Applications of mathematics to maritime search
Jinling Wei Jinming Zhang Meishuang Dong Fan Zhang Yunmo Chen Sha Jin Zhike Han
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 957-968 doi: 10.3934/dcdss.2019064

The issue of searching missing aircraft is valuable after the event of MH370. This paper provides a global optimal model to foster the efficiency of maritime search. Firstly, the limited scope, a circle whose center is the last known position of the aircraft, should be estimated based on the historical data recorded before the disappearance of the aircraft. And Bayes' theorem is applied to calculate the probability that the plane falling in the region can be found. Secondly, the drift of aircraft debris under the influence of wind and current is considered via Finite Volume Community Ocean Model(FVCOM) and Monte Carlo Method(MC), which make the theory more reasonable. Finally, a global optimal model about vessel and aircraft quantitative constraints is established, which fully considers factors including the area of sea region to be searched, the maximum speed, search capabilities, initial distance of the vessels by introducing 0-1 decision variables.

keywords: Bayes' theorem finite volume community ocean model Monte Carlo Method global optimal model
LIBOR market model with stochastic volatility
Lixin Wu Fan Zhang
Journal of Industrial & Management Optimization 2006, 2(2): 199-227 doi: 10.3934/jimo.2006.2.199
In this paper we extend the standard LIBOR market model to accommodate the pronounced phenomenon of implied volatility smiles/skews. We adopt a multiplicative stochastic factor to the volatility functions of all relevant forward rates. The stochastic factor follows a square-root diffusion process, and it can be correlated to the forward rates. For any swap rate, we derive an approximate process under its corresponding forward swap measure. By utilizing the analytical tractability of the approximate process, we develop a closed-form formula for swaptions in term of Fourier transforms. Extensive numerical tests are carried out to support the swaptions formula. The extended model captures the downward volatility skews by taking negative correlations between forward rates and their volatilities, which is consistent with empirical findings.
keywords: stochastic volatility LIBOR model Fast Fourier transform (FFT). square-root process swaptions
Multi-machine and multi-task emergency allocation algorithm based on precedence rules
Fan Zhang Guifa Teng Mengmeng Gao Shuai Zhang Jingjing Zhang
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 1501-1513 doi: 10.3934/dcdss.2019103

Aiming at the problems of asymmetric information and unreasonable emergency allocation schemes in the current cross-regional emergency operation, the emergency deployment process of multi-machine and multi-task is analyzed, and the emergency allocation model with the goal of minimizing the allocation cost and loss is established in the paper. Emergency allocation algorithm based on rule of nearest-distance-first, which allocate machinery for the nearest farmland firstly, and emergency allocation algorithm based on rule of max-ability-first, by which machinery with maximum ability to farmland is allocated firstly, are proposed. The operational data of farmland and agricultural machinery generated randomly are calculated and analyzed. The results show that when the amount of agricultural machinery is sufficient, the algorithm based on the maximum contribution capacity priority is better. When the agricultural machinery is insufficient, the calculation results of the emergency allocation algorithm based on the nearest distance priority are better. When the number of farmland is not more than 30, the average operation time of the two algorithms in this paper is not more than 3.8 seconds, and both two algorithm have good performance.

keywords: Emergency allocation algorithm precedence rules allocating strategies multi-machine and multi-task
Modelling Population Growth with Delayed Nonlocal Reaction in 2-Dimensions
Dong Liang Jianhong Wu Fan Zhang
Mathematical Biosciences & Engineering 2005, 2(1): 111-132 doi: 10.3934/mbe.2005.2.111
In this paper, we consider the population growth of a single species living in a two-dimensional spatial domain. New reaction-diffusion equation models with delayed nonlocal reaction are developed in two-dimensional bounded domains combining different boundary conditions. The important feature of the models is the reflection of the joint effect of the diffusion dynamics and the nonlocal maturation delayed effect. We consider and analyze numerical solutions of the mature population dynamics with some well-known birth functions. In particular, we observe and study the occurrences of asymptotically stable steady state solutions and periodic waves for the two-dimensional problems with nonlocal delayed reaction. We also investigate numerically the effects of various parameters on the period, the peak and the shape of the periodic wave as well as the shape of the asymptotically stable steady state solution.
keywords: time delay numerical analysis. 2-d reaction-diffusion non-local reac- tion population growth
Some new results on multi-dimension Knapsack problem
Yuzhong Zhang Fan Zhang Maocheng Cai
Journal of Industrial & Management Optimization 2005, 1(3): 315-321 doi: 10.3934/jimo.2005.1.315
We claim a conclusion on Multi-Dimensional Knapsack Problem (MKP), which extends an important proposition by Dantzig firstly, then address to a special case of this problem, and constitute a polynomial algorithm, extending Zukerman et al's work.
keywords: integer programming Knapsack problem polynomial time algorithm approximation algorithm.

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