Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems

Two stage stochastic programming is an important part in the whole area of stochastic programming, and is widely spread in multiple disciplines, such as financial management, risk management, and logistics. The two stage stochastic programming is a natural extension of linear programming by incorporating uncertainty into the model. This thesis solves the two stage stochastic programming using a novel approach. For most two stage stochastic programming model instances, both the objective function and constraints are convex but non-differentiable, e.g. piecewise-linear, and thereby solved by the first gradient-type methods. When encountering large scale problems, the performance of known methods, such as the stochastic decomposition (SD) and stochastic approximation (SA), is poor in practice. This thesis replaces the objective function and constraints with their polynomial approximations. That is because polynomial counterpart has the following benefits: first, the polynomial approximation will preserve the convexity; Second, the polynomial approximation will uniformly converge to the original objective/constraints with arbitrary accuracy; and third, the polynomial approximation will not only provide good estimation on the original objectives/functions but also their gradients/sub-gradients. All these features enable us to apply convex optimization techniques for large scale problems. Hence, the thesis applies SAA, polynomial approximation method and then steepest descent method in combination to solve the large-scale problems effectively and efficiently.

Investigation and improvement of the Mini-Max Polynomial Approximation method for solving burnup equations

This paper addresses a dynamic lot sizing problem with bounded inventory and stockout where both no backlogging and backlogging allowed cases are considered. The stockout option means that there is outsourcing in a period only when the inventory level at that period is non-positive. The production capacity is unlimited and production cost functions are linear but with fixed charges. The problem is that of satisfying all demands in the planning horizon at minimal total cost. We show that the no backlogging case can be solved in ) O(T2) time with general concave inventory holding and outsourcing cost functions where T is the length of the planning horizon. The complexity can be reduced to O(T) when the inventory holding cost functions are also linear and have some realistic properties, even if the outsourcing cost functions remain general concave functions. When the inventory holding and outsourcing cost functions are linear, the backlogging case can be solved in O(T3logT) time whether the outsourcing level at each period is bounded by the sum of the demand of that period and backlogging level from previous periods, or only by the demand of that period.

We consider an economic lot-sizing problem with a special class of piecewise linear ordering costs, which we refer to as the class of modified all-unit discount cost functions. Such an ordering cost function represents transportation costs charged by many less-than truckload carriers. We show that even special cases of the lot-sizing problem are NP-hard and therefore analyze the effectiveness of easily implementable policies. In particular, we demonstrate that there exists a zero-inventory-ordering(ZIO) policy, i.e., a policy in which an order is placed only when the inventory level drops to zero, whose total inventory and ordering cost is no more than 4/3 times the optimal cost. Furthermore, if the ordering cost function does not vary over time, then the cost of the best ZIO policy is no more than 5.6/4.6 times the optimal cost. These results hold for any transportation and holding cost functions that satisfy the following properties: (i) they are non decreasing functions, and (ii) the associated cost per unit is non increasing. Finally, we report on a numerical study that shows the effectiveness of ZIO policies on a set of test problems.

International audience

Groundwater is being contaminated rapidly due to various anthropogenic activities and geogenic sources. In this direction, assessment of water quality analysis is the basic requirement for nurturing the human being and its evolution. The Water Quality Index (WQI) parameter has been widely used in determining water quality globally. The study aims to provide the suitability of groundwater in the specified region using the polynomial approximation method for drinking and irrigation purposes along with the computation of WQI using the conventional method. Weierstrass's polynomial approximation theorem along with longitudinal and latitudinal values has been used to evaluate the polynomial regarding various physicochemical parameters. To validate the obtained results from the present approach, groundwater quality data collected and analyzed from the Pindrawan tank area in Raipur district, Chhattisgarh, India, have been used. The result is obtained, i.e., the intermediate value of the parameters obtained correctly from the mathematical modeling, with an average error of 7%. This polynomial approximation method can also be used as the substitute of inverse modeling to determine the location of the source in the two-dimensional system. The approach output can be beneficial to administrators in making decisions on groundwater quality and gaining insight into the tradeoff between system benefit and environmental requirement.

In Tl2SO2+Na2SO4+di(2‐ethylhexyl)dithiophosphoric acid+n‐C8H18+water system, the Tl+ equilibrium molalities were measured at ionic strength from 0.1 to 2.0 mol·kg‐1 containing Na2SO4 as supporting electrolyte in aqueous phase and at constant molality of extractant at temperatures from 278.15 to 303.15 K in organic phase. The standard extraction constants K0 at various temperatures were obtained by methods of extrapolation and polynomial approximation. Thermodynamic quantities for the extraction process were calculated.

In Part I of this paper, we developed an algorithm for finding planning horizons for the deterministic production smoothing problem when all demand was required to be met from regular production, under rather general assumptions for the production, production smoothing, and holding cost functions. Here, in Part II, we develop analogous planning horizons when the model is extended to include such factors as overtime, lost sales, simple subcontracting, undertime, and backlogging. Part II depends heavily on Part I. To avoid unnecessary tedium, all cost functions will be assumed to be stationary in Part II. Overtime, undertime, holding, and backlogging will be assumed to be convex increasing, all other costs linear and proportional. However, it is not difficult to see that the techniques of Part I can be utilized directly to cover more general cost functions. Part II concludes with a discussion of the application of our general approach to actual data.

