Flexible capacity strategy with multiple market periods under demand uncertainty and investment constraint

We establish a flexible capacity strategy model with multiple market periods under demand uncertainty and investment constraints. In the model, a firm makes its capacity decision under a financial budget constraint at the beginning of the planning horizon which embraces n market periods. In each market period, the firm goes through three decision-making stages: the safety production stage, the additional production stage and the optimal sales stage. We formulate the problem and obtain the optimal capacity, the optimal safety production, the optimal additional production and the optimal sales of each market period under different situations. We find that there are two thresholds for the unit capacity cost. When the capacity cost is very low, the optimal capacity is determined by its financial budget; when the capacity cost is very high, the firm keeps its optimal capacity at its safety production level; and when the cost is in between of the two thresholds, the optimal capacity is determined by the capacity cost, the number of market periods and the unit cost of additional production. Further, we explore the endogenous safety production level. We verify the conditions under which the firm has different optimal safety production levels. Finally, we prove that the firm can benefit from the investment only when the designed planning horizon is longer than a threshold. Moreover, we also derive the formulae for the above three thresholds.     

A note on “A Continuous Approach to Nonlinear Integer Programming”

Ge and Huang (1989) proposed an approach to transform nonlinear integer programming problems into nonlinear global optimization problems, which are then solved by the filled function transformation method. The approach has recently attracted much attention. This note indicates that the formulae to determine a penalty parameter in two fundamental theorems are incorrect, and presents the corrected formulae and revised theorems.     

A note on quasi-newton formulae for sparse second derivative matrices

In order to apply quasi-Newton methods to solve unconstrained minimization calculations when the number of variables is very large, it is usually necessary to make use of any sparsity in the second derivative matrix of the objective function. Therefore, it is important to extend to the sparse case the updating formulae that occur in variable metric algorithms to revise the estimate of the second derivative matrix. Suitable extensions suggest themselves when the updating formulae are derived by variational methods [1, 3]. The purpose of the present paper is to give a new proof of a theorem of Dennis and Schnabel [1], that shows the effect of sparsity on updating formulae for second derivative estimates.     

A note on asymptotic splitting and its applications

Explicit forms of the remainder terms of the asymptotic splitting formulae associated with the first order splitting, Strange's splitting, and the parallel splitting are derived. Using the identities obtained, we establish the global error estimates for the asymptotic splitting formulae. Both the theoretical investigation and numerical experiments indicate that it is a more efficient and accurate way to use the asymptotic splitting than the conventional splitting formulae.     

Updating a nonlinear discriminant function estimated from a mixture of two Weibull distributions

A procedure is presented for finding maximum likelihood estimates of the parameters of a mixture of two Weibull distributions. Estimation of a nonlinear discriminant function on the basis of small sample size is considered. Throughout simulation experiments, the total probabilities of misclassification and percentage biases are evaluated and discussed. The problem of updating a nonlinear discriminant function on the basis of two Weibull distributions is studied in situations when the additional observations are mixed or classified. The performance of all results is investigated using a series of simulation experiments by means of relative efficiencies.     

Computational tests as a means of assessing the inaccuracies of index numbers

In this paper, we propose using computational tests for appraisingthe inaccuracies of index numbers (for example, price and wageindices). This is done by programming a hypothetical situationwhere the true average index value is known and the variationof prices and quantities and the relation between them can bechosen. The indices are then calculated using the formulae ofLaspeyres, Paasche, and Fischer, and the unit and normalizedunit index formulae. By comparing the average values of theabove formulae with the true average index value, we are ableto obtain a quantitative indication of the errors in these formulaeunder various situations. The normalized unit and Fischer indexformulae were found to be consistently accurate for all ourtests, whereas the formulae of Laspeyres and Paasche were foundto have very poor accuracy in many of our tests. The unit indexwas also consistently accurate in all our tests but its useis appropriate only for a single item, thus making this formulasuitable for measuring wages.     

Corrected Euler-Maclaurin’s formulae

The aim of this paper is to derive corrected Euler-Maclaurin’s formulae, i.e. open type quadrature formulae where the integral is approximated not only with the values of the function in points (5a+b)/6, (a+b)/2 and (a+5b)/6, but also with values of the first derivative in end points of the interval. These formulae will have a higher degree of exactness than the ones obtained in [2]. Using the derived formulae, a number of inequalities for various classes of functions are obtained.     

Linearized Regression Model with Constraints of Type II

A linearization of the nonlinear regression model causes a bias in estimators of model parameters. It can be eliminated, e.g., either by a proper choice of the point where the model is developed into the Taylor series or by quadratic corrections of linear estimators. The aim of the paper is to obtain formulae for biases and variances of estimators in linearized models and also for corrected estimators.     

