Equivalence of optimal forecast combinations under affine constraints

Forecasts are usually produced from models and expert judgements. The reconciliation of different forecasts presents an interesting challenge for managerial decisions. Mean absolute deviations and mean squared errors scoring rules are commonly employed as the criteria of optimality to aggregate or combine multiple forecasts into a consensus forecast. While much is known about mean squared errors in the context of forecast combination, little attention has been given to the mean absolute deviation. This paper establishes the first-order condition and the optimal solutions from minimizing mean absolute deviation. With this result, the paper derives the conditions in which the optimal solutions for minimizing mean absolute deviation and mean squared error loss functions are equivalent. More generally, this paper derives a sufficient condition which ensures the equivalence of optimal solutions of minimizing different loss functions under the same affine constraint that each feasible solution must sum to one. A simulation study and an illustration using expert forecasts data corroborate the theoretical findings. Interestingly, the numerical analysis shows that even with skewness in the data, the equivalence is unaffected. However, when outliers are presented in the data, mean absolute deviation is more robust than the mean squared error in small samples, which is consistent with the conventional belief relating the two loss functions

On the weighting of the mean-absolute-deviation cost minimization model On the weighting of the mean-absolute-deviation cost minimization model

Abstract The mean-absolute-deviation cost minimization model, which aims to minimize sum of the mean value and the absolute deviation (AD) of the total cost multiplied by a given non-negative weighting, is one of a number of typical robust optimization models. This paper first uses a straightforward example to show that the solution obtained by this model with some weightings is not actually an optimal decision. This example also illustrates that the mean-absolute-deviation cost minimization model cannot be regarded as the conventional weighted transformation of the relevant multiobjective minimization model aiming to simultaneously minimize the mean value and AD. This paper further proves that the optimal solution obtained by the meanabsolute-deviation cost minimization model with the weighting not exceeding 0.5 will not be absolutely dominated by any other solution. This tight upper bound provides a useful guideline for practical applications. Abstract The mean-absolute-deviation cost minimization model, which aims to minimize sum of the mean value and the absolute deviation of the total cost multiplied by a given non-negative weighting, is one of the typical robust optimization models. This paper first uses a straightforward example to show that the solution obtained by this model with some weightings is actual not an optimal decision. This example also illustrates that the meanabsolute-deviation cost minimization model cannot be regarded as the conventional weighted transformation of the relevant multiobjective minimization model aiming to simultaneously minimize the mean value and absolute deviation. This paper further proves that the optimal solution obtained by the mean-absolute-deviation cost minimization model with the weighting not exceeding 0.5 will not be absolutely dominated by any other solution. This tight upper bound provides a useful guideline for practical applications

On the weighting of the mean-absolute-deviation cost minimization model

Journal of the Operational Research Society644622-628JORS

CONTAINER TRANSPORTATION NETWORK MODELING AND OPTIMIZATION

Ph.DDOCTOR OF PHILOSOPH