary	96	Bamako	8.5	2612

Source: DNSI (1991d, Tables 5a, 7, 13, 15)

The distribution of the active population per sector is given in Table 2.6. The data show that the primary sector occupies the first place, which underlines again the importance of this sector for the malian economy.

Table 2.6 Distribution of human labor force among economic sectors

sector	%
primary sector (agriculture, animal husbandry, tree plantations, fishery)	82
secondary sector (industry, mines, art, buildings and public works)	6
tertiary sector (transport, trade, service, administration, others)	12

Source: DNSI (1991e, page 40)

The statistics for the educational sector in Mali (Table 2.7) show that relatively few children go to school. 24 % for primary education is even the lowest figure of the entire West African region. In other countries in this region, the percentage is on average 61 % and varies from almost 30 % (Niger) to almost 100 % (Cap Vert). During the last decade the fraction of children going to school has had a tendency to decline. The fraction of girls going to school is only half of that of the boys.

Table 2.7 Education indicators in Mali

fraction going to school rate	%	literacy rate	%
primary education
- total	24	total	17
- boys	30	men	23
- girls	17	women	11
secondary education	6

Table 2.8 summarizes some health indicators of Mali. Child mortality is very high, 161 of 1000 children die before the age of one year. For the other sub-Saharan African countries this is 104, and in developed countries only 8. Mortality rates of children under five years of age, almost 25 %, is for over 70 % due to diseases that can be prevented with adequate medical care, like malaria, tetanus, respiratory infections, and diarrhea. Another problem is malnutrition, of which almost one of three children under five years suffers.

Table 2.8 Health and nutrition indicators in Mali

2.4.2. The PSS project

The PSS project is a cooperative project between Mali and the Netherlands. The project lasted from 1991 to 1996.

The main objectives of the project were to:
* contribute to the development of sustainable and economically viable agricultural production systems
* strengthen the local research capacity through training of national scientists

The study region is the sudano-sahelian zone, characterized by rainfall between 300 and 900 mm year^-1 in West Africa. In this zone the two main types of agricultural production, crop and animal husbandry, are strongly linked. Production of grass, leaves, and fruits on rangelands, fodder crops, crop residues and agro-industrial products serve as animal feed, while dung and animal traction, by-products of animal production activities, are used in crop production. Moreover, buying animals is a means of savings for many farmers.

In the sudano-sahelian zone cropped areas and rangelands are characterized low yields, which are due in the first place to low soil fertility, not, in this region, to lack of water ( Penning de Vries & Djitèye, 1982; Van Keulen & Breman, 1990). Water availability is nevertheless a big problem, mainly because of the great variability in rainfall (Vierich & Stoop, 1990). Soil poverty is worsened by a strong demographic growth, which leads to degradation of land and other natural resources. This corroborates even more the rural production possibilities. On cropped land the fallow periods are shortened, while the quantities of organic matter and nutritive elements don't suffice to restore soil fertility (Van de Pol, 1992). Rangelands suffer from overgrazing, causing erosion, loss of perennial species and decreasing number of trees. Animal production also suffers from low productivity, with as a key problem the lack of good quality feed, especially in the dry season (Penning de Vries & Djitèye, 1982).

Technically speaking, solutions to the various problems may exist. Van Keulen & Breman (1990) show that agricultural productivity could be boosted with the use of more nutritive elements (especially nitrogen and phosphorus), provided that the organic matter balance is in equilibrium. Higher input of nutrients would also increase the ecological sustainability of the primary production. As the quantity of organic matter used to fertilize the fields is not sufficient to maintain an equilibrium on the nutrient balances an increase in the use of chemical fertilizers seems to be a conditio sine qua non for sustainable agricultural development.

In animal production higher productivity is to be attained by an increasing use of feed supplements (crop residues, fodder crops, agro-industrial products), while lowering the pressure on the natural rangelands. This option requires a higher production of supplements. Thus, animal production systems would also profit from the use of fertilizer in primary production, either directly, by its use for fodder crops, fodder banks and improved rangeland systems, or indirectly through its use in food crops and cash crops, which would lead to an increase in quantity and in quality (higher concentration of nutrients) of crop residues.

At present the indicated solutions are not applied at a large scale. This is due to a number of factors, among which the cost of fertilizer. This shows that it is necessary to analyze not only technical but also socio-economic aspects of the problem. The systems research of the PSS project had as its objective to analyze in an integrated manner the various technical options (varying according to the climatic region) and the socio-economic conditions, taking into account the availability of resources, the need for sustainable production and the objectives of the region.

2.5. Background and motives of the case: Four perspectives for the rural areas in the European Community (Rabbinge et al., 1994a)

The European Economic Community (European Union since 1993) was founded in 1957 in Rome to encourage a general economic integration of West-Germany, the Netherlands, Belgium, Luxembourg, France and Italy. The first aim was one common market with free movement of people, commodities, services and capital. In 1968 the customs union came about: the customs duties between the six member states were gradually abolished and one common tariff against non-member states was set up. Also in 1968 a Common Agricultural Policy started. The purposes of this policy were already formulated in 1957 in the treaty of Rome:
* increasing the agricultural productivity;
* taking care for a reasonable standard of living for the agrarian population;
* stabilization of the markets;
* guaranteeing the food supply for reasonable prices for the consumers.

In 1973 the UK, Denmark and Ireland joined the EC, in 1981 Greece and in 1986 Spain and Portugal.

Currently, the EC counts 12 member states, with an area totalling 229 million ha. The largest countries are France and Spain and the smallest countries are Luxembourg, Belgium and the Netherlands. The EC counts 324 million inhabitants (4.2 % of the world population), of which most people live in the UK, Germany, France and Italy (Table 2.9). The most densely populated countries are the Netherlands and Belgium and the least densely populated countries are Ireland, Greece and Spain. The growth of the population is expected to amount less than one promille per year.

Table 2.9 The area, population, population density and the percentage of employment in agriculture in the European Community (excluding the former East Germany)

country	area (mil. ha)	population (millions)	pop./area (/km2)	% empl. in agriculture
EC-12	229	324	143	7.6
Denmark	4.3	5.1	119	5.8
UK	25	57	233	2.4
Ireland	7.0	3.5	50	15.8
FR Germany	25	61	246	4.5
Netherlands	3.5	15	368	4.8
Belgium	3.1	10	323	3.2
Luxembourg	0.3	0.4	143	3.9
France	55	56	101	7.2
Italy	31	57	190	9.8
Greece	14	10	76	26.6
Spain	52	39	77	14.3
Portugal	9.3	10	111	21.2

Source: Eurostat (1989)

Table 2.10 Total area, agricultural area, percentage agricultural area (% AA) and the percentage grassland in the EC-12 (excluding the former East Germany

country	area 106 ha	agr. area 106 ha	% AA	% grassland
EC-12	229	129	56	21
Denmark	4.3	2.8	65	5
UK	25	18	75	46
Ireland	7.0	5.7	82	66
FR Germany	25	12	48	18
Netherlands	3.5	2.0	58	33
Belgium	3.1	1.4	46	21
Luxembourg	0.3	0.1	48	27
France	55	31	57	22
Italy	31	18	57	16
Greece	14	5.7	41	13
Spain	52	27	52	13
Portugal	9.3	4.5	49	8

Source: Eurostat (1989).

In the EC there are 128 million places of work. There are large differences in employment over the different economical sectors between the different member states. On average 7.6 % of the people is employed in agriculture, 33.2 % is employed in industry and 59.2 % is employed in services. In the UK only 2.4 % is employed in agriculture, whereas in Greece more than 25 % is employed in agriculture.

The European community is located between 35 and 60 N and is surrounded by the North Sea, the Atlantic Ocean and the Mediterranean. The climate in the EC can be divided in four categories: (i) a sea climate in the north-west with moderate winters, cool summers and constant precipitation throughout the year; (ii) a land climate in the east with cold, dry winters and warm summers;

(iii) the Mediterranean climate in the regions around the Mediterranean with moderate, humid winters and warm, dry summers and (iv) the mountain areas with their own climate.

About 56 % of the total area consists of agricultural land (129 million ha - Table 2.10). This percentage varies for the member states between 41 % (Greece) and

82 % (Ireland). The percentage grassland of this agricultural area varies greatly among the different countries: 5 % in Denmark and 66 % in Ireland.

Table 2.11 gives some information about the current and potential wheat yields in the different member states. Yields are highest in the Northern states: the Netherlands, UK, Denmark, Belgium and Germany and lowest in the Southern states: Portugal, Spain, Greece and Italy. The actual average wheat yield in the EC amounts 5.1 ton ha^-1. For potatoes the average yield is 28 ton ha^-1. Between the individual countries the potato yields vary between 9.0 ton ha^-1 in Portugal and 43 ton ha^-1 in the Netherlands.

Table 2.12 shows the production of the main agricultural products. For many products self-sufficiency is more than 100 % and this was one of the reasons to start a study, carried out by the Netherlands Scientific Council for Government Policy (WRR): Four perspectives for the rural areas in the European Community.

Background of the study

In general terms the Common Agricultural Policy of the EC was rather successful: most goals were achieved. However, there are also problems. At present the situation in EC agriculture can be characterized as follows:

a. Due to an increasing productivity in the Community a situation of self-sufficiency for most agricultural products was realized (Table 2.12).

b. After self-sufficiency was reached, productivity growth continued to rise. This has led to overproduction with major budgetary consequences. The system of guaranteed prices for most agricultural products requires an increasing amount of money from the European tax payer to finance.

c. At the same time attention has grown for other goals than agricultural production. Environment, employment and farmers income are nowadays tightly linked to developments in agriculture. The considerable overuse of pesticides and plant nutrients due to their low prices, has created immense environmental problems in some parts of the EC.

d. There is a growing tension between the EC and the world market, especially the USA. In the GATT negotiations the price policy for agricultural products in the EC was a hot issue.

Table 2.11 The distribution of wheat production within the EC (excluding the former East Germany), the actual wheat yields and the actual wheat yields as percentage of the calculated Potential (PP) or Water-limited production (WLP) as a measure for the management level in the EC-12

country	production volume (% of EC-total)	wheat yield (ton ha-1)	actual production as % PP	actual production as % WLP
Denmark	2.6	6.3	67	95
UK	16.9	6.8	62	77
Ireland	0.7	3.8	62	69
FR Germany	13.4	6.2	60	67
Netherlands	1.4	7.4	74	85
Belgium	1.6	6.4	62	72
Luxembourg	0.0	4.3	40	53
France	39.8	5.9	54	67
Portugal	0.5	1.6	12	24
Spain	7.6	2.5	26	60
Italy	12.5	3.8	29	37
Greece	3.0	2.5	31	38

Source: De Koning & Van Diepen (1992).

Developments in productivity show a steady increase all over the world. In Figure 2.2 the increase in yield per hectare for wheat is shown. Both the UK and the USA show an ongoing rise in productivity especially after World War II. Of course these developments will not go on forever, although until now there is no slowing-down. When and at which level the maximum will be reached is not very clear. Because of this increasing productivity it seems possible to guarantee food security within the Community with only a relatively small number of farmers on a relatively small area. Much space and work force can be used for other aims, like nature conservation and recreation.

The study of the WRR was aimed at defining the limitations to this growth in productivity. In the end those limitations will define the possibilities of agriculture.

Table 2.12 Production and self-sufficiency percentage of some agricultural products in the EC-12 (excluding the former East Germany

product	production (mil. tons)	selfsufficiency %
wheat	72	124
barley	47	118
grain maize	26	96
total cereals	154	114
potatoes	42	103
sugar	13	136
wine	17	104
vegetables	45	106
milk	113	102 (cheese) 115 (butter)
meat	31	102
eggs	5	102

Source: Eurostat (1990).

Figure 2.2 Development of soil productivity for wheat in the UK and the USA. Source: Rabbinge & Van Latesteijn, 1992.

The study of the WRR was aimed at defining the limitations to this growth in productivity. In the end those limitations will define the possibilities of agriculture in the Community. The limitations are of three types:

1. technical limitations: there is a well defined yield maximum for each crop, given crop properties and climatic conditions. This tells us how much useful product can be produced when plants grow under optimal conditions.

2. demand limitations: now that population growth in the EC has almost come to a standstill, consumption will no longer rise.

3. limitations that stem from policy objectives: socio-economic aims in the field of nature conservation, recreation and the like.

2.6. Exercises

1. Why are explorative land use studies needed? Summarize the arguments and present them in the group.

2. On the basis of the information that you have received concerning the European study and the PSS study indicate in which ways they are similar and in which ways they are different.

3. Explain what the main differences are between explorative and predictive land use analysis studies.

3. Linear programming and multiple criteria analysis

3.1. Introduction

In the methodology for explorative land use studies operations research (mathematical programming) techniques are used, which help us to select the best option(s) from numerous alternatives. The main technique which is used is Interactive Multiple Goal Linear Programming, a Linear Programming technique. Although it is not necessary to know and understand all the ins and outs of this technique, a short introduction about the theory and principles is indispensable. If one wants to know more about operational research techniques, we advise them to study a handbook, e.g. Hendriks & Van Beek (1991; in Dutch), Hillier & Lieberman (1989; in English) and Romero & Rehman (1989; in English about multiple criteria analysis).

Before dealing with multiple criteria optimization techniques and with Interactive Multiple Goal Linear Programming in particular (3.3), we will deal with Linear Programming (3.2). The only essential difference between the two techniques is that in LP there is only one objective (e.g. financial return or numbers of hectares), that is optimized, whereas in IMGLP there are several objectives to be optimized. In this course, the theory and principles of Linear Programming will be treated by means of an example (3.2.1). The problem in this example will be solved along `three ways':
* graphically;
* algebraically;
* computer software (LP88/XPRESS).

The final part of this chapter introduces the LP88 and XPRESS software, which is used in this course (3.4).

3.2. Linear programming

3.2.1. An example

A farmer in Spain with no possibility to irrigate, wants to achieve a maximum harvestable dry matter production from two crops: wheat and potatoes. One hectare of wheat yields 2 tons dry matter (coefficient in the objective function: 2) and one hectare of potatoes yields 5 tons (coefficient in the objective function: 5).

