Chaotic solutions in dynamic linear programming

In Nishimura and Yano (1996), we demonstrate that chaos may emerge as a solution to a dynamic linear programming (LP) problem. That result is closely related to a result of Nishimura and Yano (1995), which establishes the possibility of chaotic optimal accumulation in a two-sector model of optimal capital accumulation. This study intends to survey those results and explain the basic relationship between them.

A projection and division method for problems of linear dynamic programming

We perform linear programming optimizations on the intersection of k polyhedra in R3, represented by their outer recursive decompositions, in expected time O(k log k log n + √k log k log3 n). We use this result to derive efficient algorithms for dynamic linear programming problems in which constraints are inserted and deleted, and queries must optimize specified objective functions. As an application, we describe an improved solution to the planar 2-center problem. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

It is shown that solutions of linear inequalities, linear programs and certain linear complementarity problems (e.g. those with P-matrices or Z-matrices but not semidefinite matrices) are Lipschitz continuous with respect to changes in the right-hand side data of the problem. Solutions of linear programs are not Lipschitz continuous with respect to the coefficients of the objective function. The Lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right-hand side perturbation.

In this paper, an algorithm for solving interval time-cost tradeoff transportation problemsis presented. In this problem, all the demands are defined as intervalto determine more realistic duration and cost. Mathematical methods can be used to convert the time-cost tradeoff problems to linear programming, integer programming, dynamic programming, goal programming or multi-objective linear programming problems for determining the optimum duration and cost. Using this approach, the algorithm is developed converting interval time-cost tradeoff transportation problem to the linear programming problem by taking into consideration of decision maker (DM).

The transportation problems (TPs) are a fundamental case-study topic in operations research, particularly in the field of linear programming (LP). The TPs are solved in full resolution by using two types of methods: initial basic feasible solution (IBFS) and optimal methods. In this paper, we suggest a novel IBFS method for enhanced reduction in the transportation cost associated with the TPs. The new method searches for the range in columns of the transportation table only, and selects the maximum range to carry out allocations, and is therefore referred to as the maximum range column method (MRCM). The performance of the proposed MRCM has been compared against three traditional methods: North-West-Corner (NWCM), Least cost (LCM) and Vogel’s approximation (VAM) on a comprehensive database of 140 transportation problems from the literature. The optimal solutions of the 140 problems obtained by using the TORA software with the modified distribution (MODI) method have been taken as reference from a previous benchmark study. The IBFSs obtained by the proposed method against NWCM, LCM and VAM are mostly optimal, and in some cases closer to the optimal solutions as compared to the other methods. Exhaustive performance has been discussed based on absolute and relative error distributions, and percentage optimality and nonoptimality for the benchmark problems. It is demonstrated that the proposed MRCM is a far better IBFS method for efficiently solving the TPs as compared to the other discussed methods, and can be promoted in place of the traditional methods based on its performance.

In this paper, we propose a resource allocation scheme for interference management in heterogeneous networks. We particularly consider downlink interference from a Home eNB (HeNB) to macrocell user equipments (MUEs) in the coverage of the traditional macrocell basestation (MBS). By overhearing uplink feedback information from MUEs and Home User Equipments (HUEs) together with downlink control information (DCI) from the MBS, the HeNB formulates a dynamic programming problem with the objective of maximizing the total reward over an infinite horizon. By exploiting the feedback information as well as the DCI at the HeNB, we show that the solution of the dynamic programming problem follows a greedy policy at the HeNB, which simplifies the solution of the infinite horizon dynamic programming problem. We also examine the effect of uncertainty in the feedback information on the performance of our proposed scheme.

The industry and academe broadly use linear programming to analyze the logistics or supply chain solutions, which is based on the hypothesis of fixed environment and restricted procedures. However, in the dynamic market, the supply chain driven by the market faces multi units such as customers, suppliers, manufactures, distributors and carriers and multi uncertain factors such as physical distribution, manufacturing support and purchase. Through a comparative analysis of application of linear programming and dynamic programming in supply chain policy decision, the paper concludes that dynamic programming is more suitable for logistics and supply chain system policy decision study under changing market environments.

This research is about the development of a dynamic programming model for solving fuzzy linear programming problems. Initially, fuzzy dynamic linear programming model FDLP is developed. This research revises the established dynamic programming model for solving linear programming problems in a crisp environment. The mentioned approach is upgraded to address the problem in an uncertain environment. Dynamic programming model can either be passing forward or backward. In the proposed approach backward dynamic programming approach is adopted to address the problem. It is then followed by implementing the proposed method on the education system of Pakistan. The education system of Pakistan comprises of the Primary, Middle, Secondary, and Tertiary education stages. The problem is to maximize the efficiency of the education system while achieving the targets with minimum usage of the constrained resources. Likewise the model tries to maximize the enrollment in the Primary, Middle, Secondary and Tertiary educational categories, subject to the total available resources in a fuzzy uncertain environment. The solution proposes that the enrollment can be increased by an amount 9997130, by increasing the enrollment in the Middle and Tertiary educational categories. Thus the proposed method contributes to increase the objective function value by 30%. Moreover, the proposed solutions violate none of the constraints. In other words, the problem of resources allocation in education system is efficiently managed to increase efficiency while remaining in the available constrained resources. The motivation behind using the dynamic programming methodology is that it always possesses a numerical solution, unlike the other approaches having no solution at certain times. The proposed fuzzy model takes into account uncertainty in the linear programming modeling process and is more robust, flexible and practicable.

