In-depth Understanding of MATLAB Linear Programming Sensitivity Analysis: The Impact of Model Changes on Results

# In-depth Understanding of MATLAB Linear Programming Sensitivity Analysis: Impact of Model Changes on Results

# 1. Introduction to Linear Programming**

Linear programming is a mathematical optimization technique used to maximize or minimize a linear objective function subject to a series of linear constraints. It is widely applied in fields such as economics, engineering, and operations research.

A linear programming model consists of an objective function and a set of linear constraints. The objective function represents the quantity to be optimized (maximized or minimized), while the constraints define the feasible solution space of the problem.

The standard form of a linear programming problem 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 decision variable vector
* A is the constraint matrix
* b is the constraint vector
* <= denotes less than or equal to

# 2. MATLAB Linear Programming Sensitivity Analysis**

**2.1 Concept and Importance of Sensitivity Analysis**

Sensitivity analysis is a vital technique in linear programming problems that evaluates the impact of input parameter changes on the objective function value. It aids decision-makers in understanding the robustness of the model and identifying which parameters are most sensitive to the results.

**2.2 Implementation of Linear Programming Sensitivity Analysis in MATLAB**

MATLAB provides various functions to perform linear programming sensitivity analysis.

**2.2.1 Generation of Sensitivity Analysis Reports**

The `sensitivity(model)` function generates a sensitivity analysis report containing the following information:

***Dual Variables:** Shadow prices associated with each constraint.
***Coefficient Changes:** Allowable ranges of changes to the objective function and constraint coefficients without altering the optimal solution.
***Right-Hand Side Changes:** Allowable ranges of changes to the right-hand side of constraints without altering the optimal solution.

**2.2.2 Identification of Sensitive Parameters**

MATLAB also offers the following functions to identify sensitive parameters:

* `senscoeff(model)`: Returns the sensitivity coefficients for the objective function coefficients.
* `sensrhs(model)`: Returns the sensitivity coefficients for the right-hand side of constraints.

**2.2.3 Applications of Sensitivity Analysis**

Sensitivity analysis has extensive applications in:

***Model Validation:** Determines if the model is sufficiently robust to changes in input parameters.
***Parameter Optimization:** Identifies parameters that most affect the objective function for optimization purposes.
***Risk Management:** Assesses the impact of parameter uncertainty on the objective function and develops mitigation strategies.

**Code Example:**

% Define the linear programming model
model.f = [3; 2];
model.A = [1, 1; 2, 3];
model.b = [6; 10];
model.lb = [0; 0];

% Solve the linear programming problem
[x, fval] = linprog(model.f, [], [], model.A, model.b, model.lb);

% Generate a sensitivity analysis report
sensitivity_report = sensitivity(model);

% Display the sensitivity analysis report
disp(sensitivity_report);

**Code Logic Analysis:**

* The `linprog` function solves the linear programming problem and returns the optimal solution `x` and the objective function value `fval`.
* The `sensitivity` function generates a sensitivity analysis report containing information about dual variables, coefficient changes, and right-hand side changes.
* The `disp` function displays the sensitivity analysis report.

# 3. Theoretical Foundations of Sensitivity Analysis

### 3.1 Duality Theory and Sensitivity Analysis

Duality theory is an important theoretical foundation for sensitivity analysis in linear programming. Within duality theory, a linear programming problem and its dual problem have a close relationship. The dual problem of a linear programming problem can be represented as:

Minimize c^T x
Subject to:
Ax = b
x >= 0

Where c, A, and b are the objective function coefficients, constraint matrix, and constraint vector of the linear programming problem, respectively.

The optimal solution of the dual problem represents the shadow prices of the optimal solution to the linear programming problem. Shadow prices indicate the degree of relaxation of the constraints, i.e., the amount of change in the objective function given a small change in the constraints. Therefore, through duality theory, we can obtain the sensitivity information of a linear programming problem by solving its dual problem.

### 3.2 Shadow Prices and Sensitivity Analysis

Shadow prices are a significant concept in sensitivity analysis. Shadow prices indicate the degree of relaxation of constraints, i.e., the amount of change in the objective function given a small change in the constraints.

For a linear programming problem:

Maximize c^T x
Subject to:
Ax ≤ b
x ≥ 0

The shadow price of constraint i is:

π_i = c^T x_B^-1