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Supply chain management is a multidisciplinary engineering problem. In this paper, a polymorphic uncertain equilibrium model (PUEM) is constructed to capture the joint maximization of the profits for the manufacturers and retailers in a supply chain network. To ensure applicability of the model in practice, the demand of the consumers is regarded as a continuous random variable, the holding cost of the retailer and the transaction cost between the manufacturer and retailer are described by fuzzy sets. For the PUEM, a deterministic equivalent formulation (DEF) is first derived by compromise programming approach such that the existing powerful algorithms in the standard smooth optimization are employed to find an approximate equilibrium point for the uncertain problem. Actually, the DEF turns out to be a nonlinear complementarity problem (NCP), a special variational inequality. Thus, a modified partially Jacobian smoothing algorithm is developed to solve the corresponding NCP, where the gradient information of the model is used to efficiently generate search direction. Sensitivity analysis offers a number of useful managerial implications based on practical applications of the model.

Computing traffic equilibria with a general nonadditive route cost disutility function is considered in this paper. Following the user equilibrium (UE) condition, that is, no driver can unilaterally change route to achieve less travel costs, the traffic equilibrium problem (TEP) can be formulated as a nonlinear complementary problem (NCP). In this paper, we propose a semismooth Newton method with a penalized Fischer–Burmeister (PFB) NCP function to solve the NCP formulation of the TEP, and also, we investigate the properties of the proposed method. Numerical results are provided and compared with the classical TEP with additive route cost functions. The results show the algorithm can achieved substantially better performance than the existing approaches. A sensitivity analysis is also conducted to examine the parameter of the proposed nonadditive route cost function.

The nonlinear complementarity problem can be reformulated as a nonlinear programming. For solving nonlinear programming, sequential quadratic programming (SQP) type method is very effective. Moreover, filter method, for its good numerical results, are extensively studied to handle nonlinear programming problems recently. In this paper, a modified quadratic subproblem is proposed. Based on it, we employ filter technique to tackle nonlinear complementarity problem. This method has no demand on initial point. The restoration phase, which is always used in traditional filter method, is not needed. Global convergence results of the proposed algorithm are established under suitable conditions. Some numerical results are reported in this paper.