Journal of Geographic Information and Decision Analysis, vol.3, no.2, pp. 9-17

Using an Allocation Model in Multiple Criteria Evaluation

David Pullar
Geographical Sciences and Planning, The University of Queensland, Brisbane, Australia. Q4072
D.Pullar@mailbox.uq.edu.au

Contents
1. Introduction
2. GIS And Multiple Criteria Evaluation 
3. Integrating Spatial Interaction Models and GIS
4. Solution
5. Case Study 
6. Conclusion
References		 ABSTRACT  Multiple criteria evaluation is a structured process to define objectives, to formulate criteria and to evaluate solutions in a decision problem. Relatively straight-forward procedures may be applied to perform land use evaluation within predefined land regions as long as there are no complex interactions across these spatial units. Interactions may occur because of flows or influences between neighboring regions. While there exists a large body of literature to model these geographical interactions, there are relatively few attempts to link this into the decision-making process. This paper describes a methodology to include spatial interactions, modeled as constraints for land allocations, into multiple criteria evaluation. The paper reviews tools, techniques and models for land use allocation. A solution is presented where an allocation model is coupled to multiple criteria evaluation. This is then applied to a forest resource application as a case study.
KEYWORDS:geographical information systems, spatial constraint, allocation model, resource assessment. 

1. Introduction
Multiple criteria evaluation is a structured process to define objectives, to formulate criteria and to evaluate solutions in a decision problem. There are many land-related problems requiring the evaluation of multiple criteria based upon spatial properties and preferences. For instance, land use evaluation needs to balance several competing criteria to decide the best location for a particular land activity. The land surface can be segmented into appropriate common spatial units for the task of analysis. Simple scoring techniques are then available to rate criteria based upon land suitability in each unit, and then summing these scores to indicate the most suitable locations. This procedure is relatively straight-forward as long as there are no complex interactions across spatial units. Interactions may occur because of flows or influences between neighboring regions. While there exists a large body of literature to model these geographical interactions, there are relatively few attempts to link this into the decision-making process. A solution is presented where an allocation model is coupled to multiple criteria evaluation. This is then applied to a forest resource application as a case study.
        The outline for the paper is as follows. The first section reviews tools and techniques for multiple criteria evaluation of spatial information. A subsequent section explains a common interaction model for land resource allocations. Methods for incorporating an allocation model into multiple criteria evaluation are presented. To illustrate these techniques a case study for a forest resource application is described . The objective is to balance production and environmental interests in managing commercial forests.

2. GIS And Multiple Criteria Evaluation
A geographical information system (GIS) provides the processing capability to assess spatial criteria as part of a multiple criteria evaluation (MCE) procedure (Carver, 1991). The benefit of using a GIS is that constraints can be based upon spatially related data, such as distance to a road, and the GIS is a suitable computing tool to perform the MCE analysis (Jankowski, 1995). The most prevalent procedure for integrating MCE and GIS for land suitability analysis is using a linear weighted combination (LWC) approach (Eastman et al., 1995). In this approach land information is transformed to a set of factors over the study area. These factors are combined by applying a weight to each factor, followed by an overlay summation to yield a suitability map. This map can be used directly for satisfying a single objective, or a multiple objective analysis procedure applied to allocate areas according to the highest ranked objective. It has proven to be a very popular technique because it can readily include judgements from decision-makers (as factors or weights) to influence the outcome.
The suitability score S is computed as:

        The logic that lies behind multiple criteria evaluation is to compute a combined suitability score for each location, and then rank the most suitable locations to arrive at the best solution. This is illustrated in Figure 1a. A set of standardised factors Ai and their respective weights wi are combined by additive computation to produce a suitability map S. In most applications there is an additional step to identify the best sites R using a decision rule based upon a heuristic choice, typically this is done by priority ranking the values in S and allocating the best number of sites. An example would be identifying the best amenity areas in a forest as a combination of factors for proximity to walking tracks and streams, moderate relief, and away from conflicting land uses such as logging. These factors can be computed using a cartographic model (Tomlin, 1991). Once a combined suitability score is obtained by equation (1), then a specified number of hectares are chosen from the highest ranked values for further investigation as amenity land uses.

        In summary, a linear combination technique can be used to evaluate suitable areas and to assign the best areas for an activity. Linear weighted combination has several limitations, such as the assumptions of independence and additivity of decision variables. In addition the method does not explicitly deal with spatial context in the decision evaluation, it simply performs localized computations at each spatial location. Two important spatial considerations are to generate coherent spatial areas (Brookes, 1997), and to include spatial interactions (Bailey and Gatrell, 1995). This paper focuses on the general problem of dealing with spatial dependencies in decision evaluation, and in particular on how to account for interactions between locations. The proposed approach is based upon satisfying a set of constraints in the evaluation. This is illustrated in Figure 1b. As before a suitability map S is derived with a LWC procedure, but now the decision rule weighs the choice of the best sites by considering a set of constraints. The next section reviews spatial interaction models and their integration as constraints into the decision evaluation approach.