The small-signal security region (SSSR) is an important concept in the security analysis of power systems, especially for the problems of low-frequency oscillation. In this paper, a polynomial approximation method is proposed to construct an analytical expression of the boundaries of the damping-ratio-based small-signal security region (D-SSSR) by using the implicit function approach. Compared with the conventional small-signal security region based on the Hopf bifurcation, our method is more practical and suitable for the dispatching of power systems, since it directly depicts the weak damped boundaries in the high-dimensional parameter space. The formulations of the linear and quadratic approximation are presented in detail. The effectiveness and accuracy of the algorithm is verified through case studies on a rudimentary power system.

In this paper, polynomial approximation method and theory are introduced into the research of learning search of artificial intelligence. By so doing, a learning search algorithm can, after sufficient number of problem-solving, construct a heuristic estimate function h(.) which uniformly approximates to the optimal estimate function h*(.) by arbitrary precision. One of these learning search algorithms, A-B/sub n/, is described and it is shown that, when the number of the previous problem-solving becomes large enough, the worst-case complexity of A-B/sub n/ can be reduced to O(poly(N)), where N is the length of the optimal solution path, poly(N) is a polynomial function of N.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

his paper develops a dynamic theoretical model for a polluting industry to determine the emission dynamics, internalizing damage cost function along with production cost function, considering emission factor. It turns out that the competitive output price is composed of not only marginal production cost function but also marginal damage cost of emitting one emission unit multiplied by emission factor as well as the term indicated the pollution stock effects on the difference between damage cost function and the time derivative of implicit pollution price. Natural reductions in pollution as a proportion of decay factor lead to a fluctuations in the time paths of emission and its concentration, so that increasing marginal damage cost that it follows to strengthen social health and human welfare.

It has been shown that ensilager ranke first in the cost of one K.G. of checken (Boultry) ie a bout 57% of the total cost, followed by the cost of checks which represented 17% of the total cost .it was clear that winter is characterized by an increase in the cost of one k.g. of boultry in comparison with other seasons. Function of production cost per season have been estimated since winter is significantly different regarding the cost of one k.g. of meat from other season , it has been isolated from other season with a special function for winter and another for other season. Cost cubic function was significant in expressing the relationship between the total cost and production per batch during the year of the conducting of this study and other different seasons. statistical tests showed the significance of the function. It has appeared from the factor determining R2 that more than 97% of the change in production cost were due to the illustrative variable of the estimated functions. Cost function have been economically analyzed to determine optimum production size that achieve optimum economic efficiency per each k. g. of checken meat (boultry) with the least cost .this production siz was the heigest during the year of carrying out this project. it was about 18194 k. g. of checken meat per batch with a total cost a bout 1300 I. D. per k. g. It has been clear that winter showed a difference in comparison with other seasons where the optimum production size during these seasons was 17331. Total cost function at its lowest rate was at that productive level. The cost of one k. g. of checken meat was a bout 1253 I. D. where optimum size production during winter was higher in comparison with other season. It was 20264 k. g. per batch and with a cost rate higher than its counterparts in other seasons of a bout 313 I. D. Since the main goal of the producer is to douplicate benefit or a void loss, therefore optimum production levels have been calculated at different price levels. It was clear that the increase of prices in accompanied with an increase of optimum production levels at different levels from season to season. Moreover, it was clear that the producer gets benefits when the price he gets puts his work at the stage of diseconomies of scale whereas his work will represent a loss when he works within the stage of economies of scale.

Coil motion noise is the main noise source in AEM data caused by the movement of the receiver coil in the Earth’s magnetic field. The characteristics of both time domain and frequency domain of motion noise are analyzed. In this paper we present a scheme of removing motion noise which employs polynomial to represent motion noise for each half cycle. A set of equations are set up according to the characteristics of AEM data, and Lagrange method is employed to solve minimum problem with two constraint conditions together. The polynomial coefficients are calculated by matrix inverse and finally motion noise is subtracted from the AEM data half cycle by half cycle. Polynomial approximation method is applied to the synthetic data contaminated by low frequency motion noise comparing with traditional high pass filter method. The corrected AEM data shows polynomial approximation method is more effective in dealing with motion noise.

We consider the single item lot-sizing problem with capacities that are non-decreasing over time. When the cost function is (i) non-speculative or Wagner–Whitin (for instance, constant unit production costs and non-negative unit holding costs) and (ii) the production set-up costs are non-increasing over time, it is known that the minimum cost lot-sizing problem is polynomially solvable using dynamic programming. When the capacities are non-decreasing, we derive a compact mixed integer programming reformulation whose linear programming relaxation solves the lot-sizing problem to optimality when the objective function satisfies (i) and (ii). The formulation is based on mixing set relaxations and reduces to the (known) convex hull of solutions when the capacities are constant over time. We illustrate the use and potential effectiveness of this improved LP formulation on a few test instances, including instances with and without Wagner–Whitin costs, and with both non-decreasing and arbitrary capacities over time.

Applications of Markov chain approximation methods to optimal control problems in economics