Dynamics and equilibria of ecological predator-prey networks as nature’s supply chains

In this paper, we develop a dynamic network model of ecological food webs and prove that the set of stationary points of the projected dynamical system coincides with the set of solutions of a variational inequality governing the equilibrium of predator-prey networks. We also establish the equivalence between the ecological models and supply chain network equilibrium models and highlight the connections to spatial price equilibrium problems. We propose an algorithmic scheme, provide convergence results, and apply it to a food web drawn from a fisheries application.This paper is a contribution to the interdisciplinary supply chain network literature.     

A supply chain network equilibrium model

In this paper, an equilibrium model of a competitive supply chain network is developed. Such a model is sufficiently general to handle many decision-makers and their independent behaviors. The network structure of the supply chain is identified and equilibrium conditions are derived. A finite-dimensional variational inequality formulation is established. Qualitative properties of the equilibrium model and numerical examples are given.     

Supply chain network competition in price and quality with multiple manufacturers and freight service providers

In this paper, we develop both static and dynamic supply chain network models with multiple manufacturers and freight service providers competing on price and quality. The manufacturers compete with one another in terms of price and quality of the product manufactured, whereas the freight service providers compete on price and quality of the transportation service they provide for multiple modes. Both manufacturers and freight service providers maximize their utilities (profits) while considering the consequences of the competitors’ prices and quality levels. Bounds on prices and quality levels are included that have relevant policy-related implications. The governing equilibrium conditions of the static model are formulated as a variational inequality problem. The underlying dynamics are then described, with the stationary point corresponding to the variational inequality solution. An algorithm which provides a discrete-time adjustment process and tracks the evolution of the quality levels and prices over time is proposed, and convergence results given. Numerical examples illustrate how such a supply chain network framework, which is relevant to products ranging from high value to low value ones, can be applied in practice.     

Supply chain network design under profit maximization and oligopolistic competition

In this paper, we model the supply chain network design problem with oligopolistic firms who are involved in the competitive production, storage, and distribution of a homogeneous product to multiple demand markets. The profit-maximizing firms select both the capacities associated with the various supply chain network activities as well as the product quantities. We formulate the governing Nash–Cournot equilibrium conditions as a variational inequality problem and identify several special cases of the model, notably, a generalization of a spatial oligopoly and a classical oligopoly problem to include design capacity variables. The proposed computational approach, which is based on projected dynamical systems, fully exploits the network structure of the problems and yields closed form solutions at each iteration. In order to illustrate the modeling framework and the algorithm, we also provide solutions to a spectrum of numerical supply chain network oligopoly design examples.This paper makes a contribution to game theoretic modeling of competitive supply chain network design problems in an oligopolistic setting.     

Evolutionary variational inequalities and applications to complex dynamic multi-level models

In this paper we consider a supply chain network model with three tiers of decision-makers (manufacturers, retailers, and consumers) in the case when prices and shipments are evolving on time. Moreover, we assume that excesses of production and excesses of demand of the commodity are present. For such a framework we furnish, using the infinite dimensional duality theory, the equilibrium conditions for the representatives of each tier of the supernetwork, the time-dependent variational formulation governing the complete supply chain supernetwork, and we provide some existence theorems and a numerical example.     

A note on supply chain network equilibrium models

This paper is concerned with the supply chain network equilibrium models proposed by Nagurney et al. [Nagurney, A., Dong, J., Zhang, D., 2002. A supply chain network equilibrium model. Transportation Research 38E, 281–303] and Dong et al. [Dong, J., Zhang, D., Nagurney, A., 2004. A supply chain network equilibrium model with random demands. European Journal of Operational Research 156, 194–212]. It demonstrates that these models possess the unconstrained continuously differentiable minimization formulations, whose any stationary point is the solution of the corresponding model. Accordingly, not only is the Quasi-Newton algorithm capable of finding a solution of the model, but also it can overcome the difficulty experienced by the modified projection method in choosing an appropriate predetermined step size. In addition, 11 benchmark examples are employed to show the advantage of the unconstrained minimization formulation.     