The Baum–Welch algorithm with limiting distribution constraints

We present a method of incorporating limiting distribution information in the Baum–Welch algorithm for estimating parameters of discrete-time, finite-state, Hidden Markov Models. We find that having, even imperfect, limiting distribution information can dramatically improve transition probability estimates. Additionally, we find that when (1) the underlying process is weakly correlated with the observable signal, and (2) the length of the data sequence is short, the additional information provided by the limiting distribution is substantial.     

Convergence rates in ℓ1-regularization when the basis is not smooth enough

Sparsity promoting regularization is an important technique for signal reconstruction and several other ill-posed problems. Theoretical investigation typically bases on the assumption that the unknown solution has a sparse representation with respect to a fixed basis. We drop this sparsity assumption and provide error estimates for nonsparse solutions. After discussing a result in this direction published earlier by one of the authors and co-authors, we prove a similar error estimate under weaker assumptions. Two examples illustrate that this set of weaker assumptions indeed covers additional situations which appear in applications.     

Moments of Generalized Quadratic Gauss Sums Weighted by L-Functions

The main purpose of this paper is using estimates for character sums and analytic methods to study the second, fourth, and sixth order moments of generalized quadratic Gauss sums weighted by L-functions. Three asymptotic formulae are obtained.     

The computation of Lagrange-multiplier estimates for constrained minimization

Almost all efficient algorithms for constrained optimization require the repeated computation of Lagrange-multiplier estimates. In this paper we consider the difficulties in providing accurate estimates and what tests can be made in order to check the validity of the estimates obtained. A variety of formulae for the estimation of Lagrange multipliers are derived and their respective merits discussed. Finally the role of Lagrange multipliers within optimization algorithms is discussed and in addition to other results, it is shown that some algorithms are particularly sensitive to errors in the estimates.     

Finite formulae for the premium of the general reinsurance treaty based on ordered claims

The general reinsurance treaty based on ordered claims, as defined in Kremer (1982, 1984a,b), is investigated and general premium formulae are given for a finite collective. Under additional assumptions simple formulae are stated for the net premium. The content of the paper is mainly of theoretical interest.     

Rank-one and Rank-two Corrections to Positive Definite Matrices Expressed in Product Form

It is shown that certain rank-one and rank-two corrections tosymmetric positive definite matrices may be expressed in theform of a product. This product form gives control over thepositive definiteness, determinant value and conditioning ofthe corrected matrix. An application to updating formulae ofquasi-Newton methods for unconstrained minimization is discussed.     

Convexity results and sharp error estimates in approximate multivariate integration

An interesting property of the midpoint rule and the trapezoidal rule, which is expressed by the so-called Hermite-Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite-Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. In particular, for simplices of arbitrary dimension, we present two families of integration formulae which both contain a multivariate analogue of the midpoint rule and the trapezoidal rule as boundary cases. The first family also includes a multivariate analogue of a Maclaurin formula and of the two-point Gaussian quadrature formula; the second family includes a multivariate analogue of a formula by P.C. Hammer and of Simpson's rule. In both families, we trace out those formulae which satisfy a Hermite-Hadamard inequality. As an immediate consequence of the latter, we obtain sharp error estimates for twice continuously differentiable functions.

     

Nonlocal symmetries and formulae for generation of solutions for a class of diffusion-convection equations

New formulae of nonlocal nonlinear superposition and generation of solutions are proposed for nonlinear diffusion-convection equations which are linearizable or are invariant with respect to a generalized hodograph transformation or connected by this transformation. We study in what particular ways additional Lie symmetries of diffusion-convection equations induce nonlocal symmetries of equations obtained from the initial ones by nonlocal transformations. The formulae derived are used for the construction of exact solutions.     

Asymptotic expansions of the binomial coefficients

A general formulae for the asymptotic expansion of not centered binomial coefficients are derived and some useful estimates of the binomial coefficients are presented. The sum of the binomial coefficients is also studied.     

Eigen series solutions to terminal-state tracking optimal control problems and exact controllability problems constrained by linear parabolic PDEs

Terminal-state tracking optimal control problems for linear parabolic equations are studied in this paper. The control objectives are to track a desired terminal state and the control is of the distributed type. Explicit solution formulae for the optimal control problems are derived in the form of eigen series. Pointwise-in-time L² norm estimates for the optimal solutions are obtained and approximate controllability results are established. Exact controllability is shown when the target state vanishes on the boundary of the spatial domain. One-dimensional computational results are presented which illustrate the terminal-state tracking properties for the solutions expressed by the series formulae.     

A refinement over the usual formulae for deciles

The formulae for the median (fifth decile) is well defined in the literature. But the formulae for other deciles available in the literature are conflicting in the sense that they do not divide the ordered sample observations into ten segments having the same number of observations in each. It is observed that the ranks of a particular decile for any two sample sizes with a fixed remainder, with respect to modulus10, differs by an integer. The remainder is also observed to be the number of deciles having integer ranks. Prompted by these ideas, the remainder and the usual linear interpolation are exploited to find deciles.