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1 This is the published version of a paper published in International Journal of Risk Assessment and Management. Citation for the original published paper (version of record): Laryea, R., Carling, K., Cialani, C., Nyberg, R G. (2018) Sensitivity analysis of a risk classification model for food price volatility International Journal of Risk Assessment and Management, 21(4): Access to the published version may require subscription. N.B. When citing this work, cite the original published paper. Permanent link to this version:

2 374 Int. J. Risk Assessment and Management, Vol. 21, No. 4, 2018 Sensitivity analysis of a risk classification model for food price volatility Rueben Laryea*, Kenneth Carling, Catia Cialani and Roger G. Nyberg School of Technology and Business Studies, Dalarna University, Röda vägen 3, Borlänge, Sweden rla@du.se kca@du.se cci@du.se rny@du.se *Corresponding author Abstract: A sensitivity analysis to vary the weights of an accurate predictive classification model to produce a mixed model for ranking countries on the risk of food price volatility is carried out in this paper. The classification model is a marginal utility function consisting of multiple criteria. The aim of the sensitivity analysis is to derive a mixed model to be used in ranking of country alternatives to aid in policy formulation. Since in real-life situations the data that goes into decision making could be subjected to possibilities of alterations over time, it is essential to aid decision makers to vary the weights of the criteria using both subjective and objective information to introduce imprecision and to generate relative values of the criteria with a scale to form a mixed model. The mixed model can be used to rank future relative alternative value data sets for policy formulation. Keywords: risk; sensitivity analysis; multiple criteria; weights; decision maker; classification model; imprecision; uncertainty; data; price volatility. Reference to this paper should be made as follows: Laryea, R., Carling, K., Cialani, C. and Nyberg, R.G. (2018) Sensitivity analysis of a risk classification model for food price volatility, Int. J. Risk Assessment and Management, Vol. 21, No. 4, pp Biographical notes: Rueben Laryea is currently a PhD candidate in Microdata Analysis with the School of Technology and Business Studies of Dalarna University in Sweden. He received his Master s in Computer Engineering from the Mid Sweden University in Sundsvall Sweden in He has published in the International Journal of Applied Decision Sciences and the International Journal of Decision Sciences Risk and Management. His research interest is in the area of risk and decision analysis, operations research and big data analytics. Kenneth Carling is a Full Professor in Micro Data Analysis with the School of Technology and Business Studies of Dalarna University in Sweden. He received his PhD in Statistics from the Uppsala University 1995 and has been a Visiting Researcher at Padova, Yale and Tianjin Universities over the years. He has published broadly in economics, operations research, statistics, but is presently focused on research in mobility and transportation. He is a member of INFORMS and has served as the President for the Cramer Society. Copyright 2018 Inderscience Enterprises Ltd.

3 Sensitivity analysis of a risk classification model for food price volatility 375 Catia Cialani is a Senior Lecturer of Economics at Dalarna University. She received her PhD in Economics from the Umeå University, a MSc in Economics from the Dalarna University and a MSc in Economics from the University of Rome La Sapienza. She is Coordinator of MSc in Economics and formerly Economist at the Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA) and at Italian National Council of Economy and Labor (CNEL). Roger G. Nyberg works as a Researcher and Senior Lecturer at the Department of Computer Science/Informatics at Dalarna University, Sweden. He earned his PhD at Edinburgh Napier University, Edinburgh, Scotland, UK, by defending his PhD thesis Automating condition monitoring of vegetation on railway trackbeds and embankments, in His main research interests compromise how to automate/semi-automate human decision making in different contexts. His research focus includes machine learning, machine vision, pattern recognition in data, data science/microdata analysis/data analytics, monitoring/surveillance and more. 1 Introduction The incorporation of imprecision in risk and decision analysis emanates from the fact that the value functions on criteria may vary over time and there is the need to provide adequate room for this variation in our decision making processes as real life situations permit it. Techniques have been suggested in multi-criteria decision analysis to inculcate imprecision into value functions and multi-attribute utility theorem (MAUT) (Belton, 1999; Gold, 1987) which is based on the expected utility theorem which requires stronger assumptions to arrive at additivity and has the advantage of taking uncertainty into account and injecting it directly into a decision support model. On the other hand of MAUT is the multi-attribute value theory (MAVT) which has been dubbed the compensatory technique (Beinat and Nijkamp, 1998) because it allows the weak performance by one criterion to be compensated for by the strong performance of another criterion and further aggregates the performance of all the criteria to form an overall assessment of the criteria. So for any given objective in MAVT, one or more attributes are used to measure the performance in relation to the objective with the attributes usually measured on different measurement scales (Gold, 1987). The use of scales in eliciting the weights of criteria in multi-criteria decision analysis is a dependent process and requires methods which would provide a suitable relationship between the criteria and the resulting elicited weight from the scale. This has been a problem, with several scoring techniques having been suggested for multi-attribute choice scoring models, which permit the incorporation of both objective and subjective views in decision making. Techniques such as Simple Multi-Attribute Review Technique (SMART) (Edwards and Barron, 1994; Otway and Edwards, 1977) require the assignment of weights to attributes or criteria in comparison with other criteria using a scale. It must be observed that the utility of these methods is dependent on how the scales are constructed and how properly the weights are assigned. Further techniques have been advocated to ensure a smooth dependability of weights and scales, to ensure a trusting mathematical relationship between weights and scales, notably Clemen and Reilly (2001), Keeney and Raiffa (1976) and Wright and Goodwin (2009) and others have led to

