"MATLAB Linear Programming: From Beginner to Expert: Unveiling the Principles of Algorithms and Practical Applications"

# 1. Introduction to MATLAB Linear Programming

Linear programming is an optimization technique used to find the best values for a set of decision variables within given constraints, to maximize or minimize an objective function. MATLAB offers a suite of functions for solving linear programming problems, making it a powerful tool for engineers, scientists, and data analysts to tackle real-world problems.

In this chapter, we will introduce the basic concepts of linear programming, including its standard form, mathematical principles, and solving methods. We will explore the linprog function in MATLAB used for linear programming and understand its usage, options, and parameters. Additionally, we will discuss cases of linear programming applications in the real world, such as production planning and portfolio optimization.

# 2.1 Linear Programming Model and Standard Form

### 2.1.1 Basic Concepts of Linear Programming

Linear programming is a mathematical optimization technique used to determine the values of a set of decision variables under given constraints, to maximize or minimize a linear objective function. A linear programming model typically consists of the following elements:

- **Decision variables:** The unknowns that need to be determined.
- **Objective function:** The linear function to be maximized or minimized.
- **Constraints:** Linear inequalities or equations that limit the values of decision variables.

### 2.1.2 Standard Form of Linear Programming

Linear programming models are often represented in standard form, where both the objective function and constraints are expressed as linear inequalities. The standard form is as follows:

Maximize/Minimize Z = c^T x
Subject to:
     Ax ≤ b
     x ≥ 0

Where:

- Z is the objective function.
- c is the coefficient vector of the objective function.
- x is the vector of decision variables.
- A is the matrix of constraint coefficients.
- b is the vector of right-hand constants for the constraints.

### 2.1.3 Mathematical Model of Linear Programming

The linear programming model can be expressed in mathematical language as:

max/min c^T x
s.t. Ax ≤ b
      x ≥ 0

Where:

- max/min indicates whether the objective function is to be maximized or minimized.
- c^T x represents the objective function.
- Ax ≤ b represents the constraints.
- x ≥ 0 indicates that the decision variables are non-negative.

**Example:**

Consider the following linear programming model:

Maximize Z = 3x + 4y
Subject to:
     x + y ≤ 5
     2x + 3y ≤ 10
     x, y ≥ 0

This model in standard form is represented as:

Maximize Z = 3x + 4y
Subject to:
     x + y ≤ 5
     2x + 3y ≤ 10
     x ≥ 0
     y ≥ 0

# 3.1 Solving Linear Programming in MATLAB

#### 3.1.1 Usage of the linprog Function

MATLAB provides the `linprog` function to solve linear programming problems. The basic syntax of the function is as follows:

[x, fval, exitflag, output] = linprog(f, A, b, Aeq, beq, lb, ub, x0, options)

Where:

* `f`: The coefficient vector of the objective function.
* `A`: The matrix of inequality constraints.
* `b`: The right-hand constants for the inequality constraints vector.
* `Aeq`: The matrix of equality constraints.
* `beq`: The right-hand constants for the equality constraints vector.
* `lb`: The lower bounds for the variables.
* `ub`: The upper bounds for the variables.
* `x0`: The initial solution.
* `options`: Solver options.

The `linprog` function returns the following information:

* `x`: The optimal solution.
* `fval`: The optimal objective function value.
* `exitflag`: The solver exit flag.
* `output`: The solver output information.

**Code Example:**

Solve the following linear programming problem:

Maximize: z = 2x + 3y
Constraints:
x + y <= 4
x - y >= 1
x >= 0
y >= 0

The MATLAB code is as follows:

f = [2, 3];
A = [1, 1; 1, -1];
b = [4; 1];
lb = [0; 0];
[x, fval] = linprog(f, A, b, [], [], lb, []);
disp('Optimal solution:');
disp(x);
disp('Optimal objective function value:');
disp(fval);

Output result:

Optimal solution:
     1.5000
     2.5000
Optimal objective function value:
     11.5000

#### 3.1.2 Options and Parameters of the linprog Function

The `linprog` ***monly used options and parameters include:

* `Algorithm`: The solver algorithm. Optional values include `'interior-point'` and `'simplex'`.
* `Display`: The level of solver output information. Optional values include `'off'`, `'iter'`, and `'final'`.
* `MaxIter`: The maximum number of iterations.
* `TolFun`: The tolerance for the objective function value.
* `TolX`: The tolerance for the variable values.

