INTRODUCTION TO MATHEMATICAL MODELING

Lecture Note 5

This lecture note was originally written by a student and partly modified and certified by the

course instructor.

INTRODUCTION TO MATHEMATICAL MODELING

A business mathematical model is a mathematical representation of a worded business problem. For business problems amenable to mathematical representation, a mathematical equivalent (or formulation) of the problem enables richer analysis than the worded equivalent due to its clear and concise form of representation. The first step of any mathematical formulation is the definition of so-called ‘decision variables’.

Step 0:

Define the decision variable

1. Imagine you have Ghc 10,000 to invest in two assets. Your concern is how much to invest in asset 1 and asset 2. It could be anything, depending on other information at hand. For instance, you could put Ghc 1000 in asset 1 and Ghc 9000 in asset 2. You could also (depending on other information) decide to put Ghc 6000 in asset 1 and Ghc 4000 in asset 2. The number of ways of the division could be endless (assuming no other information exist). Instead of thinking of all the possible allocations, we will rather let 𝑥1, be a variable representing the amount of money

invested in asset 1 and 𝑥2, the variable representing the amount invested asset 2.

2. A company manufactures products A and B using machine 1 and machine 2.

Suppose machine 1 is available 60 hours per week, machine 2 is available 80 hours per week. In addition, to produce a unit of product A would require 2 hours on machine 1 and 4 hours on machine 2, whereas a unit of product B would require 1 hour on machine 1 and 2 hours on machine 2. In a problem such as this, the challenge is to decide how many of product A and product B to produce taking into consideration the limitations on the availability of machine 1 and machine 2.

Solution

Let 𝑥𝐴 and 𝑥𝐵 be the variable representing the number of unit of product A and product B

respectively to produce per week. Whatever 𝑥𝐴 and 𝑥𝐵 are, the total hours spent on machine 1

cannot exceed 60 hours per week and the total hours on machine 2 cannot exceed 80 hours per week.

Using the defined variables, a mathematical model for the limitations could be written as: Alternatively, 2𝑥𝐴 + 𝑥𝐵 ≤ 60

4𝑥𝐴 + 2𝑥𝐵≤ 80

After defining the necessary variables, the other information regarding a worded problem

statement would then be translated into its equivalent mathematical form with the help of

decision variables (e.g. the representation of the variables in example 2 above). This is generally

accomplished using a combination of equations, inequalities and functions. For this course, we

will limit ourselves to only:

-

Linear equations

-

Linear inequalities

-

Linear or multi linear functions

We will call the symbols {

=, <,>, ≤

,

≥,

} found at the right-hand-side (RHS) or

left-hand-side (LHS) in the linear equations and inequalities Restrictions, Constraints or

Limitations.

Linear equation

A linear equation is of the form:

𝒂

𝒙

+ 𝒂

𝒙

+ 𝒂

𝒙

+ ⋯ + 𝒂

𝒙

= 𝒃

Where 𝒂

is a

coefficient

/

parameter,

𝒃 is a parameter, and 𝒙

is a

variable

.

Linear equations are one of the tools used in translating a worded business problem into a

mathematical form for analysis.

Examples

1. A chemist must prepare 350ml of a chemical solution made up of two parts alcohol and three parts acid. How much of each solution should be used?

Solution

If we let Let ‘n’ represent amount of one part of the solution, then:

2n +3n = 350 n = 350/5 n = 70

Therefore, amount of Alcohol =140 and amount of Acid =210

2. A company earns a before tax of GH¢100,000. It has agreed to contribute 10% of its after tax profit to the Red Cross Relief Fund. It must pay a regional tax of 5% of its profit (after the Red Cross donation) and a state tax of its profit (after the Red Cross donation and regional tax). How much does the company pay in regional tax, state tax and Red Cross donation?

Solution

To answer the question, we will first translate the worded problem into an equivalent mathematical form.

Step 0:

Let R and S be variables representing the amounts paid for regional tax and state tax respectively. Also, let C be the contribution to the Red Cross.

Step 1:

After tax profit = 100000 – (S+R) C= 0.1(100000 –S – R) R= 0.05(100000 – C) S= 0.4(100000 –C – R)

The three linear equations together does represent the worded problem above.

3. The Smith Company would like to know the total sales units that are required for the company to earn a profit of GH¢150,000 if the unit selling price is GH¢50, the variable cost per unit is GH¢25, and the total fixed cost is GH¢500000.

Solution

Step 0:

Let q be the variable representing the number of sales units required to achieve a profit of GH¢150,000.