The farmer faces three constraints:
1. he has only 6 hectares of arable land;
2. the farmer is not allowed to grow more than 4 ha wheat, because of the MacSharry rules;
3. on a certain plot, potatoes may not be grown more than once every two years.

Of course the number of hectares with wheat and potatoes are non-negative. The mathematical formulation of this problem is given in Scheme 1.

Scheme 1 An example of an LP-problem.

3.2.2. Decision variables or activities, constraints and objective function

An LP-problem consists of:
* decision variables or activities (x₁: the number of ha with wheat and x₂: the number of ha with potatoes) in the system;
* the (resource) constraints (Relation 2-5): they determine the feasible combinations of the decision variables or activities;
* objective function (Equation 1): this function describes the aim of optimization and measures how `good' a certain combination of decision variables is.

Scheme 2 The general algebraic form of an LP-problem, the canonical form.

3.2.3. Formulation

Scheme 2 shows the general algebraic form of an LP-problem. This is the canonical form of the LP-problem. The matrix A is the matrix of coefficients of the LP-problem. This matrix has the dimension: m (rows constraints) x n (columns activities).

Example: Our LP-problem can be written with matrix A (dimension 3 x 2) and vector b :

Scheme 3 The standard form of an LP-problem.

3.2.4. Transformations

Besides the canonical form of the LP-problem, there is also the standard form (Scheme 3).

In this formulation the inequalities A x < b are replaced by the equations Ax=b, by introducing so called `slack variables':

The inequality [Sigma] a_ijx_j b_i is equivalent to the equality [Sigma] a_ijx_j + y_i = b_i (y_i > 0), in which y_i is a slack variable.

Example:

Our example can be reformulated in the standard form by adding the slack variables y₁, y₂ and y₃:

max {w = 2x₁ + 5x₂ + 0y₁ + 0y₂ + 0y₃}

subject to

x₁+x₂+y₁=6
x₁+y₂=4
x₂+y₃=3
x₁, x₂, y₁, y₂ and y ₃ > 0

A number of transformations exists by means of which any LP-problem can be reformulated in its equivalent standard or canonical form:
1) max {c'x} is equivalent to max {c'x + k} and vice versa;
2) max {c'x} is equivalent to max {kc'x } (k>0) and vice versa;
3) max {c'x} is equivalent to min {-c'x } and vice versa;
4) a decision variable x_j (x_j < 0) can be replaced by x_j^* = - x_j (x_j^*>0);
5) a variable x_j which can either be negative or positive can be replaced by x_j = x_j⁺ - x_j^-, where x_j⁺ = max {x_j, 0} and x_j^- = max {-x_j , 0}, so x_j⁺ > 0 and x_j^- > 0;
6) the inequality [Sigma] a_ijx_j < b _i can be replaced by: [Sigma] a_ijx_j + y_i = b_i, where y_i > 0;
7) the inequality [Sigma] a_ijx_j > b _i can be replaced by: [Sigma] a_ijx_j - y_i = b_i, where y_i > 0.

Examples:
1) max {2x₁ + 5x₂} is equivalent to max {2x ₁ + 5x₂ + 100};
2) max {2x₁ + 5x₂} is equivalent to max {2000x₁ + 5000x₂};
3) max {2x₁ + 5x₂} is equivalent to min {-2x ₁ - 5x₂};
4) {x₁ - x₂} (x₂<0) is equivalent to {x₁ + x₂^*} (x₂^* = -x₂; x₂^*>0);
6) the inequality x₁ 4 can be replaced by x₁ + y₁ = 4, where y₁ > 0;
7) the inequality x₂ > 8 can be replaced by x ₂ - y₂ = 8, where y₂ > 0.

3.2.5. Assumptions and restrictions

There are some assumptions/restrictions underlying linear programming:
1) the objective function and the constraints must be linear in the decision variables; non-linear functions could be split up in smaller `linear' parts;
2) all parameters are assumed to have fixed and known values;
3) the variables are assumed to be real (continuous); it is possible to include integer variables (mixed integer programming), but this requires special solving methods (e.g. the Branch-and-Bound method), which can dramatically increase the computing time.

3.2.6. Solutions

Linear programming problems can be solved graphically if the problem comprises only two decision variables. When the problem comprises more than two decision variables it can only be solved algebraically, e.g. by means of the so called simplex-algorithm. The graphical method is illustrated for our example in Figure 3.1

Figure 3.1 Graphic representation of a linear programming model. The solution space (shaded) and iso-profit lines (dashed lines) Source: Rossing, 1989.

Graphically it can be understood that there are four classes of solutions of LP-problems (Figure 3.2), depending on the characteristics of the solution(s):
* unique solution (Figure 3.2a);
* alternative solutions (Figure 3.2b);
* unbounded solutions (Figure 3.2c);
* no feasible solution (Figure 3.2d).

For the classes of unique and alternative solutions it is clear that the optimal solution(s) lay(s) at the border of the feasible space (solutions that meet the constraints but which are not necessarily optimal), and an unique solution lays at an angular point of the feasible space.

Figure 3.2 Four classes of solutions of LP problems. Source: Hendriks & Van Beek (1991).

Figure 3.2.a(a): Unique solution

Figure 3.2.b (b): Alternative solutions

Figure 3.2.c (c): Unbounded solutions

Figure 3.2.d (d): No feasible solution

3.2.7. Simplex-algorithm

For problems with more than two decision variables the simplex-algorithm is the most important tool to solve linear programming problems. Computer software is indispensable to solve practical LP-problems, because they often comprise over thousand variables. The simplex-algorithm will be illustrated with our example, with just two decision variables.

The simplex tableau: The first step is to write the LP-problem in its standard form. In our example this can be done by adding the slack variables y₁, y₂ and y₃ to the constraints (2), (3) and (4) respectively (compare Schemes 1 and 4). Subsequently, the first tableau is made (Scheme 4).

The Right Hand Side (RHS) of all tableaus (the b-values) should be non-negative: if the first tableau comprises negative RHS-values, transformation(s) should be carried out. The first tableau (and all other tableaus) should comprise an unity basis, with the dimension of the number of rows. The variables `belonging to the unity vectors' are called the basic variables (in Tableau 1: y₁, y₂ and y₃). The bottom row of the tableau represents the objective function formulated in such a way that the objective value w is expressed in the non-basic variables (in other words: activities that increase the value of the objective function get a negative coefficient and activities that decrease the value of the objective function get a positive coefficient; basic variables get the zero coefficient in the objective function; the objective function in our example expressed in the non-basic variables: w-2x₁-5x₂=0).

A basic feasible solution of an LP-problem is a solution with non-negative values for all basic variables and zero values for the non-basic variables (in that way the basic feasible solution can be read from the RHS of the tableau). A basic feasible solution coincides with an angular point of the feasible space. In the iterations of the simplex-algorithm we move from one angular point to another and thus (according to one of the important propositions of linear programming) from one basic feasible solution to another, until the optimal basic feasible solution has been found (if such a solution exists). It can be proven that if an optimal feasible solution exists, an optimal basic feasible solution also exists. The simplex-algorithm tells us how to move from one angular point (basic feasible solution, tableau) to another and when to stop this procedure, because the optimum solution(s) has (have) been found. The simplex-algorithm will be explained very briefly and incompletely by means of our example.

Tableau 1 comprises the three unity vectors for the three slack variables. The two decision variables are the non-basic variables and thus zero in the basic feasible solution (angular point O in Figure 3.1). In the first iteration the best candidate for the entering basic variable is one of the current non-basic variables which increase the objective function at the fastest rate (i.e. the decision variable with the most negative coefficient in the bottom row of the tableau). The leaving basic variable is the basic variable which reaches zero first as the entering basic variable is increased. In our example, x₂ is the variable which contributes most per unit to the objective function. This variable can be increased until constraint 3 becomes limiting, i.e. x₂ = 3 (angular point A in Figure 3.1); y₃ leaves the basis. By means of a so called `pivot-operation' around the pivot (marked with an asterix in Tableau 1), Tableau 2 is generated. The value of the objective function has increased from zero to 15 (3*5).

Scheme 4 The simplex tableaus of our LP-example.

Mathematically, the pivot a_kp can be found in the column of the new basic variable x_p, such that b_k/a_kp =min {b_i/a_ip| for all a_ip>0}. In the pivot-operation the row comprising the pivot is divided by a _kp, so that the pivot gets the value 1. Subsequently, the row comprising the pivot is added to or subtracted from the other rows so that the pivot column (including the coefficient of the bottom row) represents an unity vector. In summary:

Step 1. Selection of pivot column: column with most negative value in bottom row;
Step 2. Selection of pivot row: row with smallest ratio between b-value and positive element in the pivot column;
Step 3. Derive the new pivot row: dividing each element in the pivot row by the pivot;
Step 4. Derive other rows: substracting (or adding) a suitable multiple of the pivot row from (or to) each of the other rows, such that all other elements in the pivot column equal zero.

The next step is to check whether the solution which has been found after the pivot operation is optimal or not: in other words check whether the objective function can be increased by increasing any non-basic-variable. This can be done by checking whether there are any non-basic variables with a negative coefficient in the bottom row of Tableau 2. Again, the variable with the most negative coefficient is selected and again the leaving basic variable is the basic variable which reaches zero first as the entering basic variable is increased. In our example, x₁ is the variable which now contributes most to the objective function. This variable can be increased until constraint 1 becomes limiting, i.e. x₁ = 3 (angular point B in Figure 3.1) and y₁ leaves the basis. By means of a pivot-operation around the pivot (marked with an asterix in Tableau 2) Tableau 3 is generated. The value of the objective function has become 21. Now no negative values can be found in the bottom row of Tableau 3 and the objective function cannot be further increased; the solution is optimal.

3.2.8. Post-optimal analysis or sensitivity analysis

Solving an LP-program provides more information about an optimal solution than just the value of the objective function, the levels of the decision variables and the slack or surplus in the constraints. An integral part of solving LP models is the sensitivity analysis. It is concerned with studying possible changes in the optimal solution as a result of making certain changes in the original model.

The matrix notation of an LP-problem reads:

Max {w = cx}

subject to:

Ax < b
x > 0

The coefficients a (in A), b and c of this problem are often subject to variability or uncertainty. It is important to know how the optimum solution of the LP-problem changes when coefficients change. The following analyses give information on these changes: the shadow price and right hand sides ranging for changes in the b-values, the coefficients of the objective function ranging and the reduced cost for changes in the c-values. Unfortunately, all these analyses refer only to partial changes in coefficients: a change in one coefficient simultaneously.

Shadow prices

In the optimum tableau of an LP-problem the coefficients of the bottom row in the columns of the slack variables, the so called `shadow prices', tell us the increase (or decrease in case of a negative shadow price) in the value of the objective function when the constraint is relieved with one unit. In our example: only constraints 1 and 3 have a shadow price: constraint 2 is not binding. If the right hand side of constraint 1 would be increased with one unit, the objective function would increase with 2 (the optimum solution would be: one extra unit x₁: the objective function increases with 2). If the right hand side of constraint 3 would be relieved with one unit, the objective function would increase with 3 (the optimum solution would be: one extra unit x₂ at the expense of one unit x₁: the objective function increases with 5-2=3).

Right hand sides ranging

In order to keep the same optimal basic feasible solution, the allowed changes in the right hand sides of the constraints can be deduced from the final tableau. We will only demonstrate graphically how to determine the allowed changes in the right hand sides without a change in the optimal basis (Figure 3.3):

Constraint 1: if the right hand side would be greater than 7, constraint 2 in stead of constraint 1 becomes limiting and y₁ enters the basis at the expense of y₂; if the right hand side becomes smaller than 3, y₃ enters the basis at the expense of x₁.

Constraint 2: if the right hand side decreases with more than one unit, this constraint becomes limiting and y₂ leaves the basis for y₁.

Constraint 3: for a right hand side between 2 and 6 the optimal basis does not change; beyond this range either y₂ leaves the basis for y₁, or x₁ leaves the basis for y₃ .

Coefficients of the objective function ranging

The optimal tableau also gives information about the sensitivity of the optimal solution for changes in values of the coefficients in the objective function. We will not deal with how to calculate this from the final tableau. It can be deduced from Figure 3.3 that B stays the optimal solution for objective coefficient values for x₁ between zero and 5 and for x₂ between 2 and infinity. Of course, the value of the objective function changes.

Reduced cost

Zero variables (variables that are not in the optimum solution) have so called `reduced cost'. This reduced cost indicates the amount by which the objective coefficient of the zero variable must be changed before the particular variable would have a positive value in the optimum solution. Thus, reduced cost gives the minimum change in the c-coefficient of a particular zero variable necessary to make that zero variable more attractive than a current non-zero variable. In our example both variables (x₁ and x₂) are non-zero variables (selected in the optimum solution), thus both have no reduced costs.

Figure 3.3 Sensitivity analysis of the optimal solution.

3.2.9. Example of a linear programming problem: Communal farming in Tagota Village in Mara Region, Tanzania

This example is taken from the book `Doing Mathematics in a Developing Country' published by Tanzania Publishing House, p. 63-67 (see Schweigman (1979)). The example was worked out by the former student G.C.N. Sibuti, who comes from Tagota village in Tanzania. The data in this example are based on his own experience and on interviewing in the village.

Tagota is an Ujamaa Village in Mara region, Tanzania, where a hundred families live. They grow maize as a major crop. During the growing season for maize each family uses all labour on the cultivation of maize. The people use ox-ploughs but they are discussing whether to hire a tractor. Moreover it is suggested that herbicides are applied to avoid weeding. The aim of this study is to investigate whether or not a tractor should be hired and herbicides should be applied.

Instead of considering the entire population, we will study here the farming of an average family in the village, which consists of 5 people. It may be assumed that on the average each family has an ox-plough. To study a family instead of a whole village is acceptable, because all families grow maize in the same way.

To analyze this problem we start immediately with the introduction of variables and the formulation of constraints. By setting up the formulation we will find out what type of information and statistical data we need.

