What is Optimization?
Mathematical optimization or optimization means to select the feasible element that depends on a specific standard from a set of available options.
A specific optimization problem includes minimizing or maximizing real functions efficiently by selecting input values within a given set and calculating the function’s value. It is applied in numerous areas of mathematics for specifying the theory of optimization. Optimization means examining “best available” values of the specific objective function in a defined domain including multiple types of objective functions. This article will define what is optimization, Mathematical optimization problems, why use Mathematical optimization etc.
Mathematical Optimization Problems
Now, let us look at some optimization problems. Here, you need to look for the highest or the smallest value that can be considered as a function. The constraint will be normal that can be specified by an equation.
The constraint is the quantity that has to be valid regardless of the solution. You will be looking at one quantity that is clear and has a constant value in every problem. Once you will clearly recognize the quantity to be optimized, it’s not so problematic to calculate it further.
Why use Mathematical Optimization?
Optimization is a mathematical approach that considers all the factors that influence business decisions. Optimization means careful modeling of the business, a process which itself provides valuable information. The benefits of mathematical optimizations are operational efficiency, cost minimization, performance assessment, and understanding the effects of the variation made in input data.
Important factors included in the optimization are:
Decisions- These are the things that can vary, the things we need to choose upon. For example, the number of products that can be made, how to make the product and dispatch it.
Constraints- These designs are the limitation of our decisions. For example, in a logistic problem each mode of transportation has maximum speed and payload, Operations can be controlled for many hours in a day.
Mathematical Optimization Problems in business
Here are some examples of mathematical optimization which will help you to know how mathematical optimization is helpful in business.
Portfolio Management- Mathematical optimization helps the business to manage its portfolio. With this, the entrepreneur can decide what stocks and number of stocks should be included in a portfolio that provide maximum return and minimize risk.
Stock Level Management - What stock the business should maintain and when to meet the overall cost of the stock but still meet the required supply SLAs.
Hotel Business- What should be the feasible price of the room that will maximize occupancy while considering room availability but staying within a range of prices and considering estimated take-up for the range of possible prices?
Solved Examples
Determine two positive numbers whose sum is 300 and whose product is maximum.
Step 1. The first step is to write the equation which will describe the situation.
Let us take two number p and q whose sum is 300
p + q = 300
Now we will maximize the product
A = pq
Step 2. Now, we will solve the constraint and substitute this in the above equation
q = 300 - p A(p) = p(300 - p) = 300p - p²
Step 3. Now we will find the critical points for the equation
A’(p) = 300 - 2p 300 - 2p = 0 p = 150
Step 4. As we got a single value and we can’t assume that this will provide us a maximum product. We will examine to see whether it will give us maximum value
There are multiple methods to verify this ,but in this case, we can quickly see that
A’’ (p)= -2
With this, we can conclude that the second derivative is also negative and so A(p) will always concave down and the critical point which we got in step 3 must be relatively maximum and can be a value that gives us a maximum product.
Step 5. And in this step, we will determine the value of y as we already have the value of x and that can be easily done from the constraint.
q = 300 - 150= 150
Hence, the final answer is p= 150 and q = 150
Let p and q be two positive numbers such that p + 2q and (p+1) (q+2) is a maximum.
Solution:
Step 1. As we have been given constraints of the above problem, it will be represented as
P + 2q = 50
We are asked to maximize the equation
f = (p +1) (q +2)
Now we will solve the constraints for p and q and substitute this into an product equation
p= 50- 2q → f(q) = (50 - 2q + 1)(q + 2) = (51- 2q)(q + 2) = 102 + 47q - 2q²
Step 2. In this step, we will find the critical point for the equations.
f’(q) = 47- 4q → 47- 4q = 0
Step 3. As we got a single value and we can't just assume that will provide us a maximum product. We will quickly check whether it will give us maximum value.
f’’(q) = - 4
With this, we can conclude that the second derivative is also negative and so f (p) will always be concave down and the critical point which we got in step 2 must be relatively maximum and can be a value that gives us a maximum product.
Step 4. Finally, in this step we will answer the question. We are required to provide both the values. As we already have q so we need to find p and that can be easily done from the constraint.
p = 50 - 2(47/4) = 53/2
The final answer to the question is
p= 53/2 and q = 47/4
Quiz time
Which of the following is not a step to solve the optimization problem?
Write down an equation
Answer the problem
Find the minimum or maximum value
Construct a detailed graph
Fun Facts
George Dantgi introduced the optimization to solve the LPP involving multiple equations and various variables.
FAQs on Optimization
1. What is meant by optimization in mathematics, and why is it important in real-world scenarios?
Optimization in mathematics refers to finding the best possible value (maximum or minimum) of a function within a defined set of constraints. It is important in real-world scenarios like business, engineering, and economics, as it helps in making decisions that maximize profit or efficiency and minimize costs or losses.
2. How do you identify the objective function and constraints in an optimization problem?
To solve an optimization problem, the objective function is the mathematical expression to be maximized or minimized (e.g., profit, area, volume), while constraints are equations or inequalities that limit the possible values of variables. Carefully read the problem to determine what needs optimizing and the conditions that must be met.
3. What are the main steps involved in solving an optimization problem according to the CBSE Class 12 Maths syllabus?
The main steps are:
- Define variables relevant to the problem.
- Formulate the objective function to be optimized.
- Identify and express the constraints.
- Substitute constraints into the objective function if possible.
- Differentiate the function, find critical points, and check for maxima or minima.
- Verify if the solution satisfies all constraints.
4. Can you provide an example of an optimization problem involving two variables and explain the solution process?
Suppose two positive numbers add up to 300, and their product is to be maximized. Let the numbers be p and q: p + q = 300. The product is P = pq. Substitute q = 300 – p, so P(p) = p(300 – p) = 300p – p². Differentiate and set the derivative to zero to find the critical point: dP/dp = 300 – 2p = 0 gives p = 150. Thus, p = q = 150 maximizes the product, as verified by the second derivative test.
5. What roles do the first and second derivatives play in optimization problems?
The first derivative helps locate critical points where the function may reach a maximum or minimum. The second derivative confirms the nature of these points: if it is negative at a critical point, the function has a maximum there; if positive, a minimum.
6. Why is it necessary to check boundary values in constrained optimization problems?
In some optimization problems, the maximum or minimum may occur at the boundaries of the feasible region defined by constraints, not just at critical points. Therefore, it's important to evaluate the objective function at endpoints or under extreme values as well.
7. How is mathematical optimization applied in business decision-making?
Mathematical optimization is widely used in business for resource allocation, cost minimization, revenue maximization, and performance assessment. Examples include portfolio management, determining optimal pricing, and deciding inventory or workforce levels within constraints.
8. What are common mistakes students make when attempting optimization problems in board exams?
Common mistakes include:
- Incorrectly identifying constraints or missing them.
- Not defining variables clearly.
- Applying the differentiation process inaccurately.
- Ignoring boundary or endpoint values.
- Forgetting to check if the answer meets all constraints.
9. How can students improve accuracy in solving optimization questions in the CBSE Class 12 Maths paper?
Students can improve by:
- Practicing a variety of optimization problems.
- Writing each step clearly, especially variable definition and constraint equations.
- Using proper differentiation techniques.
- Always verifying if the solution makes sense physically and mathematically.
10. In what way does optimization connect with other Class 12 Maths topics such as calculus and linear programming?
Optimization heavily relies on calculus (for identifying maxima and minima) and is closely linked to linear programming where objective functions and constraints are both linear. These connections deepen understanding and problem-solving skills across topics in the syllabus.
