On polynomial solvability of the high multiplicity total weighted tardiness problem

AbstractIn a recent paper Hochbaum developed a polynomial algorithm for solving a scheduling problem of minimizing the total weighted tardiness for a large number of unit length jobs which can be partitioned into few sets of jobs with identical due dates and penalty weights. The number of unit jobs in a set is called the multiplicity of that set. The problem was formulated in Hochbaum as an integer quadratic nonseparable transportation problem and solved, in polynomial time, independent of the size of the multiplicities and the due dates but depending on the penalty weights. In this paper we show how to solve the above problem in polynomial time which is independent of the sizes of the weights. The running time of the algorithm depends on the dimension of the problem and only the size of the maximal difference between two consecutive due dates. In the case where the due dates are large, but the size of the maximal difference between two consecutive due dates is polynomially bounded by the dimension of the problem, the algorithm runs in strongly polynomial time

Competitive Equilibrium and Trading Networks: A Network Flow Approach

Under full substitutability of preferences, it has been shown that a competitive equilibrium exists in trading networks, and is equivalent (after a restriction to equilibrium trades) to (chain) stable outcomes. In this paper, we formulate the problem of finding an efficient outcome as a generalized submodular flow problem on a suitable network. Equivalence with seemingly weaker notions of stability follows directly from the optimality conditions, in particular the absence of improvement cycles in the flow problem. Our formulation yields strongly polynomial algorithms for finding competitive equilibria in trading networks, and testing (chain) stability

Note---Efficient Heuristic Algorithms for Positive 0-1 Polynomial Programming Problems

We develop in this paper two types of heuristic methods for solving the positive 0-1 polynomial programming (PP) problem of finding a 0-1 vector x that maximizes c Tx subject to f(x) \le b where c, b \ge 0 and f is an m-vector of polynomials with non-negative coefficients. The various heuristics were first tested on randomly generated sparse problems of up to 50 variables and 50 constraints, and their performance in terms of computational time and effectiveness was investigated. The results were very encouraging. Optimal solutions were consistently obtained by some of the heuristic methods in over 50% of the problems solved. The effectiveness was on the average better than 99% and no less than 96.5%. The computational time using these heuristics is on the average 5% of the time required to solve the PP problems to optimality. Some results for very sparse problems with up to 1,000 variables and 200 constraints are also reported.programming: integer algorithms, heuristic

On competitive sequential location in a network with a decreasing demand intensity

We introduce and analyze a Hotelling like game wherein players can locate in a city, at a fixed cost, according to an exogenously given order. Demand intensity is assumed to be strictly decreasing in distance and players locate in the city as long as it is profitable for them to do so. For a linear city (i) we explicitly determine the number of players who will locate in equilibrium, (ii) we fully characterize and compute the unique family of equilibrium locations, and (iii) we show that players' equilibrium expected profits decline in their position in the order. Our results are then extended to a city represented by an undirected weighted graph whose edge lengths are not too small and co-location on nodes of the graph is not permitted. Further, we compare the equilibrium outcomes with the optimal policy of a monopolist who faces an identical problem and who needs to decide upon the number of stores to open and their locations in the city so as to maximize total profit.Location Game theory

A Parametric Analysis of a Constrained Nonlinear Inventory-production Model

We discuss in this paper some inventory-production models which can be formulated as nonlinear parametric network flow problems with one additional linear constraint. Several sensitivity analysis questions are addressed. The qualitative sensitivity results obtained provide the manager with insight to understanding qualitatively how to respond to changes in the environment such as production costs, inventory capacity, external demand, machine break-down, strike, or variations in inventory carrying cost without additional computation.nonlinear constrained networks, qualitative analysis, parametric programming, algorithm, inventory-production

Forest Management: A Multicommodity Flow Formulation and Sensitivity Analysis

We formulate the Forest Management Problem as a Multicommodity Network Flow Problem with a convex cost function. We then show how to transfer the problem to an equivalent Single-Commodity Network Flow formulation in order to address several sensitivity analysis questions using results by Granot and Veinott (Granot, F., A. F. Veinott, Jr. 1985. Substitutes, complements and ripples in network flows. Math. Oper. Res. 10 471--497.). The answers can subsequently be used to help forest managers respond to changes in the environment such as fires or changes in market prices, without any additional computation. Furthermore, the characterization of the response we provide requires no prior knowledge of optimal management policies.forestry, planning, nonlinear networks, multicommodity, parametric programming, sensitivity analysis

A Parametric Sensitivity Analysis of a Nonlinear Currency Management Model

This paper evaluates the foreign exchange currency management problem facing a multinational firm in terms of a generalized network flow problem. Qualitative sensitivity analysis is applied to the currency management problem in this form to investigate the sensitivities and interdependencies in sources of and needs for currencies. The analysis reveals several implications which would not be apparent without viewing the problem in such a context. The structure of the problem corresponds closely to that faced by many multinational firms which manage money and report performance in their home (parent) currency

Structural and algorithmic properties for parametric minimum cuts

We consider the minimum s, t-cut problem in a network with parametrized arc capacities. Following the seminal work of Gallo et al. (SIAM J. Comput. 18(1):30-55, 1989), classes of this parametric problem have been shown to enjoy the nice Structural Property that minimum cuts are nested, and the nice Algorithmic Property that all minimum cuts can be computed in the same asymptotic time as a single minimum cut by using a clever Flow Update step to move from one value of the parameter to the next. We present a general framework for parametric minimum cuts that extends and unifies such results. We define two conditions on parametrized arc capacities that are necessary and sufficient for (strictly) decreasing differences of the parametric cut function. Known results in parametric submodular optimization then imply the Structural Property. We show how to construct appropriate Flow Updates in linear time under the above conditions, implying that the Algorithmic Property also holds under these conditions. We then consider other classes of parametric minimum cut problems, without decreasing differences, for which we establish the Structural and/or the Algorithmic Property, as well as other cases where nested minimum cuts arise. © 2011 Springer and Mathematical Optimization Society