The statement that we have to determine whether a tractor should be hired or not and whether herbicides should be used, is not very quantitative. In fact we want to know how many acres have to be ploughed by a tractor and for how many acres herbicides are to be bought, if it is to be useful at all. Realizing this, the choice of variables for this example will not be surprising. We introduce:

x₁ the acreage where maize is grown, a tractor is used for ploughing and weeding is done manually;
x₂ the acreage where maize is grown, a tractor is used for ploughing, and herbicide is used;
x₃ the acreage where maize is grown, an ox-plough is used for ploughing and weeding is done manually;
x₄ the acreage where maize is grown, an ox-plough is used for ploughing and herbicide is used.

Note that the total acreage used for maize is x₁+x₂+x₃+x₄.

There is enough land, so the `land constraint' need not be included. To formulate the labour constraint, first the timing of ploughing, planting, weeding and harvesting will be considered. The timing is illustrated in the table below.

Table 3.1 Timing of agricultural activities for the cultivation of maize in Tagota village, Tanzania.

The following data on labour input have been learned from the farmers:

Ploughing and planting in October to December:

a tractor needs 1 hour to plant an acre and also 15 man-hours of family labour are needed for clearing, (1)
preparation and planting. Note: the ploughing is done twice,
the first time for clearing, the second time for planting (2)
An ox plough needs three days of 4 hours to plough an (3)
acre and 60 man-hours of family labour are needed. (4)
Note: the ploughing is done twice. (5)

Weeding in January and February:

if no herbicide is used the weeding takes 15 man-days of 9 hours =
135 man-hours per acre; (6)
if herbicide is used weeding is not necessary, but it takes 5 hours
to spray one acre during the planting period. (7)

harvesting is done manually and it takes 30 man-hours to harvest
an acre of maize. (8)

2(x₁ + x₂) < 75*8 (9)

where it is assumed that one tractor can be hired throughout the whole period, each working day for 8 hours. Use has been made of (1) and (2).

The constraint for the use of an ox-plough during the months October to December is given by:

24(x₃+x₄) < 75*4 (10)

where use has been made of (3) and (5). The oxen are only prepared to pull for 4 hours a day.

The human labour constraint during the planting, weeding and harvesting periods are given by:

30(x₁+x₂) + 120(x₃+x₄) + 5(x₂+x₄) < 75*5*9 (11)

for the period of planting from October to December. Use has been made of (1), (2), (4), (5) and (7). The people work 9 hours a day.

135(x₁+x₃) < 50*5*9 (12)

for weeding in the period January and February. Use has been made of (6).

30(x₁+x₂+x₃+x₄) < 50*5*9 (13)

for the harvesting period in April and May. Use has been made of (8).

The yield of maize per acre is 900 kg, independent of the use of a tractor or herbicides. The annual consumption of maize by the family is estimated as 720 kg. The food requirement constraint may be written as:

900(x₁+x₂+x₃+x₄) > 720. (14)

Several choices of the objective function are possible, for instance:
1) Maximization of total production:
max: 900(x₁+x₂+x₃+x₄)

2) Maximization of the net revenue of selling the surplus production:
max: 0.80*(900(x₁+x₂+x₃+x₄ )-720)-100(x ₁+x₂)-100(x₂+x₄), (15)

where use is made of the fact that the selling price of 1 kg of maize is 0.80 Shilling, the cost of hiring a tractor to plough one acre is 50 Shilling and the cost of the herbicide is 100 Shilling per acre. The costs of using the ox-plough are small in comparison with the tractor, hence these costs are omitted.

We consider objective function (15). The linear programming problem (9)-(15), where the variables have also to be non-negative, may be written as:

maximize: 620x₁+520x₂+720x₃+620x₄ -576,

where the variables x₁, x₂, x₃ and x₄ are subject to:

2x₁ + 2x₂ < 600
24x₃ + 24 x₄ < 300
30x₁ + 35x₂ +120x₃ +125x₄ < 3375
35x₁ +135x₃ < 2250 (16)
30x₁ + 30x₂ + 30x₃ + 30x₄ < 2250
900x₁ +900x₂ +900x₃ +900x₄ > 720
x_j 0 j = 1,2,3,4

The solution is given by:

x₁ = 7.4, x₂ = 58.3, x₃ = 9.3, x ₄ = 0 acres.

It follows that it seems to be beneficial to hire a tractor, but the ox-plough is to be used as well. Some herbicide should be bought but on a certain plot of (7.4 + 9.3) acres the weeding is done manually. During the planting period the ox-plough is not used the whole time, 77.8 hours are idle. All the available manual labour is fully used in the planting period, the weeding period and the harvesting time. The production of maize is much more than the consumption. The expected revenue is 41,016 Shilling. Instead of relying on these dry figures it is worth investigating some alternatives. If no tractor is used what is the impact on the revenue obtained? Does this differ very much? The same should be investigated if no herbicide is bought. Will it make much difference, if we maximize the yield instead of the net revenue? What is the income per head?

3.2.10. Software for solving LP-problems

Many computer software packages are available to solve LP-problems. The packages differ in the dimension of the problems they can solve, the required hardware and the users interface. The dimension of the problem that can be solved depends on the number of restrictions, the number of decision variables, the number of integer variables and the number of non-zero coefficients in the matrix, the so-called `density of the matrix'.

An LP software package mostly consists of several parts (Figure 3.4):
* MG-Generator (MGG). An MG-Generator is a higher programming language specifically suitable for linear programming. With an MG-Generator the LP-model can be `mathematically' described in a way closely related to the mathematical formulation which can easily be converted to computer code.
* Matrix Generator (MG). The Matrix Generator transforms data from a database to a simplex tableau by means of the mathematical formulation of the LP-model made with the MG-Generator. Checking, changing or scaling the data would be almost impossible without an MG. The result of the Matrix Generator is an MPS-file (Mathematical Programming System Format-file). This MPS-file is a standard file in which the matrix is unequivocally and compactly described. In the file a number of sections can be distinguished (see example):
* Name: the name of the problem;
* Rows: type and name of the rows (including the objective function);
* Columns: per column the name and the non-zero coefficients are given. The name or number of the row gives the right position of the coefficients;
* RHS: the Right Hand Side or the b-vector;
* Endata: the end of the file.

Figure 3.4 LP software. Source: Hendriks & Van Beek (1991).

In most recent software packages the function of the MG (linking the data to a model) is carried out by the MG-Generator (MGG) routine.

* The Solver or simplexroutine. The Solver comprises the simplex-algorithm and usually also post-optimal analysis and mixed integer programming algorithms.
* The Report-Writer (RW). The Report Writer is a program that processes the output (e.g. the report-file) of the Solver, so that the user can easily read and interpret the results.

Example of MPS-file (our LP-problem: growing wheat and potatoes):

NAME	EXAMPLE GROWING WHEAT AND POTATOES
ROWS
N 1	(objective function)
L 2
L 3	(constraints)
L 4
COLUMNS
X1 1	2.0000
X1 2	1.0000
X1 3	1.0000	(variable, number of row, c or a coefficient)
X2 1	5.0000
X2 2	1.0000
X2 4	1.0000
RHS
RHS 2	6.0000
RHS 3	4.0000	(Right Hand Sides or b-values, number of row, b coefficient)
RHS 4	3.0000
ENDATA

Table 3.2 shows several examples of LP-software packages. For this course use will be made of LP88 and XPRESS

Table 3.2 Examples of LP-software. Source: Hendriks & Van Beek (1991)

3.3. Multiple criteria optimization problems

Before dealing with the most important multiple criteria optimization techniques (3.3.4), the conceptual differences between attributes, objectives, goals and constraints are discussed (3.3.1 ), the idea of efficient or a Pareto optimal solution is introduced (3.3.2) and the concept of trade offs is presented (3.3.3).

The approach and text of sections 3.3.1-3.3.4 originates from the handbook of Romero & Rehman (1989) on Multiple Criteria Analysis for agricultural decisions.

3.3.1. Attributes, objectives, targets and goals

The stake holder can establish his preference according to various attributes , e.g. the value added or the level of employment (by a set of activities). Attributes can be measured independently from a stake holder's desires and in many cases can be expressed as a mathematical function of the decision variables. Objectives imply the maximization or minimization of (the functions representing) one or several attributes and reflect the values of the stake holder, e.g. maximizing the value added or minimizing unemployment. A target is an acceptable level of an attribute. A goal is an attribute with a certain target, e.g. the stake holder wants a value added of at least $100.000,=. In general goals take the form f(x) >/< t or f(x) = t, where t is a parameter representing the aspiration level or target value. Summarized, according to Romero & Rehman (1989), `in a farm planning problem gross margin is an attribute; to maximize gross margin, an objective; and, to achieve a gross margin of at least a certain target, a goal. Finally, a criterion is a general term comprising the three preceding concepts. That is, criteria are the attributes, objectives or goals to be considered relevant for a certain optimization problem.'

Goals and constraints have the same mathematical structure and look exactly the same as both of them are inequalities/equations. A difference between them may lay in the meaning attached to the RHS of the (in)equality: with goals the RHS is a target aspired by the stake holder, which may be achieved or not; with constraints, the RHS must be satisfied. Thus goals could be considered as soft constraints which can be violated without producing infeasible solutions. The amount of violation can be measured by introducing positive and negative deviational variables or slack variables (see linear programming). For example, the goal referring to the achievement of a value added of $100,000,= by the activities x₁ and x₂ adding $1000,= and $5000,= per unit, respectively, can be represented as follows:

1000x₁ + 5000x₂ + n - p = 100,000

The variables n and p account for deviations from the achievement of a goal from its target. For example, if the actual value added is only $75,000, then n = 25,000; if the value added is $125,000, then p = 25,000 (thus, either n or p is non-zero). Thus, a goal can be expressed as follows:

ATTRIBUTE + DEVIATIONAL VARIABLES = TARGET

f(x) + n - p = t

3.3.2. Pareto optimal or efficient solutions

Example:
A problem with the following three feasible solutions, whose performance according to three attributes is as follows:

	Gross margin	Seasonal labour	Emission
	(dollars)	(hours)	(kg N/ha)
Solution 1	80,000	500	100
Solution 2	80,000	600	100
Solution 3	90,000	700	120

The stake holder wants to maximize gross margin and to minimize seasonal labour and emission. It is clear, that the second solution is non-efficient, since it offers the same gross margin and emission, but requires more seasonal labour. The first and third solution are Pareto optimal.

According to Romero & Rehman (1989): `All the Multiple criteria optimization techniques aim to obtain solutions which are efficient in the Paretian sense. Even within the multi-objective programming approach the first step to be taken consists in obtaining the set of feasible solutions which are efficient. That is, the feasible set is partitioned into two disjoint subsets. The subset of feasible and non-efficient solutions and the subset of feasible and efficient solutions. After that the preferences of the optimizer are introduced to establish a compromise within the feasible and efficient subset.'

3.3.3. Trade-offs amongst criteria

The trade-off between two criteria (objectives) means the amount of achievement of one criterion that must be sacrificed to gain an unitary increase in the other one, e.g. the trade-off value between gross margin and seasonal labour for solution 1 and 3 in the preceding example was:

T13 = (90,000-80,000) / 700 - 500 = 50

This trade-off indicates that each hour of decrease of seasonal labour implies a mean decrease of $50,= of gross margin. The trade-offs values, besides being a good index for measuring the opportunity cost of one criterion in terms of another criterion, also play a key role in the analysis of interactive techniques.

At first the concept of trade-offs may seem rather similar to that of shadow prices. However, there is a clear difference. The trade-off between two objectives (or an objective and a constraint) is defined as the change in an objective function for a particular change of another objective/constraint. It is usually calculated by comparing results of two or more optimizations. The linearity of the trade-off within the ranges of the optimum values of the objective functions (in the preceding example: between 90,000-80,000 and 700-500) is not considered. Shadow prices are defined only for an unitary change in the right hand side of a constraint; the shadow prices are valid only for that unitary change. They can be calculated for each optimization.

3.3.4. Multiple criteria decision making techniques

The former definitions and concepts allow us to give a rough classification of the main multiple criteria optimization approaches. It is not the aim to understand all ins and outs of all techniques, but to present an overview so one will be able to recognize different techniques when reading literature. The examples have been taken from Romero & Rehman (1989).

Goal programming (GP)

Specified goals are available for each of the objectives. The deviations from the desired targets and what is actually achieved are minimized by the addition of positive and negative deviational variables permitting either the under- or over-achievement of each goal. The minimisation process can be undertaken in different ways. The most widely used in practice are:

a) Lexicographic Goal Programming (LGP) or absolute or pre-emptive goal programming (Dutch: doelprogrammering met absolute prioriteiten): absolute or pre-emptive weights to the deviational variables. There is a clear order in the priorities for the different objectives. First, it is tried to approach the goal of the most important objective. Subsequently, the less important objectives are pursued under the restriction that the approach to the goal of an objective with a lower priority does not enlarge the gap with the goal of an objective with a higher priority. See example LGP.

b) Weighed Goal Programming (WGP) or relative or non-pre-emptive goal programming (Dutch: doelprogrammering met relatieve prioriteiten): relative or non pre-emptive weights to the deviational variables. The different objectives are more or less comparable. The deviations from the different goals of the objectives are weighed such that this represents the relative importance of the different objectives. See example WGP.

Examples LGP and WGP

Given the LP-problem:

Max z = f(x1,x2) = 6250x1 + 5000x2

Subject to:

550x₁ + 400x₂ < 15,000 (c ₂ )
750x₁ + 575x₂ < 22,000 (c ₃ )
1050x₁ + 825x₂ < 29,000 (c ₄ )
1375x₁ + 1025x₂ < 36,000 (c ₅ )
120x₁ + 180x₂ < 4,000 (c ₆ )
400x₁ < 2,000 (c₇ )
450x₂ < 2,000 (c₈ )
35x₁ + 35x₂ < 1,000 (c ₉ )
and x₁, x₂ > 0

In this example x₁ and x₂ represent crop A and crop B. The objective function represents the net present value of investment of the two crops. c₂-c₉ represent constraints concerning working capital (c₂-c₅), casual labour (c₆-c₈) and mechanization (c₉).