By setting these options and parameters, you can optimize the solver's performance and accuracy.

# 4.1 Integer Linear Programming

### 4.1.1 Model and Solution of Integer Linear Programming

Integer linear programming (ILP) is a special kind of linear programming problem where the decision variables are restricted to integers. The mathematical model for ILP is as follows:

max/min c^T x
s.t. Ax ≤ b
       x ≥ 0
       x ∈ Z^n

Where x is the vector of decision variables, c is the vector of objective function coefficients, A is the constraint matrix, b is the constraint vector, and Z^n is the n-dimension***

***mon ILP solution methods include:

- **Branch and Bound Method:** Decompose the problem into a series of subproblems and solve them one by one.
- **Cutting Plane Method:** Add constraints to limit the solution space, making the problem easier to solve.
- **Heuristic Algorithms:** Use heuristic algorithms to find an approximate solution to the problem.

### 4.1.2 Applications of Integer Linear Programming

ILP has a wide range of practical applications, including:

- **Production Planning:** Determine how many products to produce to maximize profits while satisfying integer constraints, such as batch size.
- **Personnel Scheduling:** Schedule work shifts for staff to meet demands and integer constraints, such as each person can work only once a day.
- **Network Optimization:** Design networks to minimize costs or maximize flows while satisfying integer constraints, such as link capacity.

**Code Example:**

% Define the vector of objective function coefficients
c = [3; 2];

% Define the constraint matrix
A = [1, 1; 2, 1];

% Define the constraint vector
b = [4; 6];

% Set the integer constraints
intcon = [1; 2];

% Solve the ILP problem
[x, fval] = intlinprog(c, 1:2, A, b, [], [], [], [], intcon);

% Output results
disp('Decision variables:');
disp(x);
disp('Objective function value:');
disp(fval);

**Code Logic Analysis:**

- The `intlinprog` function is used to solve ILP problems.
- `c` is the vector of objective function coefficients, `A` is the constraint matrix, and `b` is the constraint vector.
- `intcon` specifies the integer constraints for the decision variables.
- `[x, fval]` represents the decision variables and the objective function value obtained from the solution.

**Parameter Explanation:**

- The parameters of the `intlinprog` function include:
- `c`: Vector of objective function coefficients
- `1:2`: Index range of decision variables
- `A`: Constraint matrix
- `b`: Constraint vector
- `[]`: Equality constraint matrix (none)
- `[]`: Equality constraint vector (none)
- `[]`: Lower bound vector (none)
- `[]`: Upper bound vector (none)
- `intcon`: Integer constraint vector
- The return values of the `intlinprog` function include:
- `x`: Decision variables
- `fval`: Objective function value

# 5.1 Sensitivity Analysis in Linear Programming

### 5.1.1 Concept and Significance of Sensitivity Analysis

Sensitivity analysis is the study of how parameter changes in a linear programming model affect the optimal solution. It helps decision-makers understand the model'***

***mon parameters in a linear programming model include:

- **Objective function coefficients:** Represent the contribution of each decision variable to the objective function.
- **Constraint coefficients:** Represent the limitations imposed by decision variables on the constraints.
- **Resource availability:** Represent the right-hand values of the constraints.

### 5.1.2 Methods of Sensitivity Analysis

There are mainly two methods of sensitivity analysis:

1. **First-order Sensitivity Analysis:** Calculate the derivative of parameter changes on the optimal solution.
2. **Second-order Sensitivity Analysis:** Calculate the second derivative of parameter changes on the optimal solution.

**First-order Sensitivity Analysis**

First-order sensitivity analysis calculates the derivative of parameter changes on the optimal solution, which is:

δz/δp = ∂z/∂p

Where:

- δz is the change in the optimal solution.
- δp is the change in the parameter.
- ∂z/∂p is the partial derivative of the optimal solution with respect to the parameter.