Profit = Revenue – Total Cost

Revenue = unit selling price x sales units = 50q

Total cost = total variable cost + total fixed cost = 500000 + 25q. Therefore, mathematically,

Profit = 50q – (500000 + 25q).

If we want to achieve a profit of GH¢150,000, we must have: 150,000 = 25q – 500000

Then we can solve for that q that will results in GH¢150,000 profit. Rearranging, we have: 25q = 650000

q= 26,000

So, the number of required sales units is 26,000 units.

4. The XYZ manufacturing company has a total fixed cost of GH¢1200, a variable cost per unit of GH¢2, and a total revenue function for selling q units of product as R(q) =100√q. Determine the break-even quantity of XYZ Manufacturing Company.

Solution

At break-even, Total Cost = Total Revenue Let ‘q’ represent quantity

Total Cost = Fixed Cost + Variable Cost Total Cost = 1200 + 2q

(1200 + 2q)2 _{= (100√q)}2

1,440,000 + 4800q + 4q2 _{= 10000q}

4q2 _{– 5200q + 1,440,000 = 0}

q= 400; q=900

Linear Inequalities

Linear inequalities are of the form:

𝒂_𝟏𝒙_𝟏+ 𝒂_𝟐𝒙_𝟐+ 𝒂_𝟑𝒙_𝟑+ ⋯ + 𝒂_𝒏𝒙_𝒏≤ 𝒃 𝒂_𝟏𝒙_𝟏+ 𝒂_𝟐𝒙_𝟐+ 𝒂_𝟑𝒙_𝟑+ ⋯ + 𝒂_𝒏𝒙_𝒏≥ 𝒃

Where

𝒂

is a

coefficient

/

parameter,

𝒃 is a parameter, 𝒙

is a

variable

, and the symbols ‘≤, ≥ ‘

stand for ‘less-than or equal-to’ and ‘greater-than or equal-to respectively.

Familiar Words and their equivalent symbols

At least/ not less than

≥

At most/ not more than

≤

Exactly/ should be/ Must be

=

Examples:

1.

A person wishes to invest GH¢20,000 in two enterprises so that total returns for the year

would be at least GH¢1,440. One enterprise pays 6% annually; the other has more risks

and so pays 8% annually. How much must be invested in each enterprise?

Solution

Step 0:

Let x1, x2 be the amount invested in enterprises 1 and 2 respectively.

Step 1:

x1 + x2 = 20,000 ( equation for total amount)

0.06x1 + 0.08x2 ≥ 1,440 (equation for total returns)

Functions

In business, objectives are mostly modeled using functions. Notable business objectives

includes:

Examples:

1.

A bank is attempting to determine where its assets should be invested during the current

year. At present, GH¢500,000 is available for investment in bonds, home loans, auto

loans and personal loans. The annual rate of return on each type of investment is known

to be: bonds 10%; home loans 16%; auto loans 13%; and personal loans 20%. To ensure

that the banks portfolio is not too risky, the bank’s investment manager has placed the

following three restrictions on the bank’s portfolio:

a.

the amount invested in personal loans cannot exceed the amount invested in

bonds;

b.

the amount invested in home loans cannot exceed the amount invested in auto

loans;

c.

no more than 25% of the total amount invested may be in personal loans.

The bank’s objective is to maximize the annual return on its investment portfolio.

Formulate a mathematical model to help the bank achieve its objectives.

Solution

Step 0:

Let x1, x2, x3, and x4 be the amounts invested in bonds, home loans, auto loans and

personal loans respectively.

Step 1:

Objective – Maximization

0.1x1 + 0.16x2 + 0.13x3 +0.2x4

Constraints/Restrictions/limitations

x1 + x2 + x3 + x4 ≤ 500,000

x4 ≤ x1

x2 ≤ x3

x4 ≤ 0.25(500,000)

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0

(non-negativity constraints)

The last part of the mathematical formulation is very important since if we don’t add this, we are

implicitly assuming one can invest a negative amount, which is not possible.

**Students will lose lots of marks for not adding the non-negativity constraints for problems that

seem obvious negative amount is not possible.

The final formulation would then be:

Max 0.1x1 + 0.16x2 + 0.13x3 +0.2x4

s.t

x1 + x2 + x3 + x4 ≤ 500,000

x4 ≤ x1

x2 ≤ x3

x4 ≤ 0.25(500,000)

Where ‘s.t’ stands for ‘subject to’.

That is, we want to achieve the objective of maximizing total annual returns taking into account

(or subject to) the constraints presented. This form of mathematical formulation where a linear

objective is subject to a set of linear constraints is known as ‘Linear programming’ or ‘linear

optimization’.