As an example of a Goal Programming (GP) model, we assume that the above cited set of inequalities is treated as a set of goals instead of a set of constraints. For each goal, two non-negative variables (the deviational variables n and p) are introduced to convert an inequality into an equation. For the objective function a target value of 200,000 is assumed:

6,250x₁ + 5,000x₂ + n₁ - p ₁ = 200,000 (g₁)
550x₁ + 400x₂ + n₂ - p ₂ = 15,000 (g₂)
750x₁ + 575x₂ + n₃ - p ₃ = 22,000 (g₃)
1,050x₁ + 825x₂ + n₄ - p ₄ = 29,000 (g₄)
1,375x₁ + 1,025x₂ + n₅ - p ₅ = 36,000 (g₅)
120x₁ + 180x₂ + n₆ - p ₆ = 4,000 (g₆)
400x₁ + n₇ - p₇ = 2,000 (g₇)
450x₂ + n₈ - p₈ = 2,000 (g₈)
35x₁ + 35x₂ + n₉ - p ₉ = 1,000 (g₉)

As an example of Lexicographic Goal Programming (LGP), assume that the stake holder's first priority is made up of goals concerning working capital (g₂-g₅). These first goals must be satisfied in an absolute and pre-emptive way; the stake holder wants to minimize p ₂ + p₃ + p₄ + p₅. The second priority is made up of goal g₉ (mechanization): the stake holder wants to minimize p₉. The third priority is made up of goal g₁ (net present value): minimize n₁. The last priority is given to the minimisation of hired casual labour: minimize p₆ + p₇ + p ₈ . The stake holder is also allowed to attach weighing factors to the goals within the same priority. Without weighing factors, the whole LGP minimisation problem is:

Min a = [(p₂ + p₃ + p₄ + p₅), (p₉), (n₁), (p₆ + p₇ + p₈)]

Using one of the possible algorithms, the optimum solution is:

x₁ = 19.18 x₂ = 9.38

n₁ = 33,250 p₁ = 0
n₂ = 699 p₂ = 0
n₃ = 2,221 p₃ = 0
n₄ = 1,122 p₄ = 0
n₅ = n6 = 0 p₅ = p₆ = 0
n₇ = 0 p₇ = 5,672
n₈ = 0 p₈ = 2,221
n₉ = 0 p₉ = 0

This solution permits complete achievement of the goals of the first two priorities. The goal of the third priority was not reached: a negative deviation of 33,250. For the goals of the last priority only g ₆ was completely satisfied.

In Weighed Goal Programming (WGP) all the goals are considered simultaneously. Assume that the stake holder considers goals g₂-g₅ as rigid constraints. Thus, we have a WGP-problem with five goals (g ₁, g₆-g₉) and four constraints (g₂-g₅). It can be calculated that the maximum net present value with the four rigid constraints g₂-g ₅ is 175,600. The variables in the objective function must represent relative deviations from the targets rather than absolute deviations because of the widely different units of measurements used for the different goals. Thus, the model minimizes the sum of the relative deviations from targets:

Minimize: W₁ * (n₁/175,600) + W ₆ * (p₆/4,000) + W₇ * (p₇/2,000) +

W₈ * (p₈/ 2,000) + W₉ * (p₉/1,000)

subject to g₂-g₅, g₁ and g₆-g₉.

W₁, ..., W₉ represent the weights attached to the deviational variables. Mathematically this is an orthodox LP-problem and requires no extension of the Simplex algorithm.

Multi-Objective Programming (MOP)

The stake holder must take his decision in a multiple objective environment where defined goals do not necessarily exist. MOP attempts to distinguish the Pareto-optimal feasible solutions from the non-Pareto ones. The elements of this efficient set are feasible solutions such that there are no other feasible solutions that can achieve the same or better performance for all the objectives and they are strictly better for at least one objective. Within the efficient set, the trade-offs between criteria can be considered. Different techniques exist to generate or approximate the efficient set: a) graphically; b) constraint method; c) weighing method; d) multi-objective simplex method. Only the graphical method is illustrated.

Example MOP

As an illustration of Multiple-Objective Programming we now suppose that the stake holder has two objectives: a) maximize the net present value (g₁) and b) minimize the number of hours of casual labour hired for harvesting (g₇ plus g₈; since min(x) is equivalent to max(-x), min(400x₁ + 450x₂) is equivalent to max (-400x₁ - 450x₂)). For the purpose of illustration a constraint representing: `minimum crop area of 10 ha' has been added. We then have the following problem:

Eff Z(x) = [Z₁(x), Z₂(x)]
(Eff: efficient set, pareto optimal solutions)
where
Z₁(x) = 6,250x₁ + 5,000x₂
Z₂(x) = - 400x₁ - 450x₂
subject to
550x₁ + 400x₂ < 15,000
750x₁ + 575x₂ < 22,000
1,050x₁ + 825x₂ < 29,000
1,375x₁ + 1,025x₂ < 36,000
120x₁ + 180x₂ < 4,000
35x₁ + 35x₂ < 1,000
x₁ + x₂ > 10
x₁,x₂ > 0

Figure 3.5 Feasible set in the decision variable space. Source: Romero & Rehman, 1989.

Figure 3.6 Image of the feasible set in the objective space. Source: Romero & Rehman, 1989.

Table 3.3 Extreme points of the feasible set F

extreme points	decision variables		objective functions
	pear trees (ha) x1	peach trees (ha) x2	Z1 (NPV) $	Z2 (casual labour) hours
A	10	0	62,500	4,000
B	26.18	0	163,625	10,472
C	19.18	9.38	166,775	11,893
D	0	22.22	111,111	10,000
E	0	10	50,000	4,500

Source: Romero & Rehman, 1989

Since the first three constraints are implied by the fourth, we can omit these constraints. This problem can be solved graphically. The feasible set F can be presented by the polygon ABCDE in Figure 3.5 and the five extreme points of this region along with the values for both objectives are shown in Table 2. The five extreme points of the example in the objective space are plotted in Figure 3.6. From this figure it can be easily deduced that the segments connecting A', B' and C' represent the efficient set in the objective space for the problem analyzed and the segments connecting A, B and C in Figure 3.5 represent the efficient set in the decision variable space. The points of F' which do not lie on A'B'C' are inferior or nonefficient because they offer less net present value and equal (or more) casual labour for harvesting, or equal (or less) net present value and more casual labour for harvesting than any point belonging to the boundary itself. The slopes of the two segments A'B' and B'C' represent the trade-offs (or opportunity costs) between the attributes being considered.

Compromise Programming (CP)

According to Romero & Rehman (1989) compromise programming can to some extent be regarded as a natural and logical complement to MOP. It determines the optimal solution from the Pareto subset. CP starts with the identification of an ideal or utopian solution and it assumes that any stake holder seeks a solution as close as possible to the ideal point. To achieve this closeness a distance function is introduced into the analysis:

in which W_j represent the weights for each of the objectives, and d_j the relative degree of closeness between the j ^th objective and its ideal. If all W_j equal one and p=2, L₂(W) equals the pythagorean concept of distance (e.g. the distance between (2,6) and (5,2): L₂(W) = [(2-5)² + (6-2)²]^1/2 = 5). In CP the distance function is minimized.

Romero & Rehman (1989) discuss the relative advantages and disadvantages of GP, MOP and CP (page 99-103).

Example CP
In CP the stake holder seeks a solution as close as possible to the ideal point (which is the point with optimal values for each of the objectives), by means of a distance function. It can be deduced from Table 3.3 that the ideal solution is the solution with NPV (=net present value) = 166,775 and Casual labour = 4,000. In Table 3.4 the relative distances between each extreme efficient point (A', B' and C') and the ideal point have been calculated for the three measures of distance L₁, L ₂ and L_.

_{As an illustration the details of calculating the relative distance between point B'and its ideal, according to the L2 metric for W1 = 3 and W2 = 1, are given below:}

_{L2(3,1) = [3²((166,775 - 163,625)/(166,775 - 62,500))² + 1²((4,000 - 10,472)/(4,000 - 11,893))²]^1/2 = 0.825}

Table 3.4 Compromise programming (discrete approximation)

		A'	B'	C'	Z*j	Z*j
NPV (Z1)		62,500	163,625	166,775	166,775	62,500
casual labour (Z2)		4,000	10,472	11,893	4,000	11,893
d1		1	0.030	0
d2		0	0.820	1
L1	W1 = 1	1	0.850	1
L2	W2 = 1	1	0.820	1
L[yen]		1	0.820	1
L2	W1 = 2	2	0.860	1
L2	W2 = 2	2	0.860	1
L[yen]		2	0.820	1
L1	W1 = 3	3	0.910	1
L2	W2 = 1	3	0.825	1
L[yen]		3	0.820	1

Source: Romero & Rehman (1989).

Table 3.4 shows that given the structure of weights W_j of the three extreme efficient points, B' is the nearest to the ideal point, whatever measure of distance is used. In other words, the point B' in the objective space or the point B in the decision variable space is the best compromise solution, according to this `Discrete approximation of the Best-compromise solution' (we will not deal with the continuous CP-technique).

Interactive Multiple Criteria Decision Making approaches

This approach implies a progressive definition of the stake holder's preferences through an interaction between him and the model. The interaction becomes a dialogue in which the model responds to an initial set of the stake holder's preferences or trade-offs, and then when this response has been examined another set is offered and so on. Thus the process proceeds in an interactive and iterative way until the stake holder has found a satisfactory solution. Most of the interactive methods can be classified according to the kind of information that is required iteratively from the stake holder during the interactive process. The kind of information can be summarized in three types of questions to the stake holder:

1) What is your trade-off between the objectives (e.g. between costs of production and nitrogen emission)?

2) Do you accept an increase in costs of $1000,=, along with a decrease in nitrogen emission of 50 kg/ha?

3) Do you accept an option with $1000,= costs, 700 hours of casual labour and 120 kg N/ha emission?

The first two questions require information (directly or indirectly) about the stake holder's trade-offs between various objectives. This type of questions, especially the first, may be hard to answer. In the third question the stake holder is asked whether he accepts a given feasible efficient solution. If the stake holder does not accept the solution, he should indicate which objectives should be improved. The method of Zionts and Wallenius is one of the most popular multiple criteria optimization approaches with the second type of questions to the stake holder. The STEM method and the Interactive Multiple Goal Linear Programming (IMGLP) method are methods with the third type of questions to the stake holder. The IMGLP approach has been used several times in land use problems.

Often, when exploring options for land use the goals for the different objectives are not very clear. Moreover, it is not clear which objective should have the highest priority. Both the goals of the objectives and the priority of the objectives depend on the policy view of the stake holder. The policy view differs between various stake holders. The IMGLP technique is a suitable tool for these kinds of multicriteria problems. De Wit et al. (1988), discussing the IMGLP technique, state: `...satisfactory solutions, from the point of view of the `user', may be obtained in subsequent iteration cycles by tightening one of the goal restrictions and repeating the iteration cycle for the other objectives. The choice of the goal restrictions and the degree to which they are tightened, reflect the specific interests of the user. This stepwise maximization of the objectives under increasingly tighter restrictions on the other goals reduces the solution space. In that way, the costs of satisfying one objective in terms of what must be sacrificed on the other objectives is expressed. At last the user is faced with a solution space in which he cannot improve on any of his objectives without sacrificing on another one, and where he has to make a choice. Hence, the user becomes aware of the possibilities of exchange between the various objectives in his own solution space, i.e. he obtains the opportunity cost of one objectives in terms of the other objectives. Of course, users with different interests and aspirations are found to end up in different corners of the initial solution space .'

The IMGLP procedure consists of a number of optimizations rounds, each round comprising several optimizations. In each optimization run, the model is optimized for one objective function, while the other objectives are used as constraints. Upper or lower bounds can be put on these `goal constraints'. In the so called `zero round' of the procedure, in subsequent runs, the model is optimized for each of the objective functions, without putting any upper or lower bounds on the goal constraints. In this zero round the feasible space (`playing field') for the objective functions is determined. These extreme values of the objective functions are important in choosing upper or lower bounds for goal constraints in scenarios. In the zero round the compatibility of objective functions is not examined. In subsequent rounds upper or lower bounds are put on relevant (from the stake holder's point of view) objective functions, and the model is optimized for an, also relevant, objective function. In this way the compatibility of and the trade offs between objective functions is investigated.

The IMGLP method is illustrated with a simple example using only two objective functions, agricultural area and the use of pesticides for agriculture in a region (after Spharim et al., 1992). The results for the zero round are given in Figure 3.7. The minimum agricultural area, without any upper limit on pesticides is 32 million ha (coinciding with an use of pesticides of 87 million kg - point A), and the minimum use of pesticides without any bound on area is 33 million kg a.i. (coinciding with an agricultural area of 43 million ha - point B). If points A and B coincide, both objectives are completely tied, so that realization of one brings with it the realization of the other. There is no conflict between both objectives.

Figure 3.7 Graphical illustration of IMGLP procedure with two objective functions (i) minimization of the use of pesticides and (ii) minimization of the agricultural area. Source: Spharim et al. (1992).

Point W1 (87, 43) represents use of pesticides when agricultural area is minimized and agricultural area when the use of pesticides is minimized. The stake holder does not have to accept higher (worse) values for these objectives. The point U1 (33, 32) combines the lowest pesticide use with the lowest agricultural area, and may be considered as an utopian solution, because it is impossible to realize the optimum of two partially conflicting objectives simultaneously.

Given the utopian solution U1 and the worst combination W1, the stake holder is asked now which of the upper values of the objectives he wants to lower and to what extent. It should be pointed out to him that he does not commit himself because any value can be reconsidered. Let us assume that he first wants to ensure that the use of pesticides does not exceed 45 million kg. The most unfavourable combination of objective achievement is then W2 (45, 43) as in Figure 3.7b. To elucidate what minimum agricultural area can be achieved with this upper bound on the use of pesticides, a second iteration of the model is necessary. This yields point C (45, 38). The utopian alternative now moves to U2 (33, 38). This is the price that has to be paid for lowering the upper bound for the use of pesticides.