4 376 R. Laryea et al. erroneous conclusions (Monat, 2009), because they tend to increase the small differences in the natural values of criteria. The work in this paper is motivated by the fact that the weights of the criteria on which the sensitivity analysis would be applied has already been developed by a utilites additives discriminantes (UTADIS) (Jacquet-Lagreze, 1999) classification methodological framework and judged to be accurate in the form of a predictive classification model in a previous research a food price volatility model for country risk classification (2017); thus the problems stated above with respect to the preference elicitation of the weights do not show up here. The previous research (a food price volatility model for country risk classification, submitted for publication, 2017) took into consideration the data on the price volatility of three food staple criteria of cassava, maize and potato. These criteria are judged to be the most consumed food staples by the United Nations of some selected countries alternatives and included the wealth Gini Index of the countries, to develop an accurate classification predictive model in the form of a marginal utility function or an additive utility model. The marginal utility function has weights that show the risk of volatility of the prices of food staple criteria. This paper would introduce imprecision into the criteria weights of the predictive classification model by using a scale to generate the relative values of the criteria using judgmental and pairwise method with the aim of deriving a mixed model to rank countries on food price volatility. The term mixed model is used in this paper because it emanates from a predictive classification model which is then infused with relative values through a sensitivity analysis process to arrive at a model for ranking country alternatives. It must be noted that the derivation of the relative values for the criteria on the scale is guided by the accurate weights of the criteria in a classification model already developed in the previous research (a food price volatility model for country risk classification, submitted for publication, 2017). Relative values for the alternatives are also generated and with the mixed model a ranking of the country alternatives is conducted for food price volatility risk. 2 Methodology The accurate predictive classification model generated by the UTADIS classification methodological framework in the paper (a food price volatility model for country risk classification, submitted for publication, 2017) is of the additive type and comes in the form of a marginal utility function with the constraint of all the weights adding up to one and has been reproduced below: u g u g u g u g The value coefficients as shown in the classification model above represent the accurate weights of the criteria maize, potato, cassava and wealth gini index. The Wealth Gini Index is a measure of income distribution to inequality in wealth distribution in a country, with 1 being complete inequality and 0 being complete equality. It helps define the gap between the rich and poor in a country. Countries may fall in a similar income category but there would exist a different balance of wealth distribution in the countries (a food price volatility model for country risk classification, submitted for publication, 2017). The food staple criteria have been judged by the United Nations to be the most consumed

