Linear Programming and Reference File Download Link

Fundamental Concepts

In an LP problem we define:

Decision variables: the quantities we want to determine.
Objective function: a linear expression of the decision variables that we want to maximize (e.g., profit) or minimize (e.g., cost).
Constraints: linear relationships that restrict the feasible values of the decision variables.

All components must be linear each variable appears to the first power and there are no products of variables.

Standard Form

The most common way to write an LP problem is the standard form:

max   cxs.t.  Ax  b      x  0

where:

c is a vector of coefficients for the objective function.
x is the vector of decision variables.
A is a matrix of constraint coefficients.
b is the righthand side vector.

Graphical Illustration (Two Variables)

When only two decision variables exist, the feasible region can be drawn on a Cartesian plane. The optimal solution lies at a corner (vertex) of this region.

Example: Diet Problem

Suppose a nutritionist wants to minimize the cost of a diet that satisfies daily requirements of protein and calories.

Food	Cost ($/unit)	Protein (g)	Calories
Food A	0.50	10	200
Food B	0.30	6	150

Decision variables: x = units of Food A, x = units of Food B.

Formulation:

min   0.5x + 0.3xs.t.  10x + 6x  50   (Protein requirement)      200x + 150x  1500 (Calorie requirement)      x, x  0

The feasible region is the intersection of the two halfplanes. The optimal point is found at the intersection of the constraints, yielding x = 2 and x = 5, costing $2.90.

Solution Methods

Simplex Method

Developed by George Dantzig in 1947, the simplex algorithm moves from one vertex of the feasible polyhedron to an adjacent one with an improved objective value, terminating at the optimal vertex. It is highly efficient in practice, though its worstcase complexity is exponential.

InteriorPoint Methods

Introduced in the 1980s, interiorpoint algorithms travel through the interior of the feasible region toward the optimum. They have polynomialtime complexity and are especially effective for very large sparse problems.

Software Packages

Commercial: IBM ILOG CPLEX, Gurobi, FICO Xpress.
Open source: COINOR CLP, GLPK, SciPys linprog.

Typical Applications

Production Planning: determining quantities of products to manufacture to maximize profit while respecting resource capacities.
Transportation & Logistics: minimizing shipping costs by optimizing routes and loads.
Finance: portfolio optimization under risk constraints.
Energy: unit commitment and economic dispatch in power systems.
Supply Chain: inventory management, facility location, and order fulfillment.

Advantages and Limitations

Advantages

Provides a global optimum for linear models.
Wellstudied theory with robust, widely available solvers.
Fast to solve for moderatesize problems.

Limitations

Requires linearity; many realworld problems are nonlinear or discrete.
Solutions are sensitive to data; small coefficient changes can shift the optimum.
Modeling complex logical conditions sometimes needs additional binary variables, turning the problem into a mixedinteger program, which is harder to solve.

Getting Started with a Simple LP Model

Below is a concise Python snippet using scipy.optimize.linprog to solve a basic production problem.

import numpy as npfrom scipy.optimize import linprog# Maximize profit: 3x + 5x   (converted to minimization by negating)c = [-3, -5]# Resource constraints:# 2x + 4x  100   (material A)# 3x + 2x  90    (material B)A = [[2, 4],     [3, 2]]b = [100, 90]# Bounds for x and x (nonnegative)bounds = [(0, None), (0, None)]result = linprog(c, A_ub=A, b_ub=b, bounds=bounds, method='highs')print('Optimal quantities:', result.x)print('Maximum profit:', -result.fun)

The script returns the optimal production levels and the corresponding maximum profit.

Conclusion

Linear programming offers a powerful, mathematically rigorous way to allocate scarce resources efficiently. Whether you are a manager planning a manufacturing schedule, a trader balancing a portfolio, or a researcher modeling a supply network, an LP model provides clear guidance. While the linearity assumption limits its direct applicability to some problems, extensions such as mixedinteger and quadratic programming broaden the scope dramatically. Mastering the basics of LP and its solution techniques equips you with a versatile tool for solving a wide variety of optimization challenges.

Linear Programming