Now suppose that after all, the stake holder is satisfied with an agricultural area of 40 million ha. The minimum use of pesticides is then 37 million kg (Point D (37,40)). If the stake holder is satisfied with this solution the procedure stops, otherwise he may continue until he arrives at an appropriate solution.

The procedure is in principle the same if more than two objectives are considered, but then the number of optimizations needed to arrive at a satisfactory solution increases rapidly with the number of objectives. The results of a zero round with maximization objective functions can be tabulated as in Table 3.5 (after Veeneklaas, 1990).

Table 3.5 Results of zero round of an IMGLP model with N objective functions (Veeneklaas, 1990)

zero round	results of the optimizations for objective			worst value	best value
	1	i	N
objective 1	b1	.	.	W1	b1
. . .	.	.	.	.	.
objective i	.	bi	.	Wi	bi
. . .	.	.	.	.	.
objective N	.	.	bN	WN	bN

The diagonal elements b_i represent by definition the best attainable values of each row in the matrix of results. The worst values w _i correspond to the lowest value of row i in the matrix of results. For each goal i no lower value then w_i needs to be accepted. The initial freedom of choice for each objective - the difference between the worst and best value - is made explicit in this way.

The next step consists of selecting out of the w-vector the objective with the worst value considered most unacceptable and formulating a higher lower bound for that objective. Let us select objective i. The optimum found for goal i in the zero round of course forms the upper limit to which the right hand side of this objective can be raised. Suppose the desired lower bound for objective i is M_i, then a new cycle of optimizations can be performed with this lower bound for objective i. The results of that optimization round are given in Table 3.6.

Table 3.6 Results of first round of an IMGLP model with N (maximization) objective functions (Veeneklaas, 1990)

first round	lower bound	results of the optimizationsfor objective			worst value	best value
		1	i	N
objective 1	> W1	B1	.	.	W1	B1
. . .		.	.	.	.	.
objective i	> Mi	.	bi	.	Mi	bi
. . .		.	.	.	.	.
objective N	> WN	.	.	BN	WN	BN

Of course the optimum for objective i is still b_i, but the optimum values for the other objective functions are probably lower, because of the required lower bound for objective i. Comparison of the best values of the zero round (Table 3.5) with those of the first round (Table 3.6) can reveal possible conflicts between objectives. No change in optimum value for a particular objective implies the absence of conflict in this stage between this objective and the one for which a lower bound has been given (objective i). In early cycles this may be possible, but in later cycles this becomes more rare.

The optimization procedure consists of subsequent optimization rounds as described above. In each step the `costs' are revealed of safeguarding a minimum level for a particular objective in terms of maximum attainable levels for the other objective functions. This information helps in deciding which minimum requirement to tighten next, and to what extent. In this way the feasible combinations of goal values can be explored until only one combination is left. In general the procedure is stopped at an earlier stage, leaving an area within which all the objective functions have acceptable values: the `window of opportunities' or `space of solutions'.

Romero & Rehman (1989) summarize the main advantages and disadvantages of the interactive approach of multiple criteria optimization:

Advantages:
a) It represents a learning process for the stake holder permitting him to better understand the system being analyzed.
b) The information required involves only the local preferences of a stake holder, that is his attitude towards a certain solution or with respect to a certain set of trade-offs.
c) In general the assumptions underlying an interactive method are much less restrictive than those required to use a non-interactive technique.

Some difficulties of the interactive approach are:
a) The effort and involvement required from the stake holder in using the model is considerably more when compared with the non-interactive methods.
b) The assumption that the stake holder makes all his decisions consistently; particularly, when inconsistencies can be common.

3.4. Optimization software used in this course

3.4.1. LP 88

The LP88 optimization software is a simple computer programme that will be used to let the course participants become acquainted with using a PC to solve optimization problems. In this Section the manner in which LP88 can be used is explained. This will be done only very briefly and will be done with the aid of a simple example.

Suppose a farmer can cultivate two crops, maize and cassava. The gross margins per ha for the maize is $1,000 and for cassava $750. The labour requirement per ha for maize is 50 mandays and for cassava 75 mandays. The size of the farm is 10 ha and the farmer has available 350 mandays. The problem we will solve using the LP 88 software is: how much land should the farmer allocate to which crop in order to obtain the highest possible level of gross margins. Here step for step it will be shown how this problem can be solved with LP88.

* Once the LP88 software has been installed the programme can be started by typing LP88.

* If your PC is connected to a printer, type the printer where it says `Destination for listings (printer or file name)'. If your printer is not connected to a printer, just type PR.txt.

* Now go to `BEGIN' and press `Enter'.
* Press F1 (= setup).
* Press F2 (= New problem).
* Enter name for new problem. This could be for example `test1'.
* Press `enter'.
* The problem is a maximization problem so type `MAX'.
* There are two constaints, so type `2'.
* There are two non-slack variables, so type `2'.
* Type `F3' so that the matrix is displayed.
* It is advisable to give your activities and constraints recognisable names. For the activities these could be for example:
X₁ = Maize
X₂ = Cassav
For the constraints, for example:
Y1 = Labour
Y2 = Land

Once you have filled in your activity matrix it will look as follows:

Return	1000	1750
Land	1	1	<	10
Labour	50	75	<	350

* After pressing `F10' twice you enter `F2'
* If you now press enter the problem will be solved. Can you interpret the results?

3.4.2. XPRESS

The second package is introduced with the use of an example. Consider a farm with an active population of 15 and a certain availability of land. Three cases, A, B and C, will be examined with an availability of land of 4, 6, and 2 ha, respectively. The problem under consideration is how to divide total area between the two crops maize (M) and cotton (c) in order to maximize net revenue. Other essential data are the following:

	unit	maize	cotton
net revenue	F cfa ha-1	35,000	50,000
labor requirement (peak period)	active ha-1	3	5

The problem for case A can be translated into the following linear programming problem:

Max :	35,000 *	AreaM	+	50,000 *	AreaC			{objective function}
subject to:		AreaM	+		AreaC	<	4	{restriction land availability}
	3 *	AreaM	+	5 *	AreaC	<	15	{restriction labor availability}
		AreaM	>	0	AreaC	>	0	{non-negativity conditions}

The decision variables are AreaM and AreaC, the areas under maize and cotton, respectively. The technical coefficients used in this problem are the coefficients of the decision variables in the objective function (35,000 and 50,000) and in the restrictions (1, 1, 3, and 5) and the right hand sides (4 and 15), that represent in this case the availability of the resources (land and labor). For cases B and C one only has to change the right hand side of the land restriction from 4 to 6 and 2, respectively.

The solutions for the three cases A, B, and C can be summarized as follows:

	A	B	C
AreaM	2.5	5	0
AreaC	1.5	0	2
Land
Availability	4	6	2
Utilization	4	5	2
Shadow price	12,500	0	50,000
Labor
Availability	15	15	15
Utilization	15	15	10
Shadow price	7500	11667	0
Value of objective function	162,500	175,000	100,000

The solution indicates the values of the decision variables, information on the restrictions and the values of the objective function. In case B it is only the labor availability that limits the value of the objective function. This function is maximized when labor productivity is maximized, which explains the choice for growing only maize. Growing this crop yields a labor productivity of 35,000/3 (F cfa active^-1), for cotton this value is 50,000/5. The land is (relatively) abundant so that only 5 of the 6 ha are used. An increase of the land area would not have any effect on the optimal solution. That can also be seen from the shadow price of the land which indicates its marginal value (in the unit of the objective function per unit of the restriction, here in F cfa ha^-1), in other words, the value with which the objective function would increase if the area would increase by one unit (ha). For case B this value is zero for the land. For the labor one finds a marginal value of 11,667 F cfa active^-1 (in the peak period). An additional active would then be used to cultivate more maize, one third of an ha, which would add 35,000/3=11,667 to the net revenue.

In case C it is labor that is relatively abundant. Its marginal value is zero. In this situation one should maximize the monetary value of the activities per unit of land, which is done by growing only cotton (50,000 ha ^-1 against 35,000 ha^-1 for maize). Note that this could change the moment that there would be more land available. The marginal value of one ha is thus 50,000 F cfa ha^-1. This value could also be interpreted as the maximum price a decision maker would be willing to pay for one additional ha of land, according to the model.

Situation A is somewhat more complex, because both resources limit the maximum value of the objective function. Both resources have a positive marginal value, but these values are lower than in cases B and C, in which the other resource is abundant. That is the reason why the value of labor and that of land are maximized simultaneously. One finds therefore a solution in which the two crops are used.

Two economic principles are thus illustrated on the basis of the results for the three situations A, B, and C:
*The law of diminishing net returns of a production factor is illustrated if one considers the marginal value of land as a function of its availability. One starts at 2 ha (case C) and moves to cases A and B (4 and 6 ha, respectively). The marginal value of land decreases from 50,000 (c) to 12,500 (A) and then becomes zero (B).
*With the increase of the availability of a production factor (here : land), as long as it is a limiting production factor, the efficiency (or productivity) of the other production factors (here : labor) increase. The marginal value of labor increases from zero (c) to 7,500 (A), and then to 11,667 (B) as a consequence of the increase in land availability. It is this principle that forms an important argument for intensification and the use of external inputs.

The modelling language

In XPRESS-MP, as in many LP packages, first the model file is read by what is called `the modeller'. On the basis of the specification of the variables, the restrictions, the objective function and the technical coefficients this program transforms the model in its proper LP form. The second phase consists of calculation of the optimal solution, which is done by the `optimizer'. The solution is then written to a text file that can be imported by any text editor, and also by spreadsheets like, for example, MS-EXCEL.

In order to introduce the LP package XPRESS-MP, it is useful to come back to the problem introduced previously and to present the problem in a form that is readable for the package.

MODEL example	! Name of the model
VARIABLES
AreaM	! Area under maize [ha]
AreaC	! Area under cotton [ha]

CONSTRAINTS
obj :	35000 *	AreaM + 50000 *	AreaC $	! objective function
land :		AreaM +	AreaC < 4	! land restriction
labor:	3 *	AreaM + 5 *	AreaC < 15	! labor restriction

This model consists of the following sections:
MODEL in this section the model is given a name
VARIABLES this section declares the variables to be used
CONSTRAINTS specifies the restrictions, including the objective function. The model recognizes an objective function, if the expression after the colon is not an equation, or an inequality, but ends with a dollar sign ($). The non-negativity constraints that all variables have to satisfy, are included automatically and do not have to be specified. The order in which the constraints and the objective function are specified is not important.

Observe the format and the objective function. Each restriction has a name, followed by a colon, and then specification of the restriction in a way that is very similar to the mathematical specification. One of the differences is that one only uses the `<', and `>' signs in stead of the mathematical symbols `>' and `<', respectively. Also observe that the text following an exclamation sign (!) is comment and does not have any impact on the model as such. For human comprehension comments in the model text are nevertheless very useful.

For an LP problem with, say, two hundred in stead of two variables, their declaration becomes tedious, and the specification of the restrictions and of the objective function can become difficult (the lines may become very long, and errors may be difficult to trace). That is why facilities are created to make it easier to specify the model. These facilities are explained by specifying the same problem again in a different form.

MODEL example	! Name of the model
LET NAcul = 2	! Number of cultural activities
LET lab_av = 15	! labor availability [man]
LET land_av = 4	! land availability [ha]
VARIABLES
Acul (NAcul)	! Agricultural activities [ha]
TABLES
lab_cul(NAcul)	! Labor input agr. activities [man ha-1]
rev_cul(NAcul)	! Net revenue for agr. activities [F cfa ha-1 ]
DISKDATA
lab_cul	= c:\xpress\data\lab_cul.dat
rev_cul	= c:\xpress\data\lab_rev.dat
CONSTRAINTS
obj :	SUM(j=1:NAcul) rev_cul(j) * Acul(j) $	! objective function
land :	SUM(j=1:NAcul) Acul(j) < land_av	! land restriction
labor:	SUM(j=1:NAcul) lab_cul(j) * Acul(j) < lab_av	! labor restr.

Consider first the familiar parts in this model specification. The MODEL part has not changed. In the VARIABLES section one finds in stead of two names just one name (Acul). This one variable name however has a dimension that is given between brackets (NAcul=2). So this one line defines two variables: Acul1 and Acul2. These variables correspond to the variables AreaM and AreaC used in the former model specification. If one would have assigned the value 200 to NAcul, using the statement LET NAcul=200, the one line in the VARIABLES section would have defined 200 variables.

In the CONSTRAINTS section de names of the restrictions and the objective function have not changed. The expressions that follow the colon, however, have. For the case of the objective function, the expression indicates the sum, for j=1 to NAcul, of the products of the coefficients rev_cul(j) and the variables Acul(j). If the values o f rev_cul(j) are properly defined (35000 and 50000), the objective function corresponds exactly to the objective function in the previous model specification. Similarly, the restrictions for land and labor can be shown to be the same as in the previous model, if also lab_cul(j) has the proper values, and if the right hand sides (land_av and lab_av) have the proper values. It can be verified that the right hand sides are indeed equal to the ones used in the former model specification.

The tables lab_cul and rev_cul are declared in the TABLES section, where it is indicated what their dimensions are. In both cases the dimension is equal to that of the variable Acul. The dimension is defined by a LET statement. This type of statement can be inserted in any of the model sections. The values of the two tables cannot be found in the model specification, because they are stored in files, that are mentioned in the DISKDATA section. If these files contain the right coefficients, than the two model specifications represent the same LP problem.

In conclusion, the new sections used are the following:

TABLES declares names and dimensions of tables in which coefficients can be stored.

DISKDATA indicates the name of the file containing the data with which a table is to be filled.

It is not the right place to go into detail as far as the possibilities of XPRESS-MP concerns. In what follows a few important extensions are given, with examples. This may serve to give the reader an impression of the package.

