PART 2

a)

b)

PART 3

Based on the equation, a table has been constructed where t represents the number of hours starting from 0 hours to 23 hours and P represents the number of people.

t/hoursP/number of people

00

1241

2900

31800

42700

53359

63600

73359

82700

91800

10900

11241

120

13241

Based on table above, graph is generated using Microsoft Excel application.

The peak hours with 3600 people in the mall is after 6 hours the mall opens 9:30 a.m. + 6 hours = 3:30 p.m. .

7:30 p.m. is 10 hours after the malls opens. Based on the graph, the number of people at the mall at 7:30 p.m. is 900 people.

By using formula ,

Thus, 9:30 a.m + 3.48 hours = 1.54 p.m.

HISTORY

George Dantzig, founder of Linear Programming.The 1940s was a time of innovation and reformation of how products were made, both to make things more efficient and to make a better-quality product. The second world war was going on at the time and the army needed a way to plan expenditures and returns in order to reduce costs and increase losses for the enemy. George B. Dantzig is the founder of the simplex method of linear programming, but it was kept secret and was not published until 1947 since it was being used as a war-time strategy. But once it was released, many industries also found the method to be highly valuable. Another person who played a key role in the development of linear programming is John von Neumann, who developed the theory of the duality and Leonid Kantorovich, a Russian mathematician who used similar techniques in economics before Dantzig and won the Nobel prize in 1975 in economics. Dantzig's original example of finding the best assignment of 70 people to 70 jobs emphasizes the praticality of linear programming. The computing power required to test all possible combinations to select the best assignment is quite large. However, it takes only a moment to find the optimum solution by modeling problem as a linear program and applying the simplex algorithm. The theory behind linear programming is to drastically reduce the number of possible optimal solutions that must be checked. In the years from the time when it was first proposed in 1947 by Dantzig, linear programming and its many forms have come into wide use worldwide. LP has become popular in academic circles, for decision scientists (operations researchers and management scientists), as well as numerical analysts, mathematicians, and economists who have written hundreds of books and many more papers on the subject. Though it is so common now, it was unknown to the public prior to 1947. Actually, several researchers developed the idea in the past. Fourier in 1823 and the well-known Belgian mathematician de la Valle Poussin in 1911 each wrote a paper describing today's linear programming methods, but it never made its way into mainstream use. A paper by Hitchcock in 1941 on a transportation problem was also overlooked until the late 1940s and early 1950s. It seems the reason linear programming failed to catch on in the past was lack of interest in optimizing. "Linear programming can be viewed as part of a great revolutionary development which has given mankind the ability to state general goals and to lay out a path of detailed decisions to take in order to 'best' achieve its goals when faced with practical situations of great complexity. Our tools for doing this are ways to formulate real-world problems in detailed mathematical terms (models), techniques for solving the models (algorithms), and engines for executing the steps of algorithms (computers and software)."

OBJECTIVES

Objectives of this folio :To apply and adapt a variety of problem-solving strategies to solve problems

To improve thinking skills

To promote effective mathematical communication

To develop mathematical knowledge through problem solving in a way that increases students interest and confidence

to use the language of the mathematical to express mathematical ideas precisely

To provide learning environment that stimulate and enhances effective learning

To develop positive attitudes towards mathematics.

FOREWARDFirst of all, would like to say Alhamdulillah thank to the God, for giving me strength and heath to do this project work.

Furthermore,I also want to give my appreciation to the internet for helping me to give me more information in depth about Linear Programming.Without it I would have not be able to finish this folio.

Besides, I would like to thank my Additional Mathematics teacher, Pn. Aslina for guiding me throughout this project. She gives a lot of guidance and information about this project. Without her I would be lost to do the project since I have never done it before.

In addition,I also want to give my appreciation to my parents for all their support in financial and moral throughout this project work. Without them standing with me, I would not have complete this folio. Last but not least, I would like to give appreciation to all my friends, who have done this project with me throughout the holidays. Also not forgotten to all my classmates and friends who are willing to share their opinion and information.

PART 1

a) MATHEMATICAL OPTIMIZATIONMathematical Optimization is a branch of mathematics that focuses on problems where scarce resources need to be allocated effectively, in complex, dynamic and uncertain conditions.

The program combines a solid foundation in math with special sequences of courses in economics, business, and management science.

As a graduate, you might enhance scheduling for airline crews and sports games, improve production and distribution efficiency for manufacturing companies, increase service quality and efficiency in healthcare administration, and develop sophisticated tools for finance and investments.

