Introduction to MCDM Slim Zekri Dept. Natural Resource Economics Sultan Qaboos University

Traditional paradigm for decision- making The traditional paradigm for analyzing decision making is based on 3 elements: 1. A decision maker 2.An array of feasible choices 3.A well-defined criterion (utility or profit) The criterion associates a number to each alternative Alternatives are ranked to find the optimal value for the criterion of choice The best (highest or lowest value) is then selected

LP shares the same paradigm Mathematical programming for decision making shares the same theoretical construct: 1.Establish a set of feasible solutions satisfying the constraints 2.Solutions are ordered according to the objective function representing the preferences of the DM 3.The optimum solution is obtained using mathematical procedure such as the simplex algorithm

Multiplicity of objectives Commercial farmer – Maximizing profit and – Minimizing Risk Subsistence farmer – Max cash – Max food security – Min Risk – Min work load

Multiple objectives are all around us Commercial farmer – Maximizing profit – Minimizing Risk – Minimize debt Subsistence farmer – Max cash – Max food security – Min Risk – Min work load Wild fish management – Maximizing rent – Ensuring sustainability – Ensuring biodiversity Engineering and design – Maximize car power – Minimize fuel consumption – Maximize safety – Comfort

Rigid constraints in LP The traditional paradigm assumes that constraints are rigid and cannot be violated under any circumstances In many situations it’s possible to accept a certain amount of violation of at least some of the constraints when the technical knowledge is not precise enough – Examples: 1.Formulation of livestock rations 2.Selection of fertilizer combinations

Economic versus technological decisions Friedman (1962) “decision problems involving a single criterion should be regarded as technological problems” Zeleny (1982) “technological problems consist only of the processes of search and measurement” – This can be undertaken using simple tools or very sophisticated methods In technological problem there is no decision making – Only search and ranking decision-making arises only when several criteria are considered

Several Criteria Single Criterion Scarce Resources Economic Problem Technological Problem No ScarcityNo problem ('Nirvana')

Wheeler and Russell (1977) were the first to introduce several goals in a farm level decision making model in agriculture. However the formalization of MCDM in the agricultural sector started in the late 1980’s

Attributes, objectives and goals An attribute is defined as a decision maker's values related to reality and is expressed as a mathematical function F(x) of the decision variables – Profit, Risk, pollution are attributes An objective represents direction of improvement of an attribute – 'the more of the attribute, the better' or Max F(x) – 'the less of the attribute, the better‘ Min F(x)

An aspiration level or a Target: is an acceptable level of achievement for an attribute – The combination of an attribute with a target gives a goal – F(x) = t; F(x) >t; F(x) <t Gross margin is an attribute, Maximize gross margin is an objective Achieve a gross margin of at least $200,000 is a goal A criterion encompasses the 3 preceding concepts: criteria are Attributes, Objectives or Goals

Distinction between a goal and a constraint We defined a goal as the association of an attribute with a target F(x) = t; F(x) >t; F(x) <t – Mathematically speaking they are the same except for the interpretation of the RHS In a constraint the RHS has to be satisfied otherwise the solution is unfeasible In a goal the RHS, t, is a target that we aspire to achieve but might or might not be achieved

Goal The amount of violation can be measured by introducing positive and negative deviational variables – Attribute + Deviational variables = Target – F(x) + n – p = t – Example: Achieve a Profit of 2,000,000 – 1 000x x 2 + n - p = 2,000,000 – Then Minimize n If n= 300,000 then the achievement is 1,700,000 If p= 200,000 then the achievement is 2,200,000

Pareto optimality Pareto optimality plays a vital role in economic theory. It is the corner stone of MCDM In MCDM there is not a single “Optimal” solution, but a set of Pareto Optimal Solutions – Feasible solutions such that no other feasible solution can achieve the same or better performance for all the criteria under consideration and strictly better for at least one criterion

Example Assume the DM Max GM; Min Labour and Min Debt Compare the 3 solutions S1; S2 and S3 below

Trade-offs A Pareto optimal solution is a feasible solution for which an increase in the value of one criterion can only be achieved by degrading the value of at least one other criterion Trade-off: the amount of achievement of one criterion that must be sacrificed to gain a unitary increase in the other criterion Trade-off S 1,3 = ₤500/hour it is an opportunity cost