* more than one dimension can be given
aa(5,10,2) !defines a 3 dimensional table or variable aa with in total 5*10*2 entries

* conditional statements (indicated by |). The example shows how in an equation the cultivated area is calculated for soiltype number one. (`| soil_cul(j)==1' means: for which soil_cul(j) equals 1)
car1 : arcul1 = SUM(j=1:NAcul | soil_cul(j)==1) Acul(j)

* with the use of indices, it is possible to generate a (large) number of restrictions in only one line. The first example shows how the cultivated area can be defined for all soil types in the model. The second example shows how one line generates Nsoil times Nnut restrictions.
car (s=1:Nsoil) : arcul(s) = SUM(j=1:NAcul | soil_cul(j)==s) Acul(j)
nut(s=1:Nsoil, n=1:Nnut) : any expression using the indices s and n

* it is possible to make calculations in order to change (part of) the coefficients. This is done in another model section: ASSIGN. The first example redefines the coefficients in the table rev_cul, with a multiplication by 1.2. The other example shows how to calculate 12 coefficients of a table (calctab) without using other tables. In all case the tables should have been declared in the TABLES section. ASSIGN

rev_cul(j=1:NAcul) = 1.2 * rev_cul(j) ! example 1

calctab(qqht=3:12) = exp(qqht^(-5.42))*12 + ln(qqht)
calctab(2) =5
calctab(1) = calctab(2)^4.23 ! ^ indicates `raised to the power'

* more than one objective function can be defined, for example:

obj1 : Netrev $
obj2 : Capcost + Labcost - SUM(ii=1:5) SALES(ii) $
obj3 : Meatprod $

3.5. Exercises

(Exercises 1-5 are taken from or based on Hendriks & Van Beek (1991))
Formulation of LP-problems

(1) Transform the following LP-problem to the standard form:

Min {20x1 - 10x2 + 30x3 + 1992}

-x₁ + 2x₂ + 3x₃ = 4
3x₁ - x₂ + 2x₃ < 9
2x₁ + x₂ - x₃ > 2
x₁ > 0, x₂ < 0, x ₃ = free

(2) Transform the following LP-problem to the standard form:

Min {w = 4x1 + 5x2 + 6x3} x₁ - 2x₂ + 3x₃ < 6
-2x₁ + 3x₂ - x₃ > 7
x₁ + x₂ + x₃ = -3
x₁ > 0, x₂ < 0, x ₃ = free.

(3) A farmer wants to draft an optimal plan for his farm for the months May until

October.

activities	labour requirement in hours ha-1						benefit $ ha-1
	May	June	July	Aug.	Sept.	Oct.
wheat	2			45			2500
potatoes	24	17	5		198		4000
sugar beats	158	98	11			120	4500
available labour (h)	350	300	250	250	400	400

He owns 60 ha.
Crop rotation constraints:

* wheat can be grown on max 1/2 of the total area,
* potatoes max 1/3 of the total area,
* sugar beets max 1/4 of the total area.

The farmer wants to maximize the benefit for the entire farm. Formulate this problem as an LP-problem.

Solving LP-problems

(4) Given the following LP-problem:

max {w = 3x₁ + x₂}

x₁ + x₂ < 5
x₁ + 2x₂ < 8
x₁ < 4
x₁ > 0, x₂ > 0

a. solve this problem graphically.
b. make the simplex tableau.
c. solve this problem with the simplex algorithm.
d. mark in the figure of a. the angular points corresponding to the basic solutions of c.
e. with how many units does the objective function increase, when
* constraint 1 is relieved with one unit;
* constraint 2 is relieved with one unit;
* constraint 3 is relieved with one unit.
f. carry out a sensitivity analysis for the right hand sides of the constraints.
g. carry out a sensitivity analysis for the coefficients of the objective function.

(5) Solve the following LP-problem with the simplex algorithm

Max {w = 6x1 - 15x2 + 8x3}

-x₁ + x₂ - x₃ < 1
x₁ - 2x₂ + x₃ < 7
* x₁ + 2x₂ + x₃ < 4

x₁, x₂, x₃ > 0

(6) Using the LP88 optimisation software

(Exercise developed by C.Schweigman (1993), pp. 55 - 57)

a) A one year farming plan has to be set up for a situation where a piece of land of size A available for farming consists of two parts with different fertility of soil. On both parts, the sizes which are A₁ and A₂, the same crops may be grown, but the yields differ. In order to make an agricultural plan a linear programming model is to be set up. As in Section 3.2.9, the variables x_j , _j =1,2,....,n are introduced; let q < n and x_j, j = 1,2,....,q refer to the crops grown on part A₁ and variables x_q+1, x_q+2,...., x_n, refer to the crops grown on part A₂. Then the following two constraints are applicable:

x₁+x₂+.....+x_q< A ₁
x_q+1+x_q+2+.....+x_n<A ₂

b) Discuss how the linear programming model of Section 3.2.9 for the planning of maize production in Tagota village, Tanzania, should be modified, if two pieces of land with different fertility are available for the production of maize. On one of them the yield of maize and all labour input data are the same as in Section 3.2.9, on the other part, however, the yield of maize is 20 % higher, but the human labour inputs in (11) and (12) are 30 %, and in (13) 20 % higher. The food requirement constraint may be left out of consideration. Show that the modified linear programming formulation is given by:

Maximise 620 x1+520 x2+720 x3+620 x4+764x5+664x6+864 x7+764x8-576

Where the variables are subjected to:

2x₁+2x₂ +2x₅ + 2x₆ < 600
24x₃ +24x₄ + 24 x₇+ 24x₈ < 300
30x₁ +35x₂+120x₃+125x₄ + 39 x₅+44x₆+156 x₇+161x₈ <3375
135x₁ +135x₃ +175.5x₅ +175.5x₇ <2250
30x₁ +30x₂+30x₃ + 30x₄ + 36 x₅+36x₆ + 36 x₇+36x₈ <2250
x_j > 0, j = 1,2...,8.

What is the precise meaning of the variables? Introducing the slack variables x₁,x₂,....,x₁₃ to replace the inequalities by equalities the constraints may be written as:

2x₁+2x₂ +2x₅ + 2x₆ +x₉ = 600
24x₃ +24x₄ + 24 x₇+ 24x₈ +x₁₀ = 300
30x₁ +35x₂+120x₃+125x₄ + 39 x₅+44x₆+156 x₇+161x₈ +x₁₁ =3375
135x₁ +135x₃ +175.5x₅ +175.5x₇ +x₁₂ =2250
30x₁ +30x₂+30x₃ + 30x₄ + 36 x₅+36x₆ + 36 x₇+36x₈ +x₁₃ =2250
x_j > 0, j = 1,2...,13.

The solution of this linear programming problem is given by:

x₃ = 7,9; x₅ = 6,7; x₆ = 49.2; x₉ = 488,2; x₁₀ = 110,3 acre.
x₁ = x₂ = x₄ = x₇ = x ₈ = x₁₁ = x₁₂ = x₁₃ = 0

Verify this by solving the linear programming problem on a computer or by making the following calculations. Express the basic variables x ₃, x₅, x₆, x₉ and x₁₀ in the non basic variables x₁, x₂, x₄, x₇ , x₈, x₁₁, x₁₂, x₁₃. Substitution of these variables into the objective function will lead to a linear function in the non-basic variables with all negative coefficients; Why does this imply that the solution has been found?

c) Verify in a similar way the solution of (16).

(7) Using the LP88 optimization software

Growing food crops or cash crops?

(Exercise developed by C.Schweigman (1985), pp. 66 - 69)

In a village in Mwanza region, Tanzania, six hundred adults participate in the farming of cotton as cash crop and maize and cassava as food crops. It will be investigated here whether it is preferable to buy maize or cassava on the local market in order to spend more time to the production of cotton. In the village cotton is planted in November or December, maize and cassava in November, December, January or February. The following variables are introduced:

x₁ the acreage for cotton planted in November and December in ha.
x₂ the acreage for maize planted in November and December in ha.
x₃ the acreage for maize planted in January and February in ha.
x₄ the acreage of cassava planted in November and December in ha.
x₅ the acreage of cassava planted in January and February in ha.

The labour inputs in the different months in mandays per ha for cotton, maize and cassava are given in table below.

Labour inputs per two months in mandays per hectare to grow cotton, maize and cassava in Mwanza region in Tanzania.

area	x1	x2	x3	x4	x5
crop	cotton	maize	maize	cassava	cassava
planting time	Nov-Dec	Nov-Dec	Jan-Feb	Nov-Dec	Jan-Feb
Jan-Feb	19	15	20	5	35
Mar-Apr	12	0	5	0	5
May-Jun	13	4	0	0	0
Jul-Aug	37	0	2	0	0
Sep-Oct	14	0	0	0	0
Nov-Dec	33	35	0	35	0

With the aid of the data in the above table and taking into account that 600 people participate in the cultivation and one month has 25 working days, the labour constraints may be formulated as:

19x₁+15x₂+20x₃+5x ₄+35x₅ < 30,000 (Jan-Feb)
12x₁+ 5x₃ + 5x₅ < 30,000 (Mar-Apr)
13x₁+ 4x₂ < 30,000 (May-Jun)
37x₁ + 2x₃ < 30,000 (Jul-Aug)
14x₁ < 30,000 (Sep-Oct)
33x₁+35x₂+ +35x₄ < 30,000 (Nov-Dec)

Yields per hectare, annual consumption and prices of maize, cassava and cotton

crop	yield per ha in kg	annual consumption in kg	purchasing or selling price per kg in Sh.
maize	1500	143,000	1
cassava	2000	72,000	1
cotton	2000		1.9

Define the variables v1 and v2 as follows:

v1 the amount of maize to be purchased annually in kg,
v2 the amount of cassava to be purchased annually in kg.

The food requirement constraints have the following form:

1500(x₂+x₃) + v1 = 143,000
2000(x₄+x₅) + v2 = 72,000

In the case of the food requirement the sign = is written instead of the > since the possibility of selling a surplus of these crops is not taken into account in this example. No capital constraints or land constraints have to be considered, so there remain only the non-negativity constraints:

x_j > 0, j = 1,2,....,5
v_i > 0, i = 1,2.

If the villagers want to maximize the annual net revenue the objective may be formulated as:

maximize 3800x₁ -v1 - v2,

where the value 3800 follows from the yield and selling price for cotton given in the above table, and the coefficients of v1 and v2 corresponds to the purchasing prices given in the same table. Note that the prices are expressed in Tanzanian shillings.

Verify that the solution of the linear programming problem is given by:

x₁ = 811; x₂ = 93; x₃ = 3; x ₅ = 36 ha
x₄ = v1 = v2 = 0

This implies that it is not beneficial to purchase maize or cassava. In the months November, December, January, February, July and August all labour is used.

(8) Using the LP88 computer software

A Pakistani farmer (developed by Professor J.M. Boussaard)

A Pakistani farmer owns 15 acres of land, and his family can supply the labour equivalent of 5 men/year. He can chose between 4 crops, with the following costs and returns:

crops	yield (t or Rs (rupees) acre-1)	cost (Rs acre-1)	price (Rs t-1 or Rs acre-1)
wheat	3	1040	1055
rice	2	1826	3276
sugar cane	6	3789	1686
fodders (and cattle raising)	5669	1669	1

At the same time the estimated number of working days for each crop is as indicated:

crop	working days per acre
wheat	13.5
rice	32.0
sugar cane	94.0
fodders (and cattle raising)	53.0

Crop water requirements are as follows:

crop	water requirements (1000 cubic feet/acre)
wheat	20
rice	80
sugar cane	60
fodders	30

The number of usable working days per worker is about 100 a year. Wheat occupies land in the winter, rice in the summer and sugar cane and fodder, all the year.

Please - using the LP88 computer software package - do the following:

(a) Build up a L.P. matrix representing the farm. Notice that the farmer is short of cash, sothat the overall cost cannot exceed 30 000 rupees. In addition, one acre of land is allowed 17 000 cubic feet of water.

Solve the problem

(b) It is possible to lease land, for a cost of 2000 rupees per year, to hire labour, for 30 rupees per day, and to get water from the neighbours, at 4.72 rupees per cubic feet.

Modify the model in order o allow these new activities. Solve it.

(c) It is possible to borrow money at a rte of 14 %. Introduce this possibility into the model, and solve it.

Conclusion?

(d) The standard deviations of receipts are as follows:

crop	standard deviations of receipts
wheat	317
rice	1431
sugar cane	1271
fodders	904

The minimum required for the farm family is about 30 000 rupees. Build up the FLCP risk matrix associated with the last model, and solve it.

Conclusion?

(9) Using the XPRESS optimisation software

Given the LP problem of our example in the syllabus:

Scheme 1. An example of an LP-problem.

a. Solve this problem in XPRESS
b. Add a crop activity sugar beet with a dry matter yield of 4 ton/ha.
c. Change the area constraint for potatoes in an area constraint for potatoes and sugar beet: x₂ + x₃ < 3.
d. Solve this new LP-problem.
e. Which crop activity has a `reduced cost'. Why? Explain this reduced cost.

(10) Solve the problem of exercise 4 and the questions under 4e, f and g with XPRESS.

(11) Solve the problem of exercise 5 with XPRESS

(12) Using the XPRESS optimisation software

Interactive Multiple Goal Linear Programming using XPRESS

Given the following IMGLP problem (compare with the `example' given in text):

A farmer wants to draft an optimal plan for his farm for the months May until October. He can grow three crops: wheat, potatoes and sugar beets. The labour requirements per month per hectare (May-October) for these three crops are given in the Table. The benefit per ha, the pesticide use, and the available labour are also given in a Table. Further the farmer faces the following constraints:

* he owns 60 ha of land;
* he wants to grow a crop on at least 3/4 of the farm area;
* potatoes can be grown on max. 1/3 of the farm area;
* sugar beets can be grown on max. 1/4 of the farm area.

The farmer wants to maximize the financial benefit for his farm.

a) Write this problem as a linear programming problem.
b) Write the LP problem in XPRESS.
c) Optimize the model.
d) Suppose that the farmer can buy extra casual labour. For which month would you advice the farmer to buy extra labour?
e) In the Table information is given on the pesticide use (kg a.i./ha) for each of the crops. Next to the financial benefit, another objective function of the farmer is the minimization of the use of pesticides on his farm. Write the problem as an IMGLP-model in XPRESS.
f) Make a Tables as in Tables 3.5 and 3.6 of your syllabus. Start with a zero round. Subsequently an upper or lower bound can be put on one of the objective functions, while optimizing the other, etc.. Proceed, until you end up with a satisfactory solution.
g) Suppose that the farmer also owns cattle. As a consequence, a third objective of the farmer for the arable part of his farm, is the minimization of labour input (in hours). Include this third objective in the model and carry out a zero and first round of this IMGLP model. Present the results as in Tables 3.5 and 3.6 of your syllabus.