The optimization of the closed-loop supply chain network

This paper develops a model of a general closed-loop supply chain network, which includes raw material suppliers, manufacturers, retailers, consumers and recovery centers. The objective of this paper is to formulate and optimize the equilibrium state of the network by using the theory of variational inequalities. Several examples and figures are used to describe the effects of parameters (the return ratio, the transformation rate of raw materials, and the transformation rate of recyclable products) on the equilibrium shipments and net revenues.     

A location-inventory-pricing model in a supply chain distribution network with price-sensitive demands and inventory-capacity constraints

This paper presents a location-inventory-pricing model for designing the distribution network of a supply chain with price-sensitive demands and inventory-capacity constraints. The supply chain has market power and uses markup pricing. An efficient Lagrangian relaxation algorithm is proposed to solve the model. Our numerical study shows that by moderately increasing the number of possible values for pricing decisions, the model can be used to find near-optimal solutions of a similar location-inventory-pricing problem with continuous pricing decisions. The approach used here to incorporate pricing decisions can be applied to other supply-chain design and planning problems with price-sensitive demands.     

A supply chain-transport supernetwork equilibrium model with the behaviour of freight carriers

This paper presents a supernetwork equilibrium model integrating supply chain networks with a transport network, namely, a supply chain-transport supernetwork equilibrium model. The model takes into account the behaviour of freight carriers and transport network users to endogenously determine the transport costs generated in the supply chain networks. The interaction between transport network and supply chain networks can also be examined. Results of the numerical tests reveal that the improvement of transport network could enhance the efficiency of supply chain networks. The paper makes contributions to modelling of supply chain networks as well as to that of transport networks.     

Dynamic supply chain network design with capacity planning and multi-period pricing

This paper addresses a new problem in designing and planning a multi-echelon and multi-product supply chain network over a multi-period horizon in which customer zones have price-sensitive demands. Based on price-demand relationships, a generic method is presented to obtain price levels for products and then, a mixed-integer linear programming model is developed. Due to the problem intractability, a simulated annealing algorithm that uses some developed linear relaxation-based heuristics for capacity planning and pricing is presented. Numerical results demonstrate the significance of the model as well as the efficiency of the solution algorithm and linear relaxation-based heuristics.     

Benders’ decomposition for concurrent redesign of forward and closed-loop supply chain network with demand and return uncertainties

This paper attempts to design a reverse supply chain network (SCN), add it to an existing multi-product forward SCN and simultaneously redesign the existing forward supply chain (SC). The problem considers uncertainty on products demand and and also returned products in multi-period context. Benders’ decomposition is applied to solve the stochastic mixed-integer model to optimality. The scenarios are generated based on the demand distribution function using Cholesky’s factorization method to consider correlation among different products’ demands. To decrease the computational effort, the number of scenarios is reduced using k-means clustering algorithm. The method is tested on a cell phone SC.     

Freight transport network design using particle swarm optimisation in supply chain–transport supernetwork equilibrium

This paper presents a discrete network design problem for optimally designing freight transport network in terms of the efficiency of supply chain. Modelling is undertaken within the framework of mathematical programmes with equilibrium constraints, which first incorporates both supply chain and transport networks explicitly. The upper level determines the best set of actions for transport network improvement, while the lower-level decision is based on a supply chain–multimodal transport supernetwork equilibrium. New variants of particle swarm optimisation are developed to approximately solve the upper level. Numerical tests reveal their superior performance and the effective freight transport-related actions.     

Reliable design of an integrated supply chain with expedited shipments under disruption risks

This paper studied the design of a two-echelon supply chain where a set of suppliers serve a set of terminals that receive uncertain customer demands. In particular, we considered probabilistic transportation disruptions that may halt product supply from certain suppliers. We formulated this problem into an integer nonlinear program to determine the optimal system design that minimizes the expected total cost. A customized solution algorithm based on Lagrangian relaxation was developed to efficiently solve this model. Several numerical examples were conducted to test the proposed model and draw managerial insights into how the key parameters affect the optimal system design.     