5 Sensitivity analysis of a risk classification model for food price volatility 377 food staples of some selected countries. We would conduct a post-optimality analysis which would generate relative values of the food staple criteria and the country alternatives to rank the country alternatives with a mixed model. Generating relative values instead of actual values allows for a greater focus on sensitivity calculations (Triantaphyllou, 2000). To carry out a sensitivity analysis on our criteria weights to begin with, it is imperative that the weights are assessed to be accurate and validated because it would form the basis of carrying out a sensitivity analysis to generate the relative criteria values and for ranking of the country alternatives. The food price volatility classification model (a food price volatility model for country risk classification, submitted for publication, 2017) developed with the UTADIS optimisation has been explained and judged to be accurate and validated from paper (a food price volatility model for country risk classification, submitted for publication, 2017); this will be used in carrying out our sensitivity analysis. The coefficients of the model which form the weights of the criteria will guide the process of generating relative values for the criteria and by means of a paired comparison together with fixed point and judgement analysis from a scale. 2.1 Relative weights for criteria Assigning the relative weights of the criteria is done with pairwise comparison together with fixed point and judgement analysis. A scale value is used to initialise the calculation, which provides the pairwise comparison value of two criteria (Triantaphyllou, 2000). Below, the scale shows the relative importance and the values assigned. Judging and determining interval information from the scale values is not a mere guess work, but is informed by the coefficient values which were obtained from the marginal utility function of the UTADIS method. Decision-makers might in many cases have more knowledge of the decision situation, even if the information is not precise. For instance, cardinal importance relation information may implicitly exist. However, these cannot be taken into account in the transformation of an ordinal rank order into weights. Some form of cardinality often exists and this information should reasonably be used when transforming orderings into weights to utilise all the information the decision-maker is able to supply (Danielson and Ekenberg, 2012). We will therefore use paired comparison together with fixed point and judgement analysis from a scale. The idea is that the decision-maker will be able to express and utilise known differences in importance between the criteria. Figure 1 Scale for generating relative criteria weight values (see online version for colours) Assume that there exists an ordinal ranking of N criteria. In order to make this order into a cardinal ranking, information should be given about how much more or less important the criteria are compared to each other. Such rankings also take care of the problem with ordinal methods of handling criteria that are found to be equally important, i.e., resisting pure ordinal ranking (Danielson and Ekenberg, 2012). In this paper, we use the

6 378 R. Laryea et al. expressions shown in the scale in Figure 1 for the strength (cardinality) of the rankings between criteria. The data in Table 1 are the percentages of the coefficients for criteria or weights in the marginal utility function produced by the UTADIS method. Table 1 UTADIS criteria weights in percentage Maize Potato Cassava Wealth Gini index The assignment of the scale values in Table 2 is carried out by comparing the values of the criteria s percentages in Table 1 and then judge the relative importance with a scale value. In this way, decision maker s preferences on relative criteria strengths is elicited from judgements on the scale well informed by the criteria weights in Table 1. So for an example maize has a weight of 28% and potato has a weight of 23.8% from Table 1. It can be seen that maize is of slightly higher importance than potato, so on the scale, in eliciting preferences on the relative importance of the two criteria, a decision maker is able to judge the position of maize as being of slightly more importance than potato and therefore a value in the range of slightly more importance on the scale is assigned for the relationship between the two criteria as shown in Table 2. The process is carried out for the relative importance of the other criteria and displayed in Table 2. Table 2 Assigning scale values Maize Potato Cassava Wealth Gini index Maize Potato Cassava Wealth Gini index Total Table 3 is a normalisation of the values in Table 2. This is carried out by dividing each column cell value by the total cell values of the column and the average of the weights is thus tabulated in the average column of Table 3. Table 3 Normalised scale values showing the mixed model in the average column Maize Potato Cassava Wealth Gini index Average Maize Potato Cassava Wealth Gini Index The average column of Table 3 presents a form of a mixed classification model which was derived by allowing the judgmental preferences of the decision maker by means of a scale. This mixed classification model would, as we would see later on in the paper would be used in ranking the country alternatives by a process as shown later in Table 6.