Table - Labour (hours)

	May	June	July	Aug	Sept	Oct
wheat (per ha)	2	2	2	5	0	3
potato (per ha)	5	2	2	2	15	2
sugar (per ha)	3	21	4	0	0	9
total available (per month)	160	320	160	200	200	200

Table - Benefits and pesticide use

	benefit ($/ha)	pesticides (kg a.i./ha)
wheat	2500	5
potato	4000	15
sugar	4500	10

4. Methodology for explorative land use studies

In this chapter various aspects of the methodology for explorative land use studies are presented and discussed. Figure 4.1 summarizes the various aspects of the methodology and is an extension of Figure 2.1 in Chapter 2.2. The basis of the quantification of input-output relations concerning agricultural activities is formed by a land evaluation (Chapter 4.2). Subsequently the basic definitions and concepts which are important in the quantification of input-output relations are presented (Chapter 4.3). Chapter 4.4 deals with quantification of the various constraints and Chapter 4.5 with the identification of policy views and objective functions. The generation of land use scenarios with the IMGLP method is presented in Chapter 4.6 and finally aspects which are important in a sensitivity analysis are discussed in Chapter 4.7. However, before a detailed description of the methodology is given, clear choices concerning the boundaries of the system to be studied should be made (Chapter 4.1).

Figure 4.1 The various aspects of the methodology for explorative land use studies.

4.1. Defining the system

The system must be defined in roughly three dimensions: time, space and influence of man. The time horizon of the study depends, of course, on the aims and motives of the study. However, generally the time horizon is at least 15 years, such that `trend breaks' are possible. The time horizon affects the choices concerning the production techniques.

The motives and aims of the study also determine the system definition in space. For studies at the regional level, the borders of the system are usually determined by geographical and administrative factors. The methodology can be applied to a region within a country (The fifth region of Mali, The Atlantic Zone of Costa Rica), to a country, to a group of countries (The European Union), or to (part of) a continent. Studies at the farm level can be carried out for example for one specific farm, for a specific farm type or for an average farm in a specific region.

Definition of the system in the dimension `influence of man' is most complex. Choices concerning economical sectors (agriculture, recreation, industry, etc.) and agricultural production systems (arable cropping, animal husbandry, horticulture, fish production, etc.) are relatively easy. However, choices concerning the social and economical factors and constraints to be involved in the study are much more difficult. What to do with (current) infrastructure, (current) level of knowledge of farmers, (current) management techniques and (current) social structures? The longer the time horizon and the more explorative the study, the more variable these factors may be. Political desire and appropriate policy may influence these factors. The longer the time horizon of the study (e.g. 25 years), the more optimal these factors could be assumed or the more they can be excluded in the model. Anyhow, it is important to be very explicit about the choices concerning definition of the system in the dimension `influence of man'.

4.2. Land evaluation

In explorative land use studies one is not primarily interested in current land use and current production techniques, but also in potential or new land use types and new production techniques which might become feasible within the time horizon of the study.

In a land evaluation the characteristics of soil and climate in spatial units are confronted with the requirements for various forms of land use. This confrontation can be a qualitative one, a spatial unit is suitable for a certain form of land use or not, or a quantitative one, which also gives information on how suitable a unit is for a certain form of land use. Special courses on land evaluation are given at the Wageningen Agricultural University (Driessen & Konijn, 1992).

Qualitative land evaluation. Relevant soil characteristics include topography, stoniness, texture and acidity. Important climate factors of course are temperature, day length and precipitation. The forms of land use can be characterized by its general purpose: e.g. agriculture, nature, landscape and recreation. If we focus on agriculture, the crops which are grown and the degree of mechanization which is used to grow these crops determine the soil and climate requirements. Different groups of crops with different requirements can be distinguished. Perennial crops and annual crops can be distinguished. Within the annual crops, generally the requirements which the soil must satisfy decline from root and tuber crops to cereals and finally grass. The requirements which the climate must satisfy, especially differs between C₃ and C₄ species, e.g. wheat and rice versus maize, millet and sorghum. The degree of mechanization has its implications for the requirements on soil and climate. In explorative studies production techniques with various degrees of mechanization should be included to show consequences of mechanization. This is particular clear for developing countries, in which mechanization is not common yet, but also in industrialized countries in which various way of weed control are possible: chemical, mechanical and with manual labor.

An important question is the scale at which the land evaluation should take place. Usually this scale is determined by the level of detail of soil maps and climatic data, administrative sub-regions within the system under study, and pragmatic arguments. Figure 4.2 illustrates that it is important to aggregate or average as late as possible: first calculate, then average.

Quantitative land evaluation. For areas suitable for a specific form of land use as e.g. growing crops, potential or water-limited crop yields (see Chapter 4.3) can be calculated by using crop growth simulation models. Crop growth models for this purpose, must be simple without too much detail. Ideally, they only require a few important parameters on soil, climate and crop characteristics.

Figure 4.2 The influence of averaging rainfall on its calculated yield response. The yield is underestimated by averaging in the lower rainfall region (A) and overestimated in the higher rainfall region (B) (De Wit & Van Keulen, 1987).

4.3. Quantification of input-output relations: Concepts and definitions

4.3.1. Introduction

In land use studies using the linear programming technique, information on land use is translated in input-output relations. Agriculture can be defined as the human activity in which energy from the sun is fixed, using other inputs as good as possible. This activity results in desirable outputs, e.g. grain, and, inevitably, in undesired outputs, such as nitrogen losses. Inputs can be applied for the realization of outputs in numerous ways.

Input-output relations can be considered from the question `what is possible' (exploration) considering the time horizon chosen or from the question `what happens'. In answering the question `what is possible', bio-physical possibilities are identified which can be curtailed by socio-economic, agronomic and environmental factors. Possibilities to grow a certain crop and the corresponding yield are determined by bio-physical factors like climate and soil and production ecological processes. In research with the central question `what happens', the current situation is described and characterized. The analysis aims at the way actual inputs and outputs are determined by production ecological, socio-economic and environmental factors.

In explorative studies input-output relations should be defined from the question `what is possible'. For a good and systematic quantification of input-output relations in explorative studies, a general terminology and some concepts are important.

4.3.2. Target-oriented approach - complete information

Inputs can be classified in various ways. An important classification in production ecology is that of substitutable and non-substitutable inputs. Non-substitutable inputs like water and nutrients are taken up by the plant, and sometimes incorporated (nutrients). They fulfil a specific and essential role; no mutual substitution between these inputs is possible. Substitutable inputs are not incorporated in the plant. They can replace each other, up to a certain degree. Examples are labor, mechanization and pesticides.

Outputs can be subdivided in desirable outputs, the crop yield or parts of it, and undesirable outputs to the environment, the emissions, such as nutrient losses and pesticides, or `immissions', such as nutrient depletion of the soil.

Both inputs and outputs are expressed in physical units. The use of inputs can be expressed per unit area or per unit desired output. The efficiency of the use of inputs is defined as the input per unit desired output. The emissions can also be expressed per unit desired output and per unit area.

As stated, inputs can be used for production of outputs in numerous ways. However, inputs are not used without any intention, but with a specific aim, related to the desired outputs, but also related to factors like risk and uncertainty. The adjustment of inputs to the desired outputs is an attractive approach in production ecological analysis of input-output relations, which is called the target oriented approach. In the definition of inputs and outputs, knowledge on the processes involved is used. The potential outputs primarily depend on the crop characteristics and the circumstances under which a crop is grown, especially temperature and radiation. By aiming at a specific output, the amounts of water and nutrient can be quantified, using the knowledge concerning uptake and use of resources. Subsequently, the required crop protection for the realization of the output is quantified.

In the target-oriented approach complete information is supposed, i.e. the output which can be realized and the required inputs are known a priori; factors like risk and uncertainty (e.g. due to weather conditions) are ignored. Further, it is assumed, that in explorative research not more water, nutrients or pesticides are used than necessary, according to the available knowledge and technique.

4.3.3. Production level, production situation, production technique and production orientation

Input-output relations according to production ecological principles can be characterized with the terms production levels, production situation, production technique and production orientation.

The production level points at the production of desired output per unit area. In production ecology various production levels can be distinguished according to three groups of production factors: growth defining, growth limiting and growth reducing factors (Figure 4.3). The growth defining factors include factors that, at optimum supply of all inputs, determine growth and production from a plant's point of view: CO₂ -concentration, radiation, temperature and crop and cultivar characteristics. The growth limiting factors comprise water and nutrients, taken up by and (partly) incorporated in the plant. The growth reducing factors include weeds, diseases, pests and pollution. The various inputs that are used in growing a crop, affect production via these production factors.

Figure 4.3 Production situation, production levels and associated principal growth factors ( Rabbinge et al., 1994b).

The three production levels that can be distinguished with these three groups of production factors are: potential production level, water and nutrient limited production (attainable) levels and actual production levels. The potential production level is dictated by growth defining factors; the crop is optimally supplied with water and nutrients and is completely protected against growth reducing factors. At water or nutrient limited production level, the production is lower due to a lack of water or nutrients. The actual production level is determined by a lack of water, nutrients and incomplete crop protection against growth reducing factors. As a matter of fact, there are many water and nutrient limited and actual production levels at one potential production level, dependent on the degree of lack of water and of nutrients, and dependent on the degree of damage due to growth reducing factors.

An input-output relation is location specific. The location can be characterized by the production situation, i.e. the conditions under which a crop activity takes place and which are more or less a given fact for that crop activity. These conditions are hard to manipulate, and they affect the potential production level and/or the required inputs to realize a certain production level. The other way around, the activity hardly affects the production situation. The production situation includes the following factors:

(i) climate, especially temperature, radiation and humidity;

(ii) soil characteristics which affect the uptake-efficiency of the non-substitutable inputs water and nutrients (such as pF-curve, clay fraction, texture, pH and organic C-concentration);

(iii) biotic factors in soil and environment, hardly affected by the crop activity itself (e.g. `pressure' of pest and diseases).

The resource pools of water and nutrients in the soil which are available for a crop, are not included in the production situation. These pools can be easily manipulated by the application of inputs.

If two production situations only differ in soil characteristics or biotic factors, the potential production levels in these production situations do not differ. With the same input levels, however, the attainable and actual production levels will be higher in the good production situation than in the bad production situation. If the climatic factors also differ between the two situations, the potential production levels will also differ (Figure 7).

The production technique stands for the complete (way of) use of non-substitutable and substitutable inputs to realize a certain production level in a certain production situation. Since some inputs are substitutable (e.g. labor, mechanization or pesticides) a certain production level in a particular production situation can be achieved with various production techniques.

A production activity (or an input-output relation) stands for the growth of a crop or cropping system fully characterized by its inputs and outputs. The same production technique applied in two different production situations results in different outputs and thus in two different production activities.

With the former concepts and the target-oriented approach, still numerous input-output relations can be defined in each production situation. We need the concept production orientation, which denotes the aim of the production activity and which directs the quantification of a particular input-output relation. Aims of production activities could be: a high soil productivity, high resource use efficiencies, low emissions per unit product, low emission per unit area, no external inputs, etc..

4.3.4. Aggregation level and time horizon of input-output relations

The input-output relations are derived for the level of crop or cropping system. For studies at the regional level, the role of the farm and the farmer are neglected; production factors like area, labor and mechanization are assumed to be variable. For studies at the farm level, the role of the farm and the farmer are kept explicit. The assumptions on farm area, labor availability and mechanization depend on the time horizon of the study.

The time horizon of an explorative study should be reflected in the time horizon of the input-output relations. Input-output relations in static models, like linear programming models, should be defined in such a way that they can hold for many years or cycles of the crop rotation. This implies that the resource pools in the soil should not change as a result of the production activity, unless explicitly desired, as for instance could be true for P-saturated soils. Further, the production situation should be maintained, e.g. the application of soil tillage to maintain or repair soil structure.

4.3.5. Quantification of non-substitutable inputs

Based on the amount of water used by the crop to produce one kg dry matter (transpiration coefficient) at a certain temperature, radiation level and humidity, the water uptake for a certain production level can be quantified (Van Keulen & Van Laar, 1986). Since each of the nutrients has a specific function within the plant, the nutrients cannot be substituted mutually and their concentrations are more or less fixed. In this way the nutrient uptake for a particular production level can also be calculated (Van Keulen, 1986; Driessen, 1986).

Not all non-substitutable inputs available in the soil are taken up by the crop. The difference between supply and uptake of water or a nutrient, or in other words, the efficiency of the uptake of the various inputs, depends on (i) the level of the other inputs (and thus the production level), because of the interaction between various inputs (De Wit, 1992), and (ii) the production situation, especially the physical and biological characteristics of the soil. The levels of the other inputs (i) are manipulated in order to realize a certain production level. An appropriate adjustment of the inputs has a positive effect on the uptake of water and nutrients. A crop which is optimally protected against pests and diseases will use water and nutrients more efficiently. In a favorable production situation (e.g. a favorable pF curve or pH), water and nutrients are used more efficiently and less inputs are required to realize a certain uptake and production level than in unfavorable production situations (ii).

The available water and nutrient pools in the soil are often insufficient to realize an uptake that meets the entire requirement for a desired production level. The resource pools, therefore, have to be supplemented with irrigation water and external nutrients. The external inputs can be applied in various ways. Nutrients can be supplied in inorganic and organic form. The frequency of input application may differ: one application at the start of the season, or split applications spread over the season. Finally the spatial distribution of applied inputs is important: broadcast or row application. The way in which the requirements of a crop are met also affects the uptake efficiency and thus the required input level. In explorative research it is assumed that not more inputs are applied than strictly necessary according to available techniques. Efficient and saving techniques are applied.

