Appendix B Linear programming model (B1-B2)

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An Optimal Spatial Resource Allocation Model has been developed for integrating livestock, crop and fuelwood production sectors within the framework of AEZ Land Productivity Assessment and its application to Potential Population Supporting Capacity and Development Planning in Kenya. This Appendix gives a mathematical description of the Linear Programming model used for formulating district level scenarios. Like with any model of this kind, the formulation gets revised and improved as new insights, needs or quantified information becomes available. The strength of the approach lies in its extensive and consistent use of spatial information for assessing agricultural land use options within the context of district development planning.

B.1 GLOSSARY

Activity:
In the context of land use, a process using a technology that combines inputs to generate particular outputs for sale, barter or household consumption. An activity is considered to be a subsystem of a farm system.

Agro-ecological Cell:
Basic unit of physical analysis in the AEZ study. Data records are unique in terms of the combination of soil, landform and climate attributes included in land resourcesinventory.

Agro-ecological Zone:
A relatively extensive area consisting of an aggregate of agroecological cells defined in terms of climatic conditions, major landform, hydrological regime, major soil groupings and/or (semi-) natural vegetation, which is suitable for a certain range of crops and cultivars.

Agro-ecological suitability:
The fitness of a given type of land for a specified land use type.

Algorithm:
A set of rules for solving a problem.

Crop Combination:
In areas where temperature and moisture allow to produce more than one crop per year, the multiple cropping activities are defined as sequential crop combinations.

Crop Combination Filter:
In addition to agronomic feasibility, a filter mechanism, testing for economic and/or nutritional value, is used to identify the most productive crop combinations in each agro ecological cell.

Crop Cultivation Factor:
Many soils cannot be continuously cultivated with annual food crops without undergoing some degradation. The crop cultivation factor expresses the number of years crop cultivation can take place within a hundred year cycle.

Crop Group:
For the purpose of formulating feed mix and nutrition constraints, crops can be grouped according to common characteristics into crop groups, e.g. cereals, root crops, oilseeds, etc.

Feed Item:
the term feed item is used to refer to products useful for feeding to animals and related to cultivation of a particular crop, grasses or forage legumes. In this study the following feed items have been considered: primary product, crop residues, and crop by-products (e.g. bran of wheat).

Feeding Period:
For the purpose of formulating feed balance and nutrition constraints and to take into account the distribution of feed supplies throughout the year, the calendar year can be sub divided into several seasons. In this study two feeding periods have been distinguished: wet season and dry season. The length of the wet season is assumed to equal the applicable LGP.

Feed Utilization Factor:
Not all potential feed supplies are actually available for feeding due to accessibility, ecological reasons, seasonality of supply, etc. Feed utilization factors are used to estimate the share of potential feed supply that is available for feeding by crop, feed item and feeding period.

Feed Depreciation Factor:
Feed supplies, when not used up entirely within a season, can potentially be carried over to the next season. To allow for modeling LUT related constraints to carrying over feed supplies as well as losses in feed storage, the model requires the specification of feed depreciation factors that express the share of unused feed supplies that is lost by carrying over to the next season. Specification is by crop, feed item and season.

GIS:
Geographical Information System; software system and specialized computer hardware to input, manipulate, analyze and present spatial information.

LGP:
Length of growing period; reference LGP is defined as duration (in days) of the period when temperature permits crop growth and soil moisture supply exceeds half potential evapotranspiration; it includes the time required to evapotranspire up to 100 mm of soil moisture storage (FAO, 1978-81). Growing periods which include a sub-period when precipitation exceeds potential evapotranspiration are termed 'normal' LGPs as compared to 'intermediate' LGPs with no such humid sub-period.

LGP, Pattern of:
Number of separate length of growing periods within a year, summarized as a historical profile of pattern of length of growing periods per year.

Livestock System:
Quantified description of livestock production activities, relating to animal type, feed requirements (quantity and quality), herd structure and management aspects.

LP:
Linear program; optimization problem involving a linear objective function and a set of linear constraints.

LRI:
Land Resources Inventory; the land resources inventory brings together several layers of information on physical environmental resources and allows the creation of unique ecological land units (agro-ecological cells) within which landform, soil and climate conditions are quantified and considered nearly homogenous.

LUT:
Land Utilization Type; to define the fundamental objects of analysis which comprise the set of alternative activities available to reach specified objectives, the first step in an AEZ application is the selection and description of land utilization types (LUT) to be considered in the study. FAO (FAO, 1984) defines LUT as follows: 'A Land Utilization Type consists of a set of technical specifications within a socio-economic setting. As a minimum requirement, both the nature of the produce and the setting must be specified'. There is a similarity between cropping system, livestock system and land use type.

Scenario:
A scenario comprises of a set of exogenous quantitative inputs to the model system that relate to the mode of operation, the levels of constraints and targets, etc. In the Kenya AEZ study, it is the description of a district planning problem to be solved using LP, in terms of specifying the objective function and the levels of constraints and targets.

Soil Mapping Unit:
Delineated area on a soil map representing an association of soil units which are grouped according to certain patterns.

The Exploratory Soil Map of Kenya (Sombroek et al., 1982), at a scale of 1:1 million, was used to compile the soil resources inventory in the Kenya study. Three hundred and ninety-two different soil map units are distinguished, describing soil associations or soil complexes composed of dominant soils, associated soils and inclusions (390 map units) or relate to water bodies and major urban areas (2 map units). A mapping unit composition table has been provided (van der Pouw, 1983) containing percentage allocation of the soil units by soil type, slope class, soil texture and soil phases. It also contains information derived from the legend of the soil map regarding landform and geology/parent material.

Thermal zone:
A classification based on the quantification of heat attributes, used to represent the prevailing termperature regime of an area. The quantification of temperature attributes has been achieved by defining reference thermal zones. As temperature seasonality effects of latitude are minor, temperature zones are closely correlated to altitude ranges (Braun, 1982). To cater for differences in temperature adaptability of crops, pasture and fuelwood species, nine thermal zones have been distinguished, based on ranges of 2.5° Celsius.

B.2 LIST OF SYMBOLS: