A Splitting Equilibration Algorithm for the Computation of Large-Scale Constrained Matrix Problems; Theoretical Analysis and Applications

In this paper we introduce a general parallelizable computational method for solving a wide spectrum of constrained matrix problems. The constrained matrix problem is a core problem in numerous applications in economics. These include the estimation of input/output tables, trade tables, and social/national accounts, and the projection of migration flows over space and time. The constrained matrix problem, so named by Bacharach, is to compute the best possible estimate X of an unknown matrix, given some information to constrain the solution set, and requiring either that the matrix X be a minimum distance from a given matrix, or that X be a functional form of another matrix. In real-world applications, the matrix X is often very large (several hundred to several thousand rows and columns), with the resulting constrained matrix problem larger still (with the number of variables on the order of the square of the number of rows/columns; typically, in the hundreds of thousands to millions). In the classical setting, the row and column totals are known and fixed, and the individual entries nonnegative. However, in certain applications, the row and column totals need not be known a priori, but must be estimated, as well. Furthermore, additional objective and subjective inputs are often incorporated within the model to better represent the application at hand. It is the solution of this broad class of large-scale constrained matrix problems in a timely fashion that we address in this paper. The constrained matrix problem has become a standard modelling tool among researchers and practitioners in economics. Therefore, the need for a unifying, robust, and efficient computational procedure for solving constrained matrix problems is of importance. Here we introduce a.n algorithm, the splitting equilibration algorithm, for computing the entire class of constrained matrix problems. This algorithm is not only theoretically justiflid, hilt l'n fi,1 vl Pnitsf htnh thP lilnlprxing s-trlrtilre of thpCp !arop-Cspe mrnhlem nn the advantages offered by state-of-the-art computer architectures, while simultaneously enhancing the modelling flexibility. In particular, we utilize some recent results from variational inequality theory, to construct a splitting equilibration algorithm which splits the spectrum of constrained matrix problems into series of row/column equilibrium subproblems. Each such constructed subproblem, due to its special structure, can, in turn, be solved simultaneously via exact equilibration in closed form. Thus each subproblem can be allocated to a distinct processor. \We also present numerical results when the splitting equilibration algorithm is implemented in a serial, and then in a parallel environment. The algorithm is tested against another much-cited algorithm and applied to input/output tables, social accounting matrices, and migration tables. The computational results illustrate the efficacy of this approach

Supply Chain Network Competition in Time-Sensitive Markets

We develop a game theory model for supply chain network competition in time-sensitive markets in which consumers respond to the average delivery time associated with the various firms’ products. The firms’ behavior is captured, along with the supply chain network topologies, with the governing equilibrium concept being that of Nash equilibrium. We derive the variational inequality formulation of the equilibrium conditions and provide illustrative examples. We also identify special cases for distinct applications. An algorithm is proposed, and the framework further illustrated through a case study in which we explore varying sensitivities to the average time delivery with interesting results

Parallel Computation of Large-Scale Dynamic Market Network Equilibria via Time Period Decomposition

In this paper we consider a dynamic market equilibrium problem over a finite time horizon in which a commodity is produced, consumed, traded, and inventoried over space and time. We first formulate the problem as a network equilibrium problem and derive the variational inequality formulation of the problem. We then propose a parallel decomposition algorithm which decomposes the large-scale problem into T + 1 subproblems, where T denotes the number of time periods. Each of these subproblems can then be solved simultaneously, that is, in parallel, on distinct processors. We provide computational results on linear separable problems and on nonlinear asymmetric problems when the algorithm is implemented in a serial and then in a parallel environment. The numerical results establish that the algorithm is linear in the number of time periods. This research demonstrates that this new formulation of dynamic market problems and decomposition procedure considerably expands the size of problems that are now feasible to solve

Parallel Computation of Large-Scale Nonlinear Network Problems in the Social and Economic Sciences

In this paper we focus on the parallel computation of large - scale equilibrium and optimization problems arising in the social and economic sciences. In particular, we consider problems which can be visualized and conceptualized as nonlinear network flow problems. The underlying network structure is then exploited in the development of parallel decomposition algorithms. We first consider market equilibrium problems, both dynamic and static, which are formulated as variational inequality problems, and for which we propose parallel decomposition algorithms by time period and by commodity, respectively. We then turn to the parallel computation of large-scale constrained matrix problems which are formulated as optimization problems and discuss the results of parallel decomposition by row/column

Supply Chain Network Sustainability Under Competition and Frequencies of Activities from Production to Distribution

In this paper, we develop a competitive supply chain network model with multiple firms, each of which produces a differentiated product by brand and weights the emissions that it generates through its supply chain network activities in an individual way. The supply chain network activities of production, transport and distribution, and storage have associated with them distinct capacities and the firms seek to determine their optimal product flows and frequencies of operation so that their utilities are maximized where the utilities consist of profits and weighted emissions. Multiple production, storage, and transport mode options are allowed. The governing equilibrium concept is that of Cournot-Nash equilibrium. We provide both path and link flow variational inequality formulations of the equilibrium conditions and then propose an algorithm, which, at each iteration, yields closed form expressions for the underlying variables.Numerical examples illustrate the generality of the model and the information provided to managerial decision-makers and policy-makers. This paper adds to the growing literature on sustainable supply chains through the development of a computable general competitive supply chain network game theory model, which brings a greater realism to the evaluation of profit and emission trade-offs through the incorporation of frequencies

Supply chain network capacity competition with outsourcing: a variational equilibrium framework

This paper develops a supply chain network game theory framework with multiple manufacturers/producers, with multiple manufacturing plants, who own distribution centers and distribute their products, which are distinguished by brands, to demand markets, while maximizing profits and competing noncooperatively. The manufacturers also may avail themselves of external distribution centers for storing their products and freight service provision. The manufacturers have capacities associated with their supply chain network links and the external distribution centers also have capacitated storage and distribution capacities for their links, which are shared among the manufacturers and competed for. We utilize a special case of the Generalized Nash Equilibrium problem, known as a variational equilibrium, in order to formulate and solve the problem. A case study on apple farmers in Massachusetts is provided with various scenarios, including a supply chain disruption, to illustrate the modeling and methodological framework as well as the potential benefits of outsourcing in this sector

Mergers and Acquisitions in Blood Banking Systems: A Supply Chain Network Approach

Blood banking systems in the United States over the past decade have been faced with a volatile demand for blood, specifically, a decrease in demand for red blood cells, for a variety of reasons. This change in the blood supply chain landscape, accompanied by an increasing emphasis on cost efficiency, is a driver of Mergers & Acquisitions between blood banks. In this paper, we first present supply chain network optimization pre- and post-merger models. The models handle perishability of the life-saving product of blood, include both operational and discarding costs of waste, capture the uncertainty associated with the demand points, as well as the expected total blood supply shortage cost and the total discarding cost at demand points. They also incorporate capacities on the links. Their solution yields the optimal path and link flows plus the frequencies of activities associated with blood collection, shipment, testing and processing, storage, and distribution, and incurred total costs. We provide a cost efficiency (synergy) measure associated with a merger or acquisition in the blood banking industry, as well as measures capturing the expected supply shortage and surplus. The methodological framework and its applicability are then illustrated via a large-scale blood supply chain network example inspired by a pending merger in the real-world in both status quo and disaster scenarios

How to increase the impact of disaster relief: A study of transportation rates, framework agreements and product distribution

Due to restricted budgets of relief organizations, costs of hiring transportation service providers steer distribution decisions and limit the impact of disaster relief. To improve the success of future humanitarian operations, it is of paramount importance to understand this relationship in detail and to identify mitigation actions, always considering the interdependencies between multiple independent actors in humanitarian logistics. In this paper, we develop a game-theoretic model in order to investigate the influence of transportation costs on distribution decisions in long-term relief operations and to evaluate measures for improving the fulfillment of beneficiary needs. The equilibrium of the model is a Generalized Nash Equilibrium, which has had few applications in the supply chain context to date. We formulate it, utilizing the construct of a Variational Equilibrium, as a Variational Inequality and perform numerical simulations in order to study the effects of three interventions: an increase in carrier competition, a reduction of transportation costs and an extension of framework agreements. The results yield important implications for policy makers and humanitarian organizations (HOs). Increasing the number of preselected carriers strengthens the bargaining power of HOs and improves impact up to a certain limit. The limit is reached when carriers set framework rates equal to transportation unit costs. Reductions of transportation costs have a consistently positive, but decreasing marginal benefit without any upper bound. They provide the highest benefit when the bargaining power of HOs is weak. On the contrary, extending framework agreements enables most improvements when the bargaining power of HOs is strong

Towards Pricing Mechanisms for Delay Tolerant Services

One of the applications of Delay Tolerant Networking (DTN) is rural networks. For this application researchers have argued benefits on lowering costs and overcoming challenging conditions under which, for instance, protocols such as TCP/IP cannot work because their underlying requisites are not satisfied. New responses are required in order to understand the true adoption opportunities of this technology. Constraints in service level agreements and viable alternative pricing schemes are some of the new issues that arise as a consequence of the particular operation mode. In this paper, we propose a novel model for pricing delay tolerant services, which adjusts prices to demand variability subject to constraints imposed by the DTN operation. With this model we also show how important parameters such as channel rental costs, cycle times of providers, and market sensitivities affect business opportunities of operators