A supply chain generalized network oligopoly model for pharmaceuticals under brand differentiation and perishability

In this paper, we construct a generalized network oligopoly model with arc multipliers for supply chains of pharmaceutical products using variational inequality theory. The model captures the Cournot competition among the manufacturers who seek to determine their profit-maximizing product flows, which can be perishable, with the consumers differentiating among the products of the firms, whether branded or generic, and the firms taking into consideration the discarding costs. The numerical examples demonstrate that a brand pharmaceutical product may lose its dominant market share as a consequence of patent rights expiration and because of generic competition.     

Analyzing the impact of intermodal-related risk to the design and management of biofuel supply chain

This study presents a mathematical model that designs a reliable multi-modal transportation network for a biofuel supply chain system, where intermodal hubs are subject to site-dependent probabilistic disruptions. The disruption probabilities of intermodal hubs are estimated by using a probabilistic model which is developed using real world data. We developed an accelerated Benders decomposition algorithm to solve this challenging NP-hard problem. Numerical analysis show that the model selects to use intermodal hubs located in areas with low disruption probabilities. In case of a disaster, the reliable solution results in 6.21% savings over the minimum cost solution.     

The equilibrium of closed-loop supply chain supernetwork with time-dependent parameters

In this paper, we develop a closed-loop supply chain supernetwork model including suppliers, manufacturers, retailers and consumers at demand market, in which the demand for product is seasonal (t), and the sensitivity (w) of demand to price is another key factor which effects consumers’ demand. Moreover, the manufacturers invest the reverse distribution channel for incenting consumers to return more used products. Based on the Evolutionary Variational Inequalities (EVI) theory and Projected Dynamical Systems (PDS), the equilibrium condition of closed-loop supply chain is formulated, and the model is verified reasonably by numerical example.     

Analysis of traffic congestion costs in a competitive supply chain

We study a competitive facility location problem with identical suppliers competing in multiple markets subject to distribution network congestion. We first analyze the solution of a symmetric competitive location game and discuss the firms’ facility location and equilibrium market supply decisions. Then, we utilize the model to provide analytical characterization of the effects of traffic congestion costs on equilibrium distribution flows. These analytical results permit qualitative characterization of how changes in congestion levels and costs affect equilibrium location and distribution decisions. The results of extensive numerical studies serve to further illustrate the effects of traffic congestion costs on location, market supply quantity, and distribution decisions.     

A hybrid decomposition algorithm for designing a multi-modal transportation network under biomass supply uncertainty

This study presents a two-stage stochastic programming model for the design and management of a biomass co-firing supply chain network under feedstock supply uncertainty. To represent a more realistic case, we generate scenarios from prediction errors of the historical and forecasted biomass supply availabilities. We solve the model using a hybrid decomposition algorithm that combines Sample average approximation with an enhanced Progressive hedging algorithm. The proposed algorithm is validated via a real-world case study using data from Mississippi and Alabama. Computational results indicate that the proposed algorithm is capable of producing high quality solutions in a reasonable amount of time.     

Optimizing inventory decisions in a multi-stage multi-customer supply chain: A note

In this paper, we propose an n-stage-multi-customer supply chain inventory model where there is a company that can supply products to several customers. We formulate the model for the simplest inventory coordination mechanism which is referred to as the same cycle time for all companies in the supply chain. We conclude that it is possible to use an algebraic approach to optimize the supply chain model without the use of differential calculus.     

Efficiency of the plate-number-based traffic rationing in general networks

Road space rationing based on vehicle plate numbers restricts vehicle access to a network based upon the license number on pre-established days. It has been used in some large cities especially when there are some major events. This paper analyzes the efficiency of road space rationing schemes by establishing the bounds of the reduction in the system cost associated with the restricted flow pattern at user equilibrium in comparison with the system cost at the original user equilibrium. The bounds are established under the general traffic equilibrium model formulated as variational inequalities and illustrated with a simple example.