The mathematics portion of the plan includes combinatorics, linear optimization, modeling, scheduling, forecasting, decision theory, and computer simulation. After first year, you'll choose one of 2 specializations: Operations Research or Business.

You can gain 20 months of paid work experience through our co-op program, the largest of its kind in the world, or fast track your degree by choosing the regular system of study.

Examples of mathematical optimization

b) GLOBAL MAXIMUM/ MINIMUMWe say that the functionf(x) has aglobal maximumatx=x0on the intervalI, if$f(x_0)\geq f(x)$for all$x\inI$. Similarly, the functionf(x) has a global minimumatx=x0on the intervalI, if$f(x_0)\leq f(x)$for all$x\inI$.Iff(x) is a continuous function on a closed bounded interval [a,b], thenf(x) will have a global maximum and a global minimum on [a,b]! (This is not easy to prove, though).On the other hand, if the interval is not bounded or closed, then there is no guarantee that a continuous functionf(x) will have global extrema. Examples:f(x)=x2does not have a global maximum on the interval$[0,\infty)$, the function$f(x)=-\frac{1}{x}$does not have a global minimum on the interval (0,1).How can we find global extrema? Unfortunately, not every global extremum is also a local extremum:

Example.Consider the functionf(x) = (x-1)2, for$x \in[0,3]$. The only critical point isx=1. And the first or second derivative test will imply thatx=1 is a local minimum. Looking at the graph (see below) we see that the rightendpointof the interval [0,3] is the global maximum.

This leads us to introduce the new concept ofendpoint extrema. Indeed, ifcis an endpoint of the domain off(x), thenf(x) is said to have an endpoint maximum atciff$f(x)\leq f(c)$for allxin the domain close toc. Similarly one can define the concept of an endpoint minimum.The news is not too bad, though. Iff(x) is differentiable on the intervalI, then:Every global extremum is a local extremum or an endpoint extremum.

This suggests the following strategy to find global extrema:Find the critical points.

List the endpoints of the interval under consideration.

The global extrema off(x) can only occur at these points! Evaluatef(x) at these points to check where the global maxima and minima are located.

Example.Let us find the global extrema of the functionf(x)=x e-xon the interval [0.1,3.5]. The functionf(x)is differentiable everywhere, its derivativef'(x)=e-x-xe-x=(1-x)e-xis zero only atx=1. Thusx=1 is the only critical point. Throw in the endpoints of the intervalx=0.1 andx=3.5, and evaluatef(x):

Thus the global minimum occurs atx=0.1, the global maximum occurs atx=1.

C) LOCAL MAXIMUM/MINIMUM

Functions can have "hills and valleys": places where they reach a minimum or maximum value.

It may not be the minimum or maximum for thewhole function, butlocallyit is.

We can see where they are,
but how do we define them?

Local MaximumFirstwe need to choose an interval:

Then we can say that a localmaximumis the point where:

Or, more briefly:f(a) f(x) for all x in the intervalIn other words, there is no height greater than f(a).Note: f(a) should be inside the interval, not at one end or the other.Local MinimumLikewise, a localminimumis:f(a) f(x) for all x in the interval

b)

FURTHER
EXPLORATION

LINEAR PROGRAMMING

(a):Linear programming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization).

More formally, linear programming is a technique for the optimization linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defines as the intersection of infinitely many help spaces, each of which is defines by a linear inequality. Its objective function is a real-valued affine function defined on the polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value such a point exist.

APPLICATION IN REAL LIFE

Crew scheduling

An airline has to assign crews to its flights. Make sure that each flight is covered. Meet regulations, eg, each pilot can only fly a certain amounteach day. Minimize costs, eg: accommodation for crews staying overnightout of town, crews deadheading. Would like a robust schedule.The airlines run on small profit margins, so saving a few percentthrough good scheduling can make an enormous difference interms of profitability.They also use linear programming for yield management.

TELECOMUNICATIONS

Call routing: Many telephone calls from New York to Los Angeles, from Houston to Atlanta, etc. How should these calls be routed through the telephone network?

Network design: If we need to build extra capacity, which links should we concentrate on? Should we build new switching stations?

