MA3264: Mathematical Modeling Weizhu Bao Department of Mathematics & Center for Computational Science and Engineering National University of Singapore Email: firstname.lastname@example.org URL: http://www.math.nus.edu.sg/~bao
Chapter 1 Introduction • Mathematical modeling • Aims: • Convert real-world problems into mathematical equations through proper assumptions and physical laws • Apply mathematics to solve real-life problems • Provide new problems for mathematicians • History: • Started by the Egyptians and other ancient civilizations • Fairly recent named as mathematical modeling & a branch of applied and computational mathematics – modeling, analysis & simulation • Rapid development in 20th centuries, especially after the computer • Mathematical modeling contest (MCM) – undergraduates & high school
Chapter 1 Introduction • Wide applications in applied sciences • In physics --- Newton’s laws of motion, quantum physics, particle physics, nuclear physics, plasma physics, …….. • In chemistry --- chemical reaction, mixing problems, first principle calculation, ……. • In engineering --- mechanical engineering (fluid flow, aircraft, Boeing 777, …), electrical engineering (semiconductor, power transport, …), civil engineering (building safety, dam analysis), …… • In materials sciences – fluid-structure interaction, new materials, quantum dots, ……. • In biology --- cell motion, cell population, plant population, …… • In social sciences --- population model, traffic flow, president election poll, casino, gambling, …… • ……..
A simple model—A saving certificate • The problem: Suppose you deposit S$10,000 into DBS bank as a fixed deposit. If the interest is accumulated monthly at 1% and paid at the end of each month, how much money is in the account after 10 years? • Solution: • Let S(n) be the amount in the account after nth month • Mathematical relation • The result
A simple model • Related question:If the interest is accumulated yearly at 12% and paid at the end of each year, how much money is in the account after 10 years? • The solution: • Let S(n) be the amount in the account after nth year • Mathematical relation • The result • Exercise question: If the year interest rate is at 12% and the interest is accumulated daily or instantly, how much money is in the account after 10 years, respectively???? 33,195 ( 33,201 )
Another example – Mortgaging a home • The problem: Suppose you want to buy a condo at $800,000 and you can pay a down payment at $160,000. You find a mortgage with a monthly interest rate at 0.3%. If you want to pay in 30 years, what is your monthly payment? If you can pay $4,000 a month, how long do you need to pay? • The solution: • Let S(n) be the amount due in the mortgage after nth month • Mathematic relation for the first part::
Another example – Mortgaging a home • The result for the first part: • Payment information • Total payment = 2909.7*360=1,047,500
Another example – Mortgaging a home • Mathematical relation for the second part: • The result for the second part:
Another example--Optimization of Profit • Economics problems: • Marco-economics: economic policy • Micro-economics: profit of a company • An example: optimization of profit • Consider an idealized company: • Object of the management: to produce the best possible dividend for the shareholders. • Assumption: The bigger the capital invested in the company, the bigger will be the profit (the net income)
Optimization of profit • Two strategies to spend the profit: • Short term management: The total profit is paid out as a dividend to the shareholders in each year. The company does not grow and shareholders get the same profit in each year. • Long term management: The total profit is divided into two parts. One part is paid out as a dividend to the shareholders and the other part is to re-invest annually in the company so that the subsequent profits in future years will increase. • Question:What part of the profit must be paid out annually as a dividend so that the total yield for the shareholders over a given period of years is a maximum ???
Optimization of profit • Variables: t: time • u(t): the total capital invested in the company in time t • w(t): total dividend in the period [0,t] to the shareholders • Parameters: • k: constant fraction of the profit which will be re-invest ( ) • a: profit rate ( profit per time per capital investment) • Assumptions • The capital and profit are continuous and the process of re-investment and dividends is also continuous. (Normally the capital and profit will be calculated at the end of the financial year of the company!!!) • The profit is directly proportional to the capital invested.
Optimization of profit • Balance equation: • Consider the time interval • Profit: • Change of the investment: • Change rate: • Rate of change:
Optimization of profit • Dividend paid to shareholders: • Change of the total dividend: • Change rate: • Rate of change: • Mathematical model:
Optimization of profit • Solution • Interpretation • If k=0:all profit is paid to shareholders, total dividend increases linearly, total investment doesn’t change & the company doesn’t grow!!! • If k=1: all profit is re-invest, total dividend is zero, total investment increases exponentially & the company grows in the fastest way. • If 0<k<1: both total investment & dividend increase
Optimization of profit • Central issue:Given a period of time [0,T], how must k be chosen so that the total dividend over the period [0,T] is a maximum? • Total dividend: • Question: Find k in [0,1] such that w(k;T) to be maximum? • For simplicity, introduce new variables: • Find x in [0,aT], such that y to be maximum
Optimization of profit • Find the derivative of y: • Different cases: • If a T=2: y is a decreasing function of x with the maximum of y at x=0 • If a T<2: y is a decreasing function of x with the maximum of y at x=0 • If a T>2: y increases and then decreases & attains its maximum at x* which is the root of
Optimization of profit • Interpretation: • If a T<=2: then k=0 produces the largest total dividend over the period of T years, which means that all the profit is paid out as a dividend. It does not pay to re-invest money in the company because either a or T or both are too small. In this case, the maximum profit is • If a T >2: there exists a unique number k=x*/a T such that k u(t) must be re-invested. In this case, the maximum profit is