Data envelopment analysis (DEA) has been widely used to measure the performance of theoperational units that convert multiple inputs into multiple outputs. In many real worldscenarios, there are systems that have a two-stage network process with shared inputs usedin both stages of productions. In this paper, the problem of evaluating the efficiency ofa set of specialized and interdependent components that make up a large DMU is considered.In these processes the first stage consists of two parallel components which are connectedserially with the process in the second stage. The paper develops a DEA approach formeasuring efficiency of decision processes which can be divided into two stages. Thisapplication of parallel-series production process involves shared resources and the paperdetermines an optimal split of shared resources among two components.

In this paper, a solution procedure is proposed to solve fuzzy linear fractionalprogramming (FLFP) problem where cost of the objective function, the resources and thetechnological coefficients are triangular fuzzy numbers. Here, the FLFP problem istransformed into an equivalent deterministic multi-objective linear fractional programming(MOLFP) problem. By using Fuzzy Mathematical programming approach transformed MOLFPproblem is reduced single objective linear programming (LP) problem. The proposedprocedure illustrated through a numerical example.

A concept of an Orderly Colored Longest Path (OCLP) refers to the problem of finding the longest path in a graph whose edges are colored with a given number of colors, under the constraint that the path follows a predefined order of colors. The problem has not been widely studied in the previous literature, especially for more than two colors in the color arrangement sequence. The recent and relevant application of OCLP is related to the interpretation of Nuclear Magnetic Resonance experiments for RNA molecules. Besides, an employment of this specific graph model can be found in transportation, games, and grid graphs. OCLP models the relationships between consecutive edges of the path, thus it appears very useful in representing the real problems with specific ties between their components. In the paper, we show OCLP’s correlation with similar issues known in graph theory. We describe the applications, three alternative models and new integer programming algorithms to solve OCLP. They are formulated by means of max flow problems in a directed graph with packing constraints over certain partitions of nodes. The methods are compared in a computational experiment run for a set of randomly generated instances.

Discrete-continuous project scheduling problems with positive discounted cash flows andthe maximization of the NPV are considered. We deal with a class of theseproblems with an arbitrary number of discrete resources and one continuous, renewableresource. Activities are nonpreemptable, and the processing rate of an activity is acontinuous, increasing function of the amount of the continuous resource allotted to theactivity at a time. Three common payment models – Lump Sum Payment, Payments at ActivityCompletion times, and payments in Equal Time Intervals are analyzed. Formulations ofmathematical programming problems for an optimal continuous resource allocation for eachpayment model are presented. Applications of two local search metaheuristics – Tabu Searchand Simulated Annealing are proposed. The algorithms are compared on a basis ofcomputational experiments. Some conclusions and directions for future research are pointedout.

This paper addresses the problem of managing a waiting list for elective surgery todecide the number of patients selected from the waiting list and to schedule them inaccordance with the operating room capacity in the next period. The waiting listprioritizes patients not only by their initial urgency level but also by their waitingtime. Selecting elective surgery patients requires a balance between the waiting time forurgent patients and that for less urgent patients. The problem is formulated as aninfinite horizon Markov Decision Process. Further, the study proposes a schedulingprocedure based on structural properties of an optimal policy by taking a sampling-basedfinite horizon approximation approach. Finally, we examine the performance of the policyunder various conditions.

This note considers an established reaction–diffusion model for a combustion system, in which there are competing endothermic and exothermic reaction pathways. A combustion front is assumed to move at constant speed through the medium. An asymptotic theory is presented for solid fuels in which material diffusion is ignored, and it allows a simple and complete analysis of the approximate system in the phase plane. Both the adiabatic and nonadiabatic cases are discussed.

Microbial competition for nutrients is a common phenomenon that occurs between species inhabiting the same environment. Bioreactors are often used for the study of microbial competition since the number and type of microbial species can be controlled, and the system can be isolated from other interactions that may occur between the competing species. A common type of competition is the so-called “pure and simple” competition, where the microbial populations interact in no other way except the competition for a single rate-limiting nutrient that affects their growth rates. The issue of whether pure and simple competition under time-invariant conditions can give rise to chaotic behaviour has been unresolved for decades. The third author recently showed, for the first time, that chaos can theoretically occur in these systems by analysing the dynamics of a model where both competing species grow following the biomass-dependent Contois model while the yield coefficients associated with the two species are substrate-dependent. In this paper we show that chaotic behaviour can occur in a much simpler model of pure and simple competition. We examine the case where only one species grows following the Contois model with variable yield coefficient while the other species is allowed to grow following the simple Monod model with constant yield. We show that while the static behaviour of the proposed model is quite simple, the dynamic behaviour is complex and involves period doubling culminating in chaos. The proposed model could serve as a basis to re-examine the importance of Contois kinetics in predicting complex behaviour in microbial competition.

