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The Shapley value, one of the most common solution concepts of cooperative game theory is defined and axiomatically characterized in different game-theoretic models. Certainly, the Shapley value can be used in interesting sharing cost/reward problems in the Operations Research area such as connection, routing, scheduling, production and inventory situations. In this paper, we focus on the Shapley value for cooperative games, where the set of players is finite and the coalition values are interval grey numbers. The central question in this paper is how to characterize the grey Shapley value. In this context, we present two alternative axiomatic characterizations. First, we characterize the grey Shapley value using the properties of efficiency, symmetry and strong monotonicity. Second, we characterize the grey Shapley value by using the grey dividends.

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This paper explains a model for analyzing and measuring the propagation of order amplifications (i.e. bullwhip effect) for a single-product supply network topology considering exogenous uncertainty and linear and time-invariant inventory management policies for network entities. The stream of orders placed by each entity of the network is characterized assuming customer demand is ergodic. In fact, we propose an exact formula in order to measure the bullwhip effect in the addressed supply network topology considering the system in Markovian chain framework and presenting a matrix of network member relationships and relevant order sequences. The formula turns out using a mathematical method called frequency domain analysis. The major contribution of this paper is analyzing the bullwhip effect considering exogenous uncertainty in supply networks and using the Fourier transform in order to simplify the relevant calculations. We present a number of numerical examples to assess the analytical results accuracy in quantifying the bullwhip effect.

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Occurrence of natural disaster inflicts irreparable injuries and symptoms on humans. In such conditions, affected people are waiting for medical services and relief commodities. Thus, quick reaction of medical services and relief commodities supply play important roles in improving natural disaster management. In this paper, a multi-objective non-linear credibility-based fuzzy mathematical programming model under uncertainty conditions is presented, which considers two vital needs in disaster time including medical services and relief commodities through location of hospitals, transfer points, and location routing of relief depots. The proposed model approaches reality by considering time, cost, failures probability in routes, and parameters uncertainty. The problem is first linearized and then global criterion method is applied for solving the multi objective model. Moreover, to illustrate model efficiency, a case study is performed on region 1 of Tehran city for earthquake disaster. Results demonstrate that if Decision-makers want to meet uncertainty with lowered risk, they have to choose a high minimum constraint feasibility degree even though the objective function will be worse.