7 Sensitivity analysis of a risk classification model for food price volatility Relative values for alternatives Table 4 is the price volatility for the countries obtained from the previous research; a food price volatility model for country risk classification. Table 4 Original price volatility values for the countries and food staples Country Maize Potato Cassava Wealth Gini index Malawi Angola Benin Burundi Cameroon Cape Verde Chad DRC Gabon Uganda The relative alternative values as presented in Table 5 are calculated as follows r min( cr) a max( cr) min( cr) (1) where r is the real alternative criteria value from Table 4, min(cr) is the minimum column value of the column that a particular criterion belongs to and max(cr) is the maximum column value of the column that a particular criterion belongs to. Thus applying equation (1) yields Table 5 Finally, the desired value and weight matrix are obtained as shown in Table 6. In this case, the average weights for each criterion are used to calculate the expected value for each alternative using equation (1). Table 5 Relative alternative values Country Maize Potato Cassava Wealth Gini index Malawi Angola Benin Burundi Cameroon Cape Verde Chad DRC Gabon Uganda

8 380 R. Laryea et al. Table 6 Relative alternative values with the mixed model of associated relative criteria average weight from Table 3 Country Maize (0.26) Potato(0.15) Cassava(0.56) Wealth Gini index(0.04) Malawi Angola Benin Burundi Cameroon Cape Verde Chad DRC Gabon Uganda Table 7 Aggregated value index and the ranking of the price volatility of the food staples for the countries Country Aggregated values Rank Malawi Angola Benin Burundi Cameroon Cape Verde Chad DRC Gabon Uganda A common approach in making informed judgements on weights for preference assessments is by specifying a set of attributes that represents the relevant aspects of the possible outcomes of a decision. Then value functions are defined over the alternatives for each attribute and a weight function is defined over the attribute set. One method is just to define a weight function by fixed numbers on a normalised scale as carried out above; and then define value functions over the alternatives, where these are mapped on fixed values as well, after which these values are aggregated and the overall score of each alternative is calculated. The most common form of value function used is the additive model from (Danielson and Ekenberg, 2012) as shown in equation (2) m Ua ( ) gu( a) (2) i 1 i i

9 Sensitivity analysis of a risk classification model for food price volatility 381 where U(a) is the aggregated value of alternative a, u i (a) is the value of the alternative under criterion i, and g i is the weight of this criterion or the mixed model value of the criteria. The criteria weights, i.e., the relative importance of the evaluation criteria, are thus a central concept in most of these methods and describe each criterions significance in the specific decision context (Danielson and Ekenberg, 2012). The resulting aggregated values for the alternatives tabulating the values in Table 6 with equation (2) is the ranking of alternatives as shown in Table 7. It can be seen from Table 7 that Gabon is ranked at the first position with an aggregated value of 0.88 followed by Uganda at the second position with an aggregated value of We have conducted this ranking by eliciting preferential judgments on criteria weights and conducting relative values calculations of the alternatives. The ordinal ranking of the country alternatives for food price volatility informs decision makers on which countries are at a higher risk of food price volatility considering imprecision in the classification model. By knowing the Gabon is the country with a very high risk of food price volatility, policies can be formulated to mitigate against this risk, moreover, the mixed model from Table 3 can be used for ranking future data sets on the relative values of alternatives as has been done in this paper and compared to see how the policies formulated in previous years has helped to changed the positions of the countries with respect to food price volatility risk ranking. 3 Conclusions This paper aims to develop a mixed model which allows imprecision in judgmental preferences from the decision maker by means of a scale. The term mixed model is used because it originates from a classification model which is re-invented with a sensitivity analysis through the generation of relative values with imprecision in the criteria weights on a scale to arrive at a model for ranking countries for food price volatility risk. The paper adopts a predictive classification model from an earlier research on food price volatility (a food price volatility model for country risk classification, submitted for publication, 2017) which consists of accurate weights of the criteria developed with the UTADIS methodological framework as explained in the previous research. This paper introduces imprecision into the criteria values by generating relative values for the criteria by means of a scale. The relative criteria values are normalised and then aggregated to derive a model of values. The relative values of the alternatives are also calculated and then scores for the alternatives are derived in the form of expected values calculated from the mixed model of the criteria values. The alternative scores are then used in ranking the alternatives to provide information to decision makers for policy formulation. The mixed model can be used to rank future data sets on the relative values of the alternatives as shown in this paper by evaluating the expected values and obtaining a score for the alternatives to be used for the ranking. Decision makers can then compare the future rankings with previous rankings to ascertain if their policies have been effective to change the positions of countries that are at high risk of food price volatility in the rankings.

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