3. Integrating Spatial Interaction Models and GIS
Location-allocation models provide a set of techniques to explicitly handle flow relationships between spatial phenomena (Fotheringham and O'Kelly, 1989). These models incorporate spatial interactions as flows between and from locations. They are used commonly in business geographics to model market catchment, or flows between supply and demand regions. They are based upon central place theory, that is they model the flow between central facilities (supply) and a set of distributed zones (demand). Conceptually a measure of supply-demand interaction can be described in terms of gravitational attraction. The model of interaction between two spatial entities is as follows: 

Figure 2 shows the relationship schematically for a retail shopping center. It assumes that shoppers choose a center in direct proportion to its attractiveness (amount of floorspace) and in inverse proportion to the distance between the origin i and each competing center j. There are many variations upon the basic interaction model to deal with contributing factors such as competition, attractiveness, and travel costs. More advanced models also deal with hierarchies of centers (Roy and Anderson, 1988).

 Figure 2. Interaction model for center (supply) and a population (demand) 

        An advantage of allocation models is that they explicitly deal with interactions, namely by using a set of heuristic rules and a distance metric to model which resources flow to designated centers. Allocation models are part of a large class of mathematical programming techniques, which focus on constrained optimisation methods to find the solution to functional relationships. These techniques are extremely popular in management science and have been applied to GIS, but they have certain limitations (Chuvieco, 1993; Arthur and Nalle, 1997). For instance, functional relationships need to be evaluated over continuous scalar value domains. In resource management we often deal with categorical values and a variety of evaluation methods including nonlinear and logical relationships. In addition, many aggregate summary values such as zonal or neighborhood operations are difficult to incorporate in a mathematical programming solution.
        The next section will present a methodology to combine a linear combination model and an allocation model. While the solution is not generic at this stage, it could be extended to include a broad set of interactions and spatial distribution models. The methodology is demonstrated in the context of a forestry application.

4. Solution
This section presents a solution to the problem of coupling an allocation model with multiple criteria evaluation. Allocations are expressed as constraints, which are satisfied as part of the decision evaluation. The principle is shown in figure 1b. A model is developed using a LWC procedure to determine suitability scores S from the decision factors Ai. This suitability map is computed as part of a cartographic model. An allocation model is then run to determine which areas satisfy one or more constraints C. The suitability scores represent the preferred locations for use as the demand. A modified distance metric is used to help determine the closest supply points to service this demand. The interaction model serves to evaluate this demand-supply constraint. A solution using an iterative approach is suggested. An initial result is computed by allocating the most suitable areas to their nearest center. This also requires satisfying a constraint on the total amount of demand, or the total amount of the resource, that may be utilised. This gives a computed allocation  for the accumulated demands that are assigned to each center. This allocation is then checked against a constraint for the desired allocation . For instance, we may wish to evenly distribute the allocation between centers. Adjustments are then made to the weights in the interaction model to satisfy the desired allocations. Because of spatial dependencies in the criteria scores, i.e. due to zonal aggregations or the way proximity measures are computed, this causes slight changes in the suitability map. Hence the evaluation is repeated in an iterative fashion until there are no significant adjustments to be made. The result R reflects the most suitable sites that satisfy the constraint allocation. Adjustments are computed as: 

The iterative approach used is summarised as follows: Step 1. Compute a set of suitability scores using Equation (1)

Step 2. Compute an allocation for sites using Equation (2) and accumulate these to give the allocations at each center 

Step 3. Test resulting allocation against the constraint for desired allocation and compute adjusted weight values using Equation (3)


          Step 4. If Dwj changes significantly then:

i) go to step 1 if the input factors need re-computation
ii) go to step 2 otherwise 5. Case Study

The technique is demonstrated with a forest management case study. Management of forest resources involves considering many economic and environmental factors. The factors include not only cost effective and sustainable production, but also conservation and preservation of environmental habitats. Determining the best sites for timber harvesting is a problem that suits both multiple criteria evaluation and spatial modeling. Multiple criteria evaluation is used for decision analysis, and spatial modeling is used to compute costs for moving logs and their allocation to mills for sawing. Figure 3 shows a high level model of the problem. Forests are managed as discrete spatial units that represent the smallest area for making a decision. There are three mills in the region and it is highly desirable to allocate the workload evenly among these mills. The process of deciding the best sites for harvesting and preservation include evaluating the following:

  • enviro-indicator that reflects environmentally sensitive areas as determined by terrain characteristics (slope), and presence of old growth forests or rainforests
  • yield indicator that reflects volume of timber harvested
  • cost indicator that reflects relative costs to extract (skid) timber to the road and then transport (haul) logs to mills
  • allocation indicator that reflects how evenly the loads are distributed to each mill, it is assumed each mill has even milling capacity.