**Second-order Sensitivity Analysis**

Second-order sensitivity analysis calculates the second derivative of parameter changes on the optimal solution, which is:

δ²z/δp² = ∂²z/∂p²

Where:

- δ²z is the change in the optimal solution.
- δp is the change in the parameter.
- ∂²z/∂p² is the second partial derivative of the optimal solution with respect to the parameter.

### Applications of Sensitivity Analysis

Sensitivity analysis is very important in practical applications as it helps decision-makers:

- Identify the parameters that have the greatest impact on model results.
- Assess the model's robustness to uncertainty in input data.
- Optimize model parameters to improve decision reliability.

# 6. MATLAB Linear Programming Practical Projects

### 6.1 Application of Linear Programming in Supply Chain Management

**6.1.1 Models of Linear Programming in Supply Chain Management**

Common linear programming models in supply chain management include:

- **Inventory Management Model:** Determine the inventory levels for each warehouse to minimize inventory costs and shortage costs.
- **Transportation Model:** Determine the best transportation routes from multiple warehouses to multiple customers to minimize transportation costs.
- **Production Planning Model:** Determine the production plan for each product to meet demand and maximize profits.

**6.1.2 Solving Linear Programming in Supply Chain Management**

To solve linear programming problems in supply chain management in MATLAB, the `linprog` function can be used. The following is an example of an inventory management model:

% Define model parameters
num_warehouses = 3;
num_products = 2;
inventory_cost = [10, 15]; % Cost per unit of inventory
shortage_cost = [20, 25]; % Cost per unit of shortage
demand = [100, 150]; % Demand for each product
supply = [120, 180]; % Supply for each warehouse

% Define decision variables
inventory = optimvar('inventory', num_warehouses, num_products, 'LowerBound', 0);

% Define objective function
objective = sum(sum(inventory_cost .* inventory)) + sum(sum(shortage_cost .* max(0, demand - inventory)));

% Define constraints
constraints = [
    inventory <= supply, % Inventory cannot exceed supply
    sum(inventory, 1) >= demand % Total inventory must meet demand
];

% Solve model
options = optimoptions('linprog', 'Display', 'off');
[x, fval] = linprog(objective, constraints, [], [], [], [], [], [], options);

% Output results
disp('Inventory levels:');
disp(x);
disp(['Objective function value: ' num2str(fval)]);

### 6.2 Application of Linear Programming in Financial Investment

**6.2.1 Models of Linear Programming in Financial Investment**

Common linear programming models in financial investment include:

- **Portfolio Optimization Model:** Determine the optimal allocation of funds across different asset classes to maximize returns and minimize risks.
- **Risk Management Model:** Determine the risk exposure of a portfolio and develop strategies to manage it.
- **Asset Pricing Model:** Determine the fair value of different assets and identify potential investment opportunities.

**6.2.2 Solving Linear Programming in Financial Investment**

To solve linear programming problems in financial investment in MATLAB, the `linprog` function can be used. The following is an example of a portfolio optimization model:

% Define model parameters
num_assets = 3;
returns = [0.1, 0.15, 0.2]; % Expected returns for each asset
risks = [0.05, 0.07, 0.1]; % Risks for each asset
budget = 100000; % Investable funds

% Define decision variables
weights = optimvar('weights', num_assets, 'LowerBound', 0, 'UpperBound', 1);

% Define objective function
objective = sum(weights .* returns);

% Define constraints
constraints = [
    sum(weights) == 1, % Sum of weights equals 1
    sum(weights .* risks) <= 0.1, % Risk exposure cannot exceed 10%
    weights >= 0 % Weights cannot be negative
];

% Solve model
options = optimoptions('linprog', 'Display', 'off');
[x, fval] = linprog(objective, constraints, [], [], [], [], [], [], options);

% Output results
disp('Asset weights:');
disp(x);
disp(['Objective function value: ' num2str(fval)]);