The purpose of the paper is to describe methods used for formulation, solution and analysis of dynamic multiobjective linear programming problems with HYBRID package (see Makowski and Sosnowski, 1987). The method adopted in HYBRID for formulation and solution of a multicriteria problem is based on the reference point approach introduced by Wierzbicki (1980). In this approach a piecewise linear scalarizing function is defined. The function parameters are aspiration level for each criterion (reference points) and — possibly — weights. Minimization of the scalarizing function subject linear constraints is substituted by solution of an equivalent single-objective linear programming problem.

In this paper, we consider the problem of dynamic programming when supremum terms appear in the objective function. Such terms can represent overhead costs associated with the underlying state variables. Specifically, this form of optimization problem can be used to represent optimal scheduling of batteries such as the Tesla Powerwall for electrical consumers subject to demand charges — a charge based on the maximum rate of electricity consumption. These demand charges reflect the cost to the utility of building and maintaining generating capacity. Unfortunately, we show that dynamic programming problems with supremum terms do not satisfy the principle of optimality. However, we also show that the supremum is a special case of the class of forward separable objective functions. To solve the dynamic programming problem, we propose a general class of optimization problems with forward separable objectives. We then show that for any problem in this class, there exists an augmented-state dynamic programming problem which satisfies the principle of optimality and the solutions to which yield solutions to the original forward separable problem. We further generalize this approach to stochastic dynamic programming problems and apply the results to the problem of optimal battery scheduling with demand charges using a data-based stochastic model for electricity usage and solar generation by the consumer.

Optimal control of parabolic systems with infinite time horizons

Linear and dynamic programming formulations are introduced for optimizing the placement of distributed best management practices (BMPs) at the watershed scale. The results of linear programming optimization of infiltration-based stormwater management BMPs are compared with the results of genetic algorithm (GA)optimization using a nonlinear distributed model. Additionally, linear and dynamic programming optimization of sediment-trapping BMPs are compared with GA optimization using a nonlinear distributed model. The results indicate that the solution to stormwater peak-flow reduction is influenced primarily by distributed-flow arrival time, and a linear programming analog to a nonlinear optimization model can efficiently reproduce much of the same solution structure. Linear and dynamic programming solutions to the storm sediment-management problem indicate natural sediment trapping is an important consideration, and a solution to the sediment-management-optimization problem can be efficiently found ...

When coefficients in the objective function cannot be precisely determined, the optimal solution is fluctuated by the realisation of coefficients. Therefore, analysing the stability of an optimal solution becomes essential. Although the robustness analysis of an optimal basic solution has been developed successfully so far, it becomes complex when the solution contains degeneracy. This study is devoted to overcoming the difficulty caused by the degeneracy in a linear programming problem with interval objective coefficients. We focus on the tangent cone of a degenerate basic feasible solution since the belongingness of the objective coefficient vector to its associated normal cone assures the solution’s optimality. We decompose the normal cone by its associated tangent cone to a direct union of subspaces. Several propositions related to the proposed approach are given. To demonstrate the significance of the decomposition, we consider the case where the dimension of the subspace is one. We examine the obtained propositions by numerical examples with comparisons to the conventional techniques.KeywordsInterval linear programmingDegeneracyPolyhedral convex coneTangent coneBasic space

Dynamic Programming Problem (DPP) is a multivariable optimization problem is decomposed into a series of stages, optimization being done at each stage with respect to one variable only. DP stands a suitable quantitative study procedure that can be used to explain various optimization problems. It deals through reasonably large as well as complex problems; in addition, it involves creating a sequence of interconnected decisions. The technique offers an efficient procedure for defining optimal arrangement of decisions. Throughout this chapter, solving procedure completely deliberate about as Fuzzy Dynamic Programming Problem for single additive constraint with additively separable return with the support of trapezoidal membership functions and its arithmetic operations. Solving procedure has been applied from the approach of Fuzzy Dynamic Programming Problem (FDPP). The fuzzified version of the problem has been stated with the support of a numerical example for both linear and nonlinear fuzzy optimal solutions and it is associated to showing that the proposed procedure offers an efficient tool for handling the dynamic programming problem instead of classical procedures. As a final point the optimal solution with in the form of fuzzy numbers and justified its solution with in the description of trapezoidal fuzzy membership functions.