    •
    Figure 3: Forest study area and indicative factors used to evaluate areas for harvesting.

            A cartographic model was used for the computation of factors and indicators (McKendry et al., 1995). While these criteria and the subsequent factor development represent only a subset of what would be considered in a complete application, one can appreciate the amount of logical and computational modeling that takes place. The main aspect of the model we will focus on is the allocation for the mill. This is shown in Figure 4.

    Figure 4. Model for allocation
              The attractiveness of mills is weighted so loads are evenly distributed to each mill. Initially the weights are set equal. After computing loads the weights are adjusted appropriately. This cannot be computed in a functional sense, because the results depend upon accumulating individual loads along a road network. An iterative process is used where weights are adjusted proportionally, until the solution converges to the desired result. The weights are adjusted according to equation (3) where k = 2, in this case the weights are adjusted by half of what is indicated from proportioning the loads. Table 1 shows the results of the analysis, the calculations converge to a satisfactory solution after three iterations.

    Table 1: Iterative evaluation of constraint and subsequent adjustment of interaction model
    weights to satisfy constraint to evenly allocate resource to each mill.

    6. Conclusion
    The advantage of linear combination is its simplicity and flexibility to include spatial modeling in a multiple criteria evaluation. However it is still an essentially point-based evaluation method. That is, it computes a score at each location and as a post-procedure it priority ranks the highest valued locations to make up a prerequisite total land area. Linear combination is unable to accommodate constraints that involve spatial relationships and dependencies. For instance, a constraint to evenly allocate land resources to a set of target distribution centers. These problems are normally handled by mathematical programming solutions. But it would be desirable to include such constraints in the linear combination method. Two significant classes of spatial dependencies are desirable: 1) space allocation models, and 2) spatial distributions. Spatial constraints of this nature can be solved using a sub-optimal procedure that iteratively evaluates the models against a constraint until a stable balance is achieved.
            The paper describes a method to include spatial constraints for a resource allocation problem. The process is applied to a relatively complex case study for balancing production and environmental interests in managing commercial forests. The objective of production is to get the maximum timber yield from the forest in the most cost effective manner. Regulations control some environmental aspects of what can be harvested, e.g. terrain slopes, old growth habitat and hydrology. Costs relate to harvesting and transporting logs to the closest mill. Utilisation of the mills is also important so the milling workload must be approximately distributed evenly to several mill locations. The MCE analysis evaluates a cartographic model to determine suitable areas for logging, and then selects areas that satisfy a constraint to evenly allocate the resource to a number of mills.

    References
    Arthur J.L. and Nalle D.J. (1997) Clarification of the Use of Linear Programming and GIS For Land-Use Modelling, International Journal of Geographical Information Systems, 11(4), 397-402.

    Bailey T.C. and Gatrell A.C. (1995) Interactive Spatial Data Analysis, Longmann Scientific & Technical, Essex.

    Brookes C.J. (1997) A Parameterized Region-Growing Programme for Site Allocation on Raster Suitability Maps. International Journal of Geographical Information Systems, 11(4), 375-396.

    Carver S.J. (1991) Integrating Multi-Criteria Evaluation with Geographic Information Systems, International Journal of Geographical Information Systems, 5(3), 321-339.

    Chuvieco E. (1993) Integration of Linear Programming and GIS for Land-Use Modelling. International Journal of Geographical Information Systems, 7(1), 71-83.

    Eastman J.R., Jin W., Kyem A.K. and Toledano J. (1995) Raster Procedures for Multi-Criteria / Multi-Objective Decisions. Photogrammetric Engineering & Remote Sensing 61(5), 539-547.

    Fotheringham A.S. and O'Kelly M.E. (1989) Spatial Interaction Models : Formulations and Applications, Kluwer Academic Publishers, Dordrecht, Holland.

    Jankowski P. (1995). Integrating Geographical Information Systems and Multiple Criteria Decision-Making Methods. International Journal of Geographical Information Systems, 9(3), 251-273.

    McKendry J.E., Eastman J.R., St. Martin K. and Fulk M.A. (1995) Explorations in Geographical Information Systems Technology, UNEP/UNITAR, Clark University, Worchester.

    Roy J.R. and Anderson M. (1988) Assessing Impacts of Retail Development and Redevelopment, In: P. Newton, et al., Desktop Planning, Hargreen Publishers, Melbourne.

    Tomlin C.D. (1991) Cartographic Modelling, In: D. Macguire, M Goodchild and D. Rhind, Longman (eds) Geographical Information Systems, 1, 361-374.


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