A goal programming based heuristic method to obtaining interval weights in analytic form from interval multiplicative comparison matrices

This paper focuses on how to judge and measure the quality of interval judgments in an interval multiplicative comparison matrix (IMCM) and how to obtain optimal interval weights in analytic form from consistent IMCMs and heuristic based interval weights in analytic form from inconsistent IMCMs. The paper first analyzes two commonly used interval weight normalization models and illustrates that it is problematic to link any of the two models with interval weights obtained from an IMCM. A novel framework of normalized interval multiplicative weights (NIMWs) is then put forward and used to define consistent IMCMs. The paper presents important properties for uncertainty indices of consistent IMCMs and their associated NIMWs. Two goal programming (GP) models are subsequently developed to find NIMWs from consistent IMCMs, and the uniquely analytic solution of optimal NIMWs is obtained by the Lagrangian multiplier method. By devising heuristic rules and modifying constraints of the second GP model, two visualized and concise formulas are proposed to determine the lower and upper bounds of heuristic NIMWs from any IMCM. The paper shows that this heuristic based solution is the uniquely optimal solution if an IMCM is consistent. The obtained heuristic NIMWs in analytic form are further used to introduce a geometric consistency index for measuring the quality of interval judgments in an IMCM. Two numerical examples and comparative analyses are supplied to illustrate the rationality and effectiveness of the developed models.