Internet traffic: For example, there was a great deal of construction of new networks for carrying internet traffic a few years ago

HOW IT STARTED

Leonid Vitalyevich Kantorovich

The problem of solving a system of linear inequalities dates back at least as far as Fourier, who in 1827 published a method for solving them, and after whom the method of FourierMotzkin elimination is named. The first linear programming formulation of a problem that is equivalent to the general linear programming problem was given by Leonid Kantorovich in 1939, who also proposed a method for solving it. He developed it during World War II as a way to plan expenditures and returns so as to reduce costs to the army and increase losses incurred by the enemy. About the same time as Kantorovich, the Dutch-American economist T. C. Koopmans formulated classical economic problems as linear programs. Kantorovich and Koopmans later shared the 1975 Nobel prize in economics. In 1941, Frank Lauren Hitchcock also formulated transportation problems as linear programs and gave a solution very similar to the later Simplex method; Hitchcock had died in1957 and the Nobel prize is not awarded posthumously. During 1946-1947, George B. Dantzig independently developed general linear programming formulation to use for planning problems in US Air Force. In 1947, Dantzig also invented the simplex method that for the first time efficiently tackled the linear programming problem in most cases. When Dantzig arranged meeting with John von Neumann to discuss his Simplex method, Neumann immediately conjectured the theory of duality by realizing that the problem he had been working in game theory was equivalent. Dantzig provided formal proof in an unpublished report "A Theorem on Linear Inequalities" on January 5, 1948. Postwar, many industries found its use in their daily planning. Dantzig's original example was to find the best assignment of 70 people to 70 jobs. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the observable universe. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the simplex algorithm. The theory behind linear programming drastically reduces the number of possible solutions that must be checked. The linear-programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior-point method for solving linear-programming problems.

(b) i) (a) I. Cost : 100x + 200y 1400 Space : 0.6x + 0.8y 7.2

III. Volume = 0.8x + 1.2y

(b) I.

x024681214

y7654310

II.

x024681012

y97.564.531.50

(ii) Maximum storage volume Method 1 Test using corner point of Linear Programming Graph (8, 3), (0, 7), and (12, 0) Volume = 0.8x + 1.2y Coordinate 1 (8,3)Volume = 0.8(8) + 1.2(3) Volume = 10 cubic meter

Coordinate 2 - (0,7) Volume = 0.8(0) + 1.2(7)Volume = 8.4 cubic meter

Coordinate 3 - (12,0) Volume = 0.8(12) + 1.2(0) Volume = 9.6 cubic meter

Thus the maximum storage volume is 10 cubic meter.

Method 2-Using simultaneous equation

Applying the value of x and y in formula, Volume=0.8x+1.2y Thus, the maximum storage volume is 10 cubic meter

(iii)Cabinet xCabinet yTotal Cost (RM)

461600

551500

641400

731300

831400

921300

(iv) I would choose (8,3) , 8 cabinet x and 3 cabinet y . This is because the total cost does not exceed the limit amount this is RM1400 and the choice provided the biggest space 10m

REFLECTION

Ive found a lot of information while conducting this Additional Mathematics project. Ive learnt the uses of function in our daily life.

Apart from that, Ive learnt some moral values that can be applied in our daily life. This project has taught me to be responsible and punctual as I need to complete this project in a week. This project has also helped in building my confidence level. We should not give up easily when we cannot find the solution for the question.

Then, this project encourages students to work together and share their knowledge. This project also encourages students to gather information from the internet, improve their thinking skills and promote effective mathematical communication.

Lastly, I think this project teaches a lot of moral values, and also tests the students understanding in Additional Mathematics. Let me end this project with a poem;

In math you can learn everything,Like maybe youll like comparing,You have to know subtraction,a.k.a brother of addition,You might say I already simplified,so now your work aint jankedified,So now dont think negative,Its better to think positive,Dont stab yourself with a fork,But its better to show your work,My math grades are fat,But not as fat as my cat,Lets get typical,And use a pencil,Add Math is fun!

INTRODUCTION

What is FUNCTION?

Inmathematics, a functionis arelationbetween asetof inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real numberxto its squarex2. The output of a function f corresponding to an inputxis denoted byf(x) (read "fofx"). In this example, if the input is 3, then the output is 9, and we may writef(3) = 9. Likewise, if the input is 3, then the output is also 9, and we may writef(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The inputvariable(s)are sometimes referred to as the argument(s) of the function.Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by aformulaoralgorithmthat tells how to compute the output for a given input. Others are given by a picture, called thegraph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as theinverseto another function or as a solution of a differential equation.The input and output of a function can be expressed as anordered pair, ordered so that the first element is the input (ortupleof inputs, if the function takes more than one input), and the second is the output. In the example above,f(x) =x2, we have the ordered pair (3, 9). If both input and output arereal numbers, this ordered pair can be viewed as theCartesian coordinatesof a point on the graph of the function.

b) METHODS IN FINDING MAXIMUM OR MINIMUM VALUE OF QUADRATIC FUNCTION