Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Linear Programming and Optimization ↓


Integer Programming


Integer programming is a branch of optimization or mathematical programming. It means finding the best solution from a set of possible solutions. While linear programming deals with linear equations and some constraints, integer programming is a special type where the solution variables must be integers. This constraint makes integer programming more complex and challenging, but also more applicable in real scenarios where some resources may only be whole units.

What is integer programming?

Integer programming involves mathematical problems that aim to optimize a given objective function. At first glance it may look like linear programming, but there is an important difference: in integer programming, the probability of a function being optimized is constant unless some or all of the variables are integers. are limited only.

Integer programming can be used to solve a variety of optimization problems. Some examples include planning and scheduling, resource allocation, and decision making in industries such as logistics, manufacturing, telecommunications, and finance.

Mathematical formulation

The general form of a linear programming problem can be expressed as follows:

Max or Min: c1*x1 + c2*x2 + ... + cn*xn

subject to:

a11*x1 + a12*x2 + ... + a1n*xn (<=, =, or >=) b1
a21*x1 + a22*x2 + ... + a2n*xn (<=, =, or >=) b2
,
am1*x1 + am2*x2 + ... + amn*xn (<=, =, or >=) bm

x1, x2, ..., xn >= 0

Here, c denotes the coefficients of the objective function, a denotes the coefficients of the constraints, and b denotes the right-hand side constants. The goal is to find the values of x1, x2, ..., xn that maximize or minimize the objective function. Do the minimum.

In integer programming, one or more of the variables x1, x2, ..., xn are constrained to take only integer values. When all variables are constrained to be integers, the problem is called a pure integer programming problem. When only some of the variables are integers, this is a mixed-integer programming problem.

Example

Simple integer programming problem

Let's look at a simple integer programming example to understand these concepts. Suppose a company wants to decide what is the optimal number of products A and B to produce. The profit from one unit of product A is $3, and is $5 from product B. The company cannot produce more than 7 units of A and 4 units of product B jointly.

The integer programming model can be set up as follows:

Max: 3A+5B

subject to:

a + b <= 7
b <= 4
A, B >= 0 and integer

Here, both A and B must be integers because the company cannot make a fraction of a product. This requirement is where integer programming becomes more complex than simple linear programming.

Visualization of integer solutions

For our example, let's visualize the feasible region and the solution. Consider the restrictions plotted on the graph by specially marking the integers:

A B 2,4 3,4 4,3

In the SVG example, the line A + B = 7 and the line B = 4 are sketched with integer points. These points have (2,4), (3,4), and (4,3) are included.

Since our goal is to maximize our objective function 3A + 5B for the solution points (3,4) and (4,3), the objective values are:

  • For the point (3,4): 3 * 3 + 5 * 4 = 9 + 20 = 29
  • For the point (4,3): 3 * 4 + 5 * 3 = 12 + 15 = 27

Thus, the maximum profit is obtained at (3,4), producing 3 units of A and 4 units of B.

Challenges in integer programming

Integer programming has inherent difficulties that general linear programming does not have. Linear programming problems can be solved using methods such as the simplex algorithm, which navigates efficiently through feasible regions.

However, adding integer constraints complicates the search for a solution. Since the mathematical spaces involved in integer programming are not nice convex shapes, special algorithms are needed for visualization and computation such as:

  • Branch-and-bound: repeatedly divide the problem into sub-problems ("branches"), find feasible regions, and solve easier linear programs.
  • Cutting plane: This is an addition to the standard linear programming relaxation. It adds constraints to create a feasible region showing only integer solutions.
  • Branch-and-Cut: It combines the above methods, providing efficiency and greater practicality in solving mixed-integer and integer programs.

Applications of integer programming

Due to its nature, integer programming finds wide applications in various fields. These functions solve problems rather than exploring theoretical parts.

Following are some of the major application areas:

  • Manufacturing: contracting, machine utilization, blending and distribution.
  • Telecommunications: Bandwidth allocation, network design, and routing.
  • Logistics: inventory management, vehicle routing, and personnel scheduling.
  • Finance: Portfolio selection and capital budgeting.

Each area makes use of factors favoring integer programming, such as the discrete nature and explicit choice optimization among finite possibilities.

Conclusion

Integer programming serves as a key asset in the field of optimization. Its functionality adds to the practicality, as the constraints often seen in reality are naturally addressed by its discrete solutions. Being complex in nature and computation, integer programming is a very simple and costly process. Regardless, existing methods make it feasible and widely applicable. The potential of optimization continues to grow in adoption as industries advance, driven in part by advances in and understanding of integer programming.


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Integer Programming

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