Understanding R Optimization: Best Possible Solutions

When faced with a problem—whether it’s deciding how many products to manufacture, how to allocate resources, or even how to minimize costs—there are usually many possible solutions. But only one (or a few) solutions can be considered the “best.” This process of finding the most effective solution is called optimization.

In simple terms, optimization helps you make the smartest possible choice given your goals and limitations. It does this using mathematical models and algorithms that weigh different possibilities until they arrive at the best outcome.

For example, a business might want to:

• Maximize profits

• Minimize costs

• Reduce production time

• Increase sales

Each of these goals can be framed as an objective in an optimization problem. Once the objective is defined, optimization techniques can help determine the best path forward.

Types of Optimization

There are many types of optimization, but let’s start with two commonly used approaches: unconstrained optimization and linear programming.

1. Unconstrained Optimization

In unconstrained optimization, the variables can take any value within their range. There are no strict limits (like resource caps or time restrictions) placed on the solution.

For example, in R, the optim() function can be used to find the minimum or maximum of a given function. Suppose we define a mathematical function that we want to minimize:

f <- function(x) 4 * (x[1]-1)^2 + 7 * (x[2]-3)^2 + 30

By using optim(), R runs through different possible values of x and finds the point where the function reaches its minimum. In this case, the function converges to a value of 30 when the inputs are approximately (1, 3).

The key idea: optimization helps pinpoint the sweet spot where your function performs best.

2. Linear Programming (LP)

Linear programming is one of the most powerful and widely used optimization techniques. It deals with problems where both the objective function and the constraints are linear.

Think of LP as answering this question:

“Given limited resources, what’s the best way to maximize my outcome (like profit) or minimize my downside (like costs)?”

A popular definition (from Techopedia) describes LP as a method to determine the best outcome from a set of requirements represented as linear relationships. Businesses use LP to allocate scarce resources like money, labor hours, or raw materials.

Example 1: A Company Producing Two Products

Imagine a company that produces two products, A and B.

• Product A sells for $25 and requires 20 resource units.

• Product B sells for $20 and requires 12 resource units.

• The company has 1,800 resource units available per day.

• Both products take 4 minutes to produce, and there are only 8 working hours (480 minutes) available.

The question is: How many units of A and B should be produced to maximize profits?

We can write this problem as:

• Objective function (maximize sales):
  25y1 + 20y2
  where y1 = units of A, y2 = units of B

• Constraints:
  20y1 + 12y2 <= 1800 (resource limit)
  4y1 + 4y2 <= 480 (time limit)
  y1, y2 >= 0

Using R’s lpSolve package, we find that the optimal solution is to produce 45 units of A and 75 units of B, leading to maximum sales of $2,625.

This is optimization at work—turning a resource allocation problem into a clear, data-backed decision.

Example 2: A Farmer’s Dilemma

Consider a farmer with 75 acres of land who can plant wheat or barley.

• Planting wheat costs $120 per acre and barley $210.

• The farmer has $15,000 available.

• Storage capacity is 4,000 bushels.

• Each acre of wheat produces 110 bushels, barley produces 30.

• Profit is $1.30 per bushel of wheat, $2 per bushel of barley.

The farmer wants to maximize profit.

The LP model becomes:

• Maximize profit:
  143x + 60y
  where x = acres of wheat, y = acres of barley

• Constraints:
  120x + 210y <= 15000 (budget)
  110x + 30y <= 4000 (storage)
  x + y <= 75 (land)
  x, y >= 0

When solved (either in Excel Solver or R), the optimal solution is to plant about 22 acres of wheat and 53 acres of barley, resulting in a maximum profit of around $6,316.

Solving LP Problems: Tools You Can Use

There are several ways to solve optimization problems:

• Graphical Method – Useful for problems with only two variables.

• Simplex Method – A widely used algorithm for larger, more complex problems.

• Software Tools:

  • R packages like lpSolve and lpSolveAPI

  • Excel Solver, which provides an intuitive, no-code approach

For small business owners, Excel Solver can be an excellent starting point. For data analysts, R offers flexibility and scalability to solve bigger problems programmatically.

Why Optimization Matters

Optimization isn’t just about math—it’s about smarter decision-making. Organizations use it to:

• Allocate budgets more effectively

• Optimize supply chains

• Improve production schedules

• Reduce costs while maintaining quality

• Plan workforce requirements

In short, optimization translates complexity into clarity. It allows leaders to base decisions not just on intuition but on hard numbers, ensuring they make the best possible choice under given constraints.

Conclusion

Optimization is a powerful tool for tackling real-world problems—from deciding how many products to produce, to how a farmer should allocate land for crops. Whether through simple functions in R, advanced linear programming models, or even Excel’s Solver, optimization gives us a structured way to achieve the best outcome.

Next time you’re faced with a decision involving limited resources, think about how optimization techniques could help. Chances are, the “best” solution is waiting to be discovered mathematically.

Our mission is to empower organizations to unlock the full power of their data and automation. With over 20 years of experience, we’ve helped more than 100 clients—from Fortune 500 enterprises to fast-growing firms—tackle complex challenges and drive meaningful outcomes. Our capabilities include Chatbot Consulting to elevate user engagement, expert support from an Excel VBA Programmer to automate and streamline workflows, and advanced services from an Excel Expert to optimize reporting and analytics clarity.