A Fuzzy Multidimensional Multiple-Choice Knapsack - Dr. Madjid ...

464 Ann Oper Res (2013) 206:449–483Fig. 5The DMU representation of a non-dominated solution3.3 Phase III: Pruning procedureThe selection of the final solution from the resultant non-dominated solutions is a difficulttask. In the previous section, both methods were used to generate a set of non-dominatedsolutions. In circumstances where the DM has no knowledge of the properties of the objectivesand the posterior articulation of the DM’s preferences, a systematic approach is neededto facilitate the final selection process. A pruning method such as Data Envelopment Analysis(DEA) can be used to decrease the final number of non-dominated solutions. DEA is amethodology for evaluating and measuring the relative efficiencies of a set of decision makingunits (DMUs) that use multiple inputs to produce multiple outputs. The DEA method isbased on the economic notion of Pareto optimality, which states that a DMU is considered tobe inefficient if some other DMUs can produce at least the same amount of output with lessof same input and not more of any other inputs. Otherwise, a DMU is considered to be Paretoefficient. Due to its solid mathematical basis and wide applications to real-world problems,much effort has been devoted to the DEA models since the pioneering work of Charnes et al.(1978). Each non-dominated solution of the modified NSGA-II algorithm is considered as aDMU with two inputs (i.e. cost and time) and one output (i.e. profit). Figure 5 presents theschematic view of a DMU.Let us assume that there are a set of n DMUs (DMU j , j = 1,...,n) producing s outputs(y rj , r = 1,...,s) by consuming m inputs (x ij , i = 1,...,m). We use the followingadditive model proposed by Charnes et al. (1982) to check whether a given DMU (i.e.,a non-dominated solution) is efficient.m∑ s∑max S − ip +i=1S rp+r=1s.t.n∑λ j x ij + S − ip = x ip,j=1i = 1, 2,...,mn∑λ j y rj − S rp + = y rp,j=1n∑λ j = 1j=1λ j ,S − ip ,S+ rp ≥ 0, r = 1, 2,...,s(15)where, S − ip and S+ rp are the input and output slacks and DMU p is efficient under the additiveModel (15) if and only if the optimal value of its objective function is zero.