4.3.6. Quantification of substitutable inputs

Next to the non-substitutable inputs, also substitutable inputs are required to realize a certain production level. The crop must be protected against weeds, diseases and pests by for instance crop protection techniques and crop rotation. Since it does not matter for a crop how it is protected, as long as it is protected, we speak about substitutable inputs: herbicides or weeding machines, pesticides or crop rotation. Not all pesticides, however, can be substituted by other inputs, for instance some fungicides.

The choice for substitutable inputs depends on the production orientation. In economical aims, the mutual price ratios determine the optimum mix of substitutable inputs. In explorative research other aims may be important as well, for instance ecological or social aims; inputs which are favorable from an economical point of view (e.g. pesticides) can be substituted by labor.

4.3.7. Emissions and immissions

Undesired outputs are inevitable in production activities. If we suppose an equilibrium situation in which the available nutrient pools in the soil do not change as a result of a production activity, then it is clear that the difference between the input of nutrients (thus available nutrients minus soil pools) and the uptake by the crop disappears from the system (leaching, denitrification, volatilization, etc.). Emission also occurs when using pesticides. This may apply to the applied compound itself, or to its degradation products.

With similar input levels, in good production situations higher yields can be achieved than in bad production situations. Therefore, at similar input levels per hectare, the emissions per unit product and per unit area are lower in a good production situation than in a bad production situation.

Next to emission, `immission' may occur; if not all nutrients taken up by the plant are supplied by external inputs, the nutrient pools in the soil decrease.

Not always, equilibrium situations are pursued. Sometimes emissions or `immissions' are desired. It can be desired to improve soil fertility by building up soil pools. In the case of saturated soil, for instance with phosphorus, `immission' can be desirable.

4.3.8. Production functions or technology sets

Relations between one input and one output can be represented mathematically and graphically in a production function. The concept of production functions is however limited, because generally several inputs are varied simultaneously and some inputs have a discontinuous nature. Because of positive interaction between various inputs, generally it is not rational to consider relations between just one input and an output, but to tune a mix of inputs to realize a certain output. Some inputs have a discontinuous nature, for instance the use of pesticides or the use of some kind of machine. The use of such an input, opens the possibility of realizing a higher production level with the right mix of the other inputs; a technology jump. It is important to include essential technology sets in the linear programming model for the land use study.

Examples of technology jumps (and sets) at crop or cropping system level can be found by comparing yields and belonging inputs for the growth of some kind of crop some decades ago and the growth of that crop nowadays. Table 4.1 gives the technology sets of 1972, 1982 and 1992 for the growth of sugar beets in two areas in the Netherlands. Yields and a selection of inputs are given. In both areas yields have increased, whereas the N rates did not change or decreased. Different cultivars were used and the crop protection has been fine tuned. The number of spraying increased, but the amount of active ingredient per spraying decreased. Chemical crop protection against weeds was partly replaced by mechanical techniques.

Table 4.1 also illustrates the difference in production situation. In both regions the output/input ratio was improved in the course of the decades, but this was more true for the `Centraal kleigebied' than for the `Veenkoloniën'. With the same or less inputs, higher yields are realized in the better production situation (`Centraal kleigebied') than in the worse production situation (`Veenkoloniën').

4.4. Identification and quantification of constraints

The constraints in the linear programming model can be roughly classified in resource constraints and product and demand constraints.

The resource constraints comprise e.g. area constraints, water constraints, nutrient/manure constraints and labor constraints. The area, and sometimes the water constraints, are a result of the land evaluation.

The product constraints `regulate' the conversion of arable cropping products or roughage to industrial products or feed, and the conversion of feed to animal products. The demand constraints `regulate' the demand for agricultural products produced in the system, which is related to several factors:
* the trade situation and the policy views on the future trade situation (free trade or autarky);
* population and population growth;
* dietary patterns of the population.

The dietary patterns depend on the increase in real disposable income and on inflation. As the living standard rises, the consumption of basic food requirements flattens out, in both absolute and per capita terms. The total consumption of food, however, continues to rise on account of a shift in demand towards more expensive, high-protein products such as meat and cheese.

Table 4.1 The outputs and a selection of inputs for the growth of sugar beets in 1972, 1982 and 1992, in two production situations (`Centraal kleigebied' and `Veenkoloniën')

outputs/inputs	1972	1982	1992
'Centraal kleigebied'
yield (ton ha-1)	56	59	68
cultivar (-)	Kawepoly	Monohil	Univers
fertilizer use (kg ha-1):
N	160	160	140
P2O5	100	100	100
K2O	80	310	145
crop protection:
pesticides (kg a.i. ha-1):
herbicides	10	7	7
others	2.6	0.5	1.2
treatments (number):
spraying	4	3	6
weeding	0	1	2
'Veenkoloniën'
yield (ton ha-1)	45	48	51
cultivar (-)	Kawepoly	Monohil	Univers
fertilizer use (kg ha-1):
N	160	160	160
P2O5	120	90	50
K2O	200	285	200
crop protection:
pesticides (kg a.i. ha-1):
herbicides	13	18.5	13
others	1	4.5	1
treatments (number):
spraying	4	5	7
weeding	0	2	3

4.5. Policy views and objective functions

In the subsequent step in the methodology various policy views concerning land use problems should be identified, e.g. views emphasizing self-sufficiency of food, free market and trade, nature conservation, environmental issues, etc. They can be distilled from policy documents issued by the relevant governments or donor agencies and from interviews with policy makers and representatives of societal organizations.

Policy views can be operationalized by means of objective functions that are minimized or maximized in the IMGLP-procedure. For instance, the policy view with great concern for environmental issues can be operationalized by the objective functions `the use of nutrients or pesticides per hectare or per unit product', that are minimized.

The objectives must be quantifiable, each in their own dimensions and the quantification must be linked to land use in the system. More over, the objective functions must be mutually conflicting, at least up to a certain extent and not totally. If the objectives form extensions of one another the model cannot generate alternative allocations. If on the other hand the objectives are totally mutually conflicting, the results will be meaningless since the loss in terms of one objective will automatically mean the gain in terms of another.

Not all policy views can be easily quantified with objective functions, e.g. nature development or conservation. These kinds of views have strong spatial components, that are hard to catch in a mathematical function. Such views should be confronted with the generated scenarios in various evaluation steps (ex post analysis).

4.6. Interactive multiple goal programming

In the former steps of the methodology an IMGLP model is formulated. In the following step the technical coefficients about the production of and the demand for agricultural products are confronted with the objective functions representing the different policy views. This is the most important interactive part of the methodology. Of course stakeholders can interact earlier in the process, e.g. by assisting in defining the system, identifying production techniques and identifying policy views, but particularly in this part they are confronted with the consequences of choices in priorities they make.

Firstly, the `playing field' is determined by optimizing each of the objective functions, without putting (heavy) restrictions on the other objective functions. These are called the zero rounds. Worst and best values are determined for each of the objective functions. The initial freedom of choice for each objective -the difference between the best and worst value- is made explicit to the stakeholder in this way.

In the next step the stakeholder has to select the objective with the worst value which he considers as most unacceptable. A tighter bound for that objective has to be formulated. Subsequently the stakeholder is confronted with the results of a new series of optimization runs and has to select again an objective with a value which is unacceptable to him. This procedure continues until the stakeholder is satisfied with the results.

Ideally the procedure should follow the steps described above, but is not followed in both case studies. In the EC-study, ideally the different groups, e.g. farmers organizations, nature conservation organizations and so on, should have been invited. In the Mali-study, local and national authorities and representatives of the different donor agencies were only invited once. They were confronted with several scenarios, differing because emphasis was put on different objectives. However, they had no opportunity to make other choices than the scientists executing the study had made for them.

Apparently it is difficult in such land use studies to organize an interactive step with the stakeholders. This might be due to the numerous stakeholders involved at different levels of a decision process. Furthermore the high hierarchical levels of both studies imply that several groups acting at lower hierarchical levels, are less interested. For them, the methodology is difficult to understand and, additionally, the LP software-packages are not user friendly. It also takes much time to run the models.

4.7. Generating land use scenarios

In giving more or less weight to some of the policy objectives in relation to others different scenarios can be developed representing the different policy views. In developing the scenarios trade-off between the different goals become clear.

In both studies the executing scientists made a choice to create scenarios related to the different policy views in stead of following a real interactive procedure with stakeholders. The stakeholders at higher integration levels are politicians. Since their policy views are commonly written down in programs, the scientists executing the studies have selected and earmarked them according to the objective which is getting the highest priority in that policy view.

The differences between the scenarios in both studies indicate that there is room for policy change.

4.8. Sensitivity analysis

The results of an analysis with the IMGLP technique can be sensitive to various aspects. In each of the steps in the methodology, we make choices and use `guestimates'. We deliberately ignore complex issues. For instance, we average values and thus eliminate variation in technical coefficients in time and in space and we neglect stochastic aspects, subjects we can not deal with up till now. Evidently, a sensitivity analysis should be part of the land use studies. However, given the enormous number of technical coefficients used and choices made throughout the process, sensitivity analysis was always limited to only some aspects in the studies carried out so far. We will illustrate the need for a sensitivity analysis by going through the examples found in literature.

In defining the system, choices are made on the spatial unit within the system, for which a land evaluation is carried out, and for which specific input-output coefficients are defined. This implies that within these units heterogeneous sub-units or sub-regions are aggregated. An example of the effects of aggregation on results with land use models is given by Rabbinge & Van Ittersum (1994), showing that using aggregated data in an LP-model results in less extreme values after optimization. They supposed, that a particular region is divided in four land evaluation units of 1000 ha each. The wheat yields with a certain production technique in these units are supposed to be 8, 6, 4 and 2 t ha^-1 for unit 1, 2, 3, and 4, respectively. The objective of the optimization is to minimize the area in the region required to produce 10 000 ton wheat. In Figure 4.4 the effect of aggregating units 1 and 2 and units 3 and 4 is shown. Without aggregation, the minimum area is 1333 ha (1000 ha in unit 1 with 8 t ha^-1 and 333 ha in unit 2 with 6 t ha ^-1 ), whereas after aggregation the minimum area is 1429 ha (1429 ha in unit 1 + 2 with an average yield of 7 t ha^-1). This example shows that the level of aggregation has an effect on the results of the optimization.

The choices which production techniques, products and policy views are included in the study are extremely crucial. The sensitivity of changing policy views are commonly tested by changing e.g. the targets of objective functions. If scenario building is really done in an interactive way, thus in close collaboration with stakeholders, the outcome of the land use study should not result in a big surprise. Part of our exercises deals with this type of sensitivity analysis. In other exercises it is shown that the choice to add one or more production techniques does influence the results in the way that the `solution space' or `window of opportunities' can be enlarged.

Figure 4.4 The effect of aggregation of four land evaluation units of each 1000 ha to two units of 2000 ha each on the minimum agricultural area required to produce 10000 ton wheat in the region (4000 ha total) (see text).

All work on system definition, choices for production techniques, products and policy views results in an LP-model which can be simply represented by:

MIN/MAX (c' x)

subject to:

A(x) < b

These LP-models are sensitive to:
* the input-output relations (technical coefficients, the values for a):
* the physical or normative boundaries (the values for b);
* the coefficients of the objective functions (the values for c).

LP-software has a standard sensitivity analysis for part of these values (shadow prices, Right Hand Side ranging and ranging of the coefficients of the objective functions; see Chapter 3.2.8). However, in defining the technical coefficients, we use an enormous number of values, which can be criticized. Testing the model for sensitivity to changes in these values is hardly done and is mainly limited to some values as e.g. prices for inputs (fertilizers) or outputs (cereals). Bessembinder (in press) argues that more attention should be paid to the uncertainty of other technical coefficients as e.g. the fertilizer use efficiency. The sensitivity of a model should not only be tested for changes in technical coefficients but also for changes in assumed restrictions (RHS=Right Hand Side values).

In estimating technical coefficients often average values are used. In practice values can vary in time and in space. This is e.g. the case for production estimates at different rainfall regimes. Figure 4.2 shows the relation between yield and rainfall. Since the relation is curvilinear, yields are underestimated in the lower rainfall region and overestimated in the higher rainfall region.

Nevertheless, stochastic aspects are completely neglected. In practice the distribution or succession of low and high values in time is also very important. E.g. in livestock production based on natural pastures two subsequent dry years are much more disadvantageous than two years alternated by a normal or wet year.

Whether we deal with heterogeneity in space (see example of aggregating land units) or in time (see example with production levels at different rainfall regimes), the credo `first calculate and then average' is valid. Heterogeneity and curvilinearity in input relations should be retained as long as possible and their consequences should be included in the evaluation phase.

4.9. General discussion

The stages and steps that are included in a land use analysis study, generally speaking, are shown in Table 4.2

Table 4.2 Stages and steps included in land use analysis studies

This Chapter has discussed the methodological aspects of explorative land use studies. Very often these studies leave insufficient time for the utilization of the IMGLP model with stakeholders in the use of land. Consequently, results are generated, however, the interpretation of these results and their translation for policy making is limited. IMGLP then becomes MGLP. In this regard LUA studies leave room for improvement. This can be found particularly in the field of quick and efficient generation of data, required for the study.

LUA studies have a tendency to be very data intensive. This makes it extra necessary that the data requirements are made explicit at a very early stage of a LUA project. The expressed data requirements have to be very carefully evaluated considering the objectives of the study. Each time when conducting a LUA study the following questions should be asked:
* what is the purpose of the LUA study that we are going to conduct?
* what implications does that have for the type of data that we are going to need?; and
* how can we obtain that data in the quickest and most efficient manner?

Only when these questions are given serious consideration, a situation can be avoided where too much time is spent on stage 1 of a LUA study and too little time on stage 3 as is often the case with LUA studies.

4.10. Exercises

1) Give five different examples of a system, clearly indicate the boundaries.

2) Indicate what the differences are between qualitative and quantitative land evaluation.

3) Write down as briefly as you can what the differences are between production level, production situation, production technique and production orientation.

4) Why is a distinction made between substitutable and non-substitutable inputs.

5) For the region that you live in formulate at least four policy views concerning land use.

6) Considering the answers that you gave to question (5) indicate which type of sensitivity analysis you consider most important.

7) In which ways is a playing field (also called a potency matrix) different from a scenario?

Chapter 5 and 6