A new formula for Adomian polynomials is introduced and applied to obtain truncated series solutions for fractional initial value problems with nondifferentiable functions. These kinds of equations contain a fractional single term which is examined using Jumarie fractional derivatives and fractional Taylor series for nondifferentiable functions. The property of nonlocality of these equations is examined, and the existence and uniqueness of solutions are discussed. Convergence and error analysis for the Adomian series solution are also studied. Numerical examples show the accuracy and efficiency of this formula for solving initial value problems for high-order fractional differential equations.

Inequalities for spatial competition verify the pair approximation of statistical mechanics introduced to theoretical ecology by Matsuda, Satō and Iwasa, among others. Spatially continuous moment equations were introduced by Bolker and Pacala and use a similar assumption in derivation. In the present article, I prove upper bounds for the $k\mathrm{th} $ central moment of occupied sites in the contact process of a single spatial dimension. This result shows why such moment closures are effective in spatial ecology.

We propose and analyse a method based on the Riccati transformation for solving the evolutionary Hamilton–Jacobi–Bellman equation arising from the dynamic stochastic optimal allocation problem. We show how the fully nonlinear Hamilton–Jacobi–Bellman equation can be transformed into a quasilinear parabolic equation whose diffusion function is obtained as the value function of a certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence and uniqueness and derive useful bounds of classical Hölder smooth solutions. Furthermore, we construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit travelling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 index as an example of the application of the method.

We study the sequence of monic polynomials orthogonal with respect to inner product

$$\begin{eqnarray*}\langle p, q\rangle = \int \nolimits \nolimits_{0}^{\infty } p(x)q(x){e}^{- x} {x}^{\alpha } \hspace{0.167em} dx+ Mp(\zeta )q(\zeta )+ N{p}^{\prime } (\zeta ){q}^{\prime } (\zeta ),\end{eqnarray*}$$

$\alpha \gt - 1$

$M\geq 0$

$N\geq 0$

$\zeta \lt 0$

$p$

$q$

This paper broadens the exponential utility function commonly used by risk-averse investors to the linear plus double exponential utility function, which is applicable in most cases. Thus it is of essential and supreme significance to conduct a research on its optimal investment portfolio in securities investment. This paper, by means of the non-difference curve method, carries out a research into the optimal portfolio decision-making by investors who have this type of utility function. The optimal decision-making and the ratio of optimal portfolio investment are derived. Finally, an actual case is given to verify the relevant results.

A simple idea used in many combinatorial algorithms is the idea ofpivoting. Originally, it comes from the method proposed by Gauss in the19th century for solving systems of linear equations. This method had been extended in1947 by Dantzig for the famous simplex algorithm used for solving linear programs. Fromsince, a pivoting algorithm is a method exploring subsets of a ground set and going fromone subset σ to a new one σ′ by deleting anelement inside σ and adding an element outside σ:σ′ = σ\ { v}  ∪  {u},with v ∈ σ and u ∉ σ.This simple principle combined with other ideas appears to be quite powerful for manyproblems. This present paper is a survey on algorithms in operations research and discretemathematics using pivots. We give also examples where this principle allows not only tocompute but also to prove some theorems in a constructive way. A formalisation isdescribed, mainly based on ideas by Michael J. Todd.

We study the value of European security derivatives in the Black–Scholes model when the underlying asset $\xi $ is approximated by random walks ${\xi }^{(n)} $. We obtain an explicit error formula, up to a term of order $ \mathcal{O} ({n}^{- 3/ 2} )$, which is valid for general approximating schemes and general payoff functions. We show how this error formula can be used to find random walks ${\xi }^{(n)} $ for which option values converge at a speed of $ \mathcal{O} ({n}^{- 3/ 2} )$.

We review the work of the present authors to employ variational calculus to formulate continuous models for the connections between various carbon nanostructures. In formulating such a variational principle, there is some evidence that carbon nanotubes deform as in perfect elasticity, and rather like the elastica, and therefore we seek to minimize the elastic energy. The calculus of variations is utilized to minimize the curvature subject to a length constraint, to obtain an Euler–Lagrange equation, which determines the connection between two carbon nanostructures. Moreover, a numerical solution is proposed to determine the geometric parameters for the connected structures. Throughout this review, we assume that the defects on the nanostructures are axially symmetric and that the into-the-plane curvature is small in comparison to that in the two-dimensional plane, so that the problems can be considered in the two-dimensional plane. Since the curvature can be both positive and negative, depending on the gap between the two nanostructures, two distinct cases are examined, which are subsequently shown to smoothly connect to each other.

The Kohlrausch functions $\exp (- {t}^{\beta } )$, with $\beta \in (0, 1)$, which are important in a wide range of physical, chemical and biological applications, correspond to specific realizations of completely monotone functions. In this paper, using nonuniform grids and midpoint estimates, constructive procedures are formulated and analysed for the Kohlrausch functions. Sharper estimates are discussed to improve the approximation results. Numerical results and representative approximations are presented to illustrate the effectiveness of the proposed method.