Simplex algorithm

From binaryoption
Jump to navigation Jump to search
Баннер1
  1. Simplex Algorithm

The Simplex algorithm is a popular and widely-used method for solving linear programming (LP) problems. These problems involve optimizing (maximizing or minimizing) a linear objective function, subject to linear equality and inequality constraints. It's a cornerstone of operations research, management science, and economics, with applications ranging from resource allocation and production planning to network optimization and financial modeling. This article provides a detailed, beginner-friendly explanation of the Simplex algorithm, its underlying principles, steps involved, and limitations.

What is Linear Programming?

Before diving into the Simplex algorithm, it's crucial to understand linear programming. Essentially, LP aims to find the best possible solution (the optimal solution) to a problem where both the objective and the constraints are expressed as linear equations or inequalities.

A typical LP problem can be formulated as follows:

Maximize (or Minimize): z = c1x1 + c2x2 + ... + cnxn

Subject to:

a11x1 + a12x2 + ... + a1nxn ≤ b1 a21x1 + a22x2 + ... + a2nxn ≤ b2 ... am1x1 + am2x2 + ... + amnxn ≤ bm

x1, x2, ..., xn ≥ 0

Where:

  • z is the objective function – the quantity we want to optimize.
  • x1, x2, ..., xn are the decision variables – the quantities we can control.
  • c1, c2, ..., cn are the coefficients of the objective function.
  • aij are the coefficients of the constraints.
  • b1, b2, ..., bm are the right-hand side values of the constraints.
  • The non-negativity constraints (xi ≥ 0) ensure that the decision variables are non-negative.

For example, consider a simple problem:

Maximize: z = 3x1 + 2x2

Subject to:

2x1 + x2 ≤ 18 2x1 + 3x2 ≤ 42 x1, x2 ≥ 0

This could represent maximizing profit (z) by producing two products (x1 and x2), subject to limited resources (constraints).

The Geometry of Linear Programming

The feasible region of an LP problem is the set of all points that satisfy all the constraints. This region is a convex polyhedron. The optimal solution *always* occurs at a corner point (vertex) of this polyhedron. This is a fundamental principle that the Simplex algorithm exploits. Imagine a candlestick pattern – the optimal solution lies at one of the extreme points. The algorithm systematically explores these corner points until it finds the one that yields the best value for the objective function. Understanding this geometric intuition clarifies why the Simplex algorithm works. Similar to how Fibonacci retracement helps identify potential turning points, the Simplex algorithm searches for the optimal vertex within the feasible region.

From Standard Form to Initial Basic Feasible Solution

The Simplex algorithm requires the LP problem to be in a specific format called *standard form*. This involves:

1. **Converting inequalities to equalities:** Introduce *slack variables* (si) for "less than or equal to" constraints and *surplus variables* (ei) for "greater than or equal to" constraints. For example, 2x1 + x2 ≤ 18 becomes 2x1 + x2 + s1 = 18. These variables represent unused resources. Thinking about this like Bollinger Bands - the slack variable represents the distance from the upper band. 2. **Non-negativity constraints:** All variables (including slack and surplus variables) must be non-negative.

Once in standard form, we need to find an *initial basic feasible solution* (IBFS). This is a solution that satisfies all the constraints and has a limited number of basic variables (equal to the number of constraints). Basic variables are those that are not set to zero. A common way to find an IBFS is to set all non-basic variables to zero and solve for the basic variables.

For our example, converting to standard form:

Maximize: z = 3x1 + 2x2 + 0s1 + 0s2

Subject to:

2x1 + x2 + s1 = 18 2x1 + 3x2 + s2 = 42 x1, x2, s1, s2 ≥ 0

An IBