Step 7. Sensitivity and robustness analysis
3.3 Multi-criteria decision analysis techniques
3.3.1 Multiobjective optimisation
Multiobjective optimisation is an MCDA technique for solving problems with multiple objective functions that must be maximized or minimized. The method generates a set of answers that define the optimum compromise between competing goals. According to Miettinen (2012), there are four types of multiobjective optimisation techniques: no preference, ! #$%&$%,
! #&'()$%&$% and interactive multiobjective optimisation techniques.
In ! #$%&$%, the decision-maker expresses preference information prior to the optimisation.
Common examples of ! #$%&$% techniques include the lexicographic, goal programming and the utility function. (Hwang and Masud, 2012). In a ! #&'()$%&$%, the decision-maker expresses preference information after being informed about the trade-offs among non-dominated solutions (Ibid). Using the interactive technique, the objective functions and constraints and their prioritization are obtained by requesting user feedback on preferences at multiple points during the execution of an algorithm (Ibid).
In literature, a crucial distinction is made between implicit and explicit approaches in multi- objective optimisation (Andonegi et al., 2021). The implicit approach uses information obtained from the decision-maker(s) during the solution process (Geoffrion et al., 1972). The implicit approach is based on of ! #$%&$% technique discussed above, where the decision-maker(s) defines the aggregation function (Geoffrion et al., 1972). A well-known implicit technique is the GP approach.
Accounting for multiple objectives in combinatorial problems and at the same time finding Pareto-optimal solutions is hugely challenging. The use of explicit approaches is regarded as one of the ways available to deal with this challenge. Joseph et al. (2009) and Carwardine (2012) encouraged the use of explicit approaches such as the value function technique by empirically asking experts to avoid the mathematical challenge associated with implicit approaches (Lahdelma et al., 2000; Mendoza and Martins, 2006).
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46 3.3.2 Goal programming techniques
GP models are premised on a decision analyst working with the decision-maker(s) to determine the alternatives that are considered closest to the achievement of a pre-determined goal (Hwang and Yoon, 1981; Yoon, 1987; Hwang et al., 1993). Figure 13 presents a graphical representation of GP topics according to Tamiz et al. (1995). A key element of GP formulations is the achievement function that measures the degree of minimization of the unwanted deviation variables of the goals considered in the model. Each type of achievement function leads to a different GP variant. The three oldest and still most widely used forms of achievement functions are the weighted (Archimedean), pre-emptive (lexicographic) and MINMAX (Chebychev) (Tamiz et al., 1995).
Figure 13: Graphical representation of goal programming topics
Source: Tamiz et al, 1995
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47 GP technique is an extension of standard linear programming in which desirable levels of performance (labelled *!) are specified by decision-makers for each objective + (+ = 1, … +) given set of constraints (Charnes and Cooper, 1961; Rifai, 1996). In obtaining any feasible solution, non-negative deviational variables are introduced, defined as 1!" and 1!#. The objective in GP is to minimize the (weighted) sum of undesirable deviations as 1!" or 1!# (depending on the goal). For each goal (*!), at least one of the deviational variables must be equal to “0”, such that 2!+ 1!# − 1!"= *!; with z representing objectives k representing criteria/objectives which may be scenario related. The 1# and the 1" superscripts indicate whether a goal is to be minimised, over, or underachievement of the goal priority. An optimal solution is attained when all the goals are reached as close as possible to their aspiration level while satisfying a set of constraints (Charnes and Cooper, 1961).
In some cases, decision analysts incorporate the simplified Chebychev Theorem into the analysis. Proponents (Charnes and Cooper, 1957; Benayoun et al., 1971; Yu and Zeleny 1975; Hwang et al. 1993), of the GP models, posit that the approach is best suited for the use of interactivity. In the linear programming technique postulated by Benayoun et al. (1971), a specific goal determines the ideal solution for each criterion. The technique uses Chebychev Theorem formulations. The formulation uses a relative range of normalised values for the weights converted to a convenient scale, e.g., 1-10.
Benayoun et al. (1971) advocated that the concept of optimum solution be replaced with the concept of best compromise solution. This technique is referred to as the STEP technique (STEM). The Chebychev distance between an ideal point and the criterion space is reduced using the STEM technique. To alter the parameters of the distance formula and the feasible space, a normalized weighting strategy based on the decision-preferences decision-makers in the prior solution can be utilized. The STEM technique allows the decision-maker to find solutions and the relative importance of the goals.
According to Belton and Gear (1997), relative importance is the per cent improvement over the most important predictor. Relative importance is calculated by dividing each variable importance score by the variables' largest importance score, then multiplying by 100 per cent (Ibid). At each iteration, the decision-maker can improve some objectives while sacrificing others. In addition, the decision-maker must state the maximum amount by which the goal functions can be compromised, while trade-offs on other factors must be considered.
Another variation of the GP approach is the Technique for Order Preference by Similarity to Ideal Solutions (TOPSIS) technique. Hwang and Yoon (1981) first proposed the TOPSIS
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48 technique in 1981. The essential principle in TOPSIS is that the chosen alternative should be the furthest from the perfect solution and the closest to the negative-ideal solution (Figure 14).
Figure 14 : Basic concept of TOPSIS method
Key: A+: Ideal point, A−: Negative-ideal point Source: Yoon (1980); Hwang and Yoon (1981)
The non-ideal option adjoining the ideal alternative is selected. The GP procedure is well- suited for the use of interactivity. Because of these properties, the technique has been used in unravelling energy resource planning problems (Ramanathan and Ganesh, 1994; Cherni et al. (2007). The GP technique is capable of dealing with large-scale issues. The technique is also suitable in cases where the number of alternatives available is very large, and the primary goal is narrowing the search for fewer options. GP models, therefore, can handle large-scale problems.