Minimal pairs of polytopes and their number of vertices

We define what is called Blaschke difference for polytopes as an inverse operation to Blaschke addition. Using this operation we give a new algorithm to reduce and find a minimal pair of polytopes from the given class of the Rådström-Hörmander lattice containing a pair of polytopes in IR2. This method gives a better algorithmic insight and easy to handle than the one given by Handschug (1989). We also prove that a pair of polytopes in the plane is minimal if and only if the sum of the number of their vertices is minimal in the class. However, it is shown in the paper that, this last statement does not hold true in general for higher dimensional spaces

Gradient-based Instantaneous Traffic Flow Optimization on a Roundabout

In this article we focus on instantaneous traffic fl ow optimization on a roundabout using a macroscopic approach. The roundabout is modeled as a concatenation of 2x2 junctions with one main lane and secondary incoming and outgoing roads. We consider a cost functional that measures the total travel time spent by drivers on the roundabout and compute its gradient with respect to the priority parameters at junctions. Then, through numerical simulations, the traffic behavior is studied on the whole roundabout. The numerical approximations compare the performance of a roundabout for instantaneous optimization of the priority parameters and fixed constant parameters

Macroscopic traffic flow optimization on roundabouts

The aim of this paper is to propose an optimization strategy for traffic flow on roundabouts using a macroscopic approach. The roundabout is modeled as a sequence of 2 × 2 with one mainline and secondary incoming and outgoing roads. We consider two cost the total travel time and the total waiting time, which give an estimate of the time by drivers on the network section. These cost functionals are minimized with respect to the of way parameter of the incoming roads. For each cost functional, the analytical expression given for each junction

Climate-dependent malaria disease transmission model and its analysis

Malaria infection continues to be a major problem in many parts of the world including Africa. Environmental variables are known to significantly affect the population dynamics and abundance of insects, major catalysts of vector-borne diseases, but the exact extent and consequences of this sensitivity are not yet well established. To assess the impact of the variability in temperature and rainfall on the transmission dynamics of malaria in a population, we propose a model consisting of a system of non-autonomous deterministic equations that incorporate the effect of both temperature and rainfall to the dispersion and mortality rate of adult mosquitoes. The model has been validated using epidemiological data collected from the western region of Ethiopia by considering the trends for the cases of malaria and the climate variation in the region. Further, a mathematical analysis is performed to assess the impact of temperature and rainfall change on the transmission dynamics of the model. The periodic variation of seasonal variables as well as the non-periodic variation due to the long-term climate variation have been incorporated and analyzed. In both periodic and non-periodic cases, it has been shown that the disease-free solution of the model is globally asymptotically stable when the basic reproduction ratio is less than unity in the periodic system and when the threshold function is less than unity in the non-periodic system. The disease is uniformly persistent when the basic reproduction ratio is greater than unity in the periodic system and when the threshold function is greater than unity in the non-periodic system.The Department of Mathematics at Addis Ababa University, the International Science Program (ISP), the DST/NRF Centre of Excellence in Epidemiological Modelling and Analysis (SACEMA) at Stellenbosch University, South Africa, the Asendabo Health Center, the National Meteorological Agency of Ethiopia and the DST/NRF SARChI Chair in Mathematical Models and Methods in Biosciences and Bioengineering at the University of Pretoria.https://www.worldscientific.com/worldscinet/ijb2020-11-01hj2020Mathematics and Applied Mathematic

Gradient-based Instantaneous Traffic Flow Optimization on a Roundabout

In this article we focus on instantaneous traffic fl ow optimization on a roundabout using a macroscopic approach. The roundabout is modeled as a concatenation of 2x2 junctions with one main lane and secondary incoming and outgoing roads. We consider a cost functional that measures the total travel time spent by drivers on the roundabout and compute its gradient with respect to the priority parameters at junctions. Then, through numerical simulations, the traffic behavior is studied on the whole roundabout. The numerical approximations compare the performance of a roundabout for instantaneous optimization of the priority parameters and fixed constant parameters

An iterative method for tri-level quadratic fractional programming problems using fuzzy goal programming approach

Abstract Tri-level optimization problems are optimization problems with three nested hierarchical structures, where in most cases conflicting objectives are set at each level of hierarchy. Such problems are common in management, engineering designs and in decision making situations in general, and are known to be strongly NP-hard. Existing solution methods lack universality in solving these types of problems. In this paper, we investigate a tri-level programming problem with quadratic fractional objective functions at each of the three levels. A solution algorithm has been proposed by applying fuzzy goal programming approach and by reformulating the fractional constraints to equivalent but non-fractional non-linear constraints. Based on the transformed formulation, an iterative procedure is developed that can yield a satisfactory solution to the tri-level problem. The numerical results on various illustrative examples demonstrated that the proposed algorithm is very much promising and it can also be used to solve larger-sized as well as n-level problems of similar structure

Optimal Control Strategy for TB-HIV/AIDS Co-Infection Model in the Presence of Behaviour Modification

A mathematical model for a transmission of TB-HIV/AIDS co-infection that incorporates prevalence dependent behaviour change in the population and treatment for the infected (and infectious) class is formulated and analyzed. The two sub-models, when each of the two diseases are considered separately are mathematically analyzed. The theory of optimal control analysis is applied to the full model with the objective of minimizing the aggregate cost of the infections and the control efforts. In the numerical simulation section, various combinations of the controls are also presented and it has been shown in this part that the optimal combination of both prevention and treatment controls will suppress the prevalence of both HIV and TB to below 3% within 10 years. Moreover, it is found that the treatment control is more effective than the preventive controls

Hierarchical multilevel optimization with multiple-leaders multiple-followers setting and nonseparable objectives

Hierarchical multilevel multi-leader multi-follower problems are non-cooperative decision problems in which multiple decision-makers of equal status in the upper-level and multiple decision-makers of equal status are involved at each of the lower-levels of the hierarchy. Much of solution methods proposed so far on the topic are either model specific which may work only for a particular sub-class of problems or are based on some strong assumptions and only for two level cases. In this paper, we have considered hierarchical multilevel multi-leader multi-follower problems in which the objective functions contain separable and non-separable terms (but the non-separable terms can be written as a factor of two functions, a function which depends on other level decision variables and a function which is common to all objectives across the same level) and shared constraint. We have proposed a solution algorithm to such problems by equivalent reformulation as a hierarchical multilevel problem involving single decision maker at all levels of the hierarchy. Then, we applied a multi-parametric algorithm to solve the resulting single leader single followers problem

Approximate solution algorithm for multi-parametric non-convex programming problems with polyhedral constraints

In this paper, we developed a novel algorithmic approach for thesolution of multi-parametric non-convex programming problems withcontinuous decision variables. The basic idea of the proposedapproach is based on successive convex relaxation of each non-convexterms and sensitivity analysis theory. The proposed algorithm isimplemented using MATLAB software package and numericalexamples are presented to illustrate the effectiveness andapplicability of the proposed method on multi-parametric non-convexprogramming problems with polyhedral constraints