Structure of a Linear Programming Problem

Optimization lies at the heart of modern analytics, operations research, and decision science. From minimizing operational costs to maximizing profits or efficiency, optimization techniques help organizations make the best possible decision under constraints.
In analytics-driven enterprises, optimization models power pricing engines, supply chain planning, workforce scheduling, portfolio allocation, and AI-driven decision systems.

What Is Optimization?
Optimization is a mathematical technique used to identify the best possible solution from a set of feasible solutions, given an objective and a set of constraints.
At its core, every optimization problem has three components:
Decision Variables – the values we want to determine
Objective Function – what we want to maximize or minimize
Constraints – limitations or rules that restrict feasible solutions
Common Optimization Objectives
Maximize profit or revenue
Minimize cost, time, or risk
Optimize resource utilization
Improve operational efficiency
Optimization uses rigorous mathematical models to ensure solutions are not just intuitive—but provably optimal.

Types of Optimization Techniques
Optimization problems broadly fall into several categories:
Unconstrained Optimization – no restrictions on variable values
Constrained Optimization – variables must satisfy equality or inequality conditions
Linear Programming (LP) – linear objective and constraints
Nonlinear Programming – nonlinear relationships
Integer Programming – variables must be integers
In this article, we focus on Unconstrained Optimization and Linear Programming, with hands-on implementations in R and Excel.

Unconstrained Optimization in R
In unconstrained optimization, decision variables can vary freely within their domain. These problems are common in statistical modeling, curve fitting, and machine learning.
The optim() Function in R
The optim() function is a general-purpose optimizer that supports both one-dimensional and multi-dimensional problems.
General syntax:
optim(par, fn, gr = NULL, method = "Nelder-Mead",
lower = -Inf, upper = Inf, control = list(), hessian = FALSE)

Example: Minimizing a Quadratic Function
Let’s define an objective function:
f <- function(x) 4 * (x[1]-1)^2 + 7 * (x[2]-3)^2 + 30
f

This function has a global minimum when:
x1 = 1
x2 = 3
Now define initial values:
c <- c(1, 1)

Run the optimization:
r <- optim(c, f)
r

Output Interpretation
$par
[1] 0.9999207 3.0001660

$value
[1] 30

$convergence
[1] 0

$par → Optimal values of decision variables
$value → Minimum value of the objective function
$convergence == 0 → Optimization succeeded
This confirms the optimizer converged to the global minimum.

Linear Programming (LP)
What Is Linear Programming?
Linear Programming is used when:
The objective function is linear
Constraints are linear inequalities or equalities
Resources are finite
LP is widely used in:
Supply chain optimization
Production planning
Workforce allocation
Financial portfolio optimization
Definition (Technopedia):
Linear programming determines the best outcome—such as maximum profit or minimum cost—under linear constraints.

Structure of a Linear Programming Problem
General form:
Maximize or Minimize:
f(y1, y2) = g1·y1 + g2·y2

Subject to:
g11·y1 + g12·y2 ≤ p1
g21·y1 + g22·y2 ≤ p2
y1 ≥ 0, y2 ≥ 0

Example 1: Product Mix Optimization
Business Problem
A company produces Product A and Product B.
ParameterProduct AProduct B
Profit per unit
$25
$20
Resource usage
20 units
12 units
Time required
4 min
4 min
Constraints:
1800 total resource units/day
8 working hours/day (480 minutes)

Step 1: Define the Objective Function
Maximize Sales = 25y1 + 20y2

Where:
y1 = units of Product A
y2 = units of Product B

Step 2: Define Constraints
20y1 + 12y2 ≤ 1800 (Resource)
4y1 + 4y2 ≤ 480 (Time)
y1, y2 ≥ 0

Solving LP in R Using lpSolve
Step 1: Load Package and Define Objective
require(lpSolve)
objective.in <- c(25, 20)

Step 2: Define Constraint Matrix
const <- matrix(c(20, 12, 4, 4), nrow=2, byrow=TRUE)

Step 3: Define RHS and Direction
rhs <- c(1800, 480)
direction <- c("<=", "<=")

Step 4: Solve the LP
optimum <- lp(direction="max", objective.in, const, direction, rhs)

Results
optimum$solution
[1] 45 75

optimum$objval
[1] 2625

Optimal decision:
Produce 45 units of Product A
Produce 75 units of Product B
Maximum profit = $2625

Example 2: Agricultural Optimization Problem
A farmer has 75 acres to plant wheat and barley.
Profit Function
Maximize g = 143x + 60y

Constraints
120x + 210y ≤ 15000 (Budget)
110x + 30y ≤ 4000 (Storage)
x + y ≤ 75 (Land)
x, y ≥ 0

Solving Using Excel Solver
Excel Solver uses the Simplex algorithm and is ideal for business users.
Key steps:
Define decision cells
Use SUMPRODUCT() for constraints
Set objective cell
Apply constraints
Choose Simplex LP
Maximize objective
This approach is useful for:
Non-programmers
Rapid prototyping
Business demonstrations

Solving Using lpSolveAPI in R
Step 1: Create LP Model
require(lpSolveAPI)
lprec <- make.lp(0, 2)
lp.control(lprec, sense="max")

Step 2: Define Objective and Constraints
set.objfn(lprec, c(143, 60))
add.constraint(lprec, c(120, 210), "<=", 15000)
add.constraint(lprec, c(110, 30), "<=", 4000)
add.constraint(lprec, c(1, 1), "<=", 75)

Step 3: Solve and Extract Results
solve(lprec)
get.objective(lprec)
get.variables(lprec)

Output
Profit: 6315.625
Wheat: 21.875 acres
Barley: 53.125 acres

Real-World Applications of Optimization
Optimization techniques power critical enterprise decisions such as:
Supply chain network design
Inventory replenishment
Pricing and revenue optimization
Workforce scheduling
Risk-adjusted portfolio allocation
Our AI consulting company frequently leverages these optimization models to support supply chain planning, pricing intelligence, and decision automation at enterprise scale.

Conclusion
Optimization is not just a mathematical exercise—it is a decision-making superpower.
By mastering:
Unconstrained optimization
Linear programming
Solver-based implementations in R and Excel
You unlock the ability to transform raw data into optimal, defensible decisions.
The next step is to integrate these models into automated pipelines, AI agents, and decision intelligence systems—where optimization moves from theory into continuous execution.
At Perceptive Analytics, our mission is “to enable businesses to unlock value in data.” For over 20 years, we’ve partnered with more than 100 clients—from Fortune 500 companies to mid-sized firms—to solve complex data analytics challenges. Our services include working with experienced advanced analytics consultants and collaborating with trusted AI consulting firms, turning data into strategic insight. We would love to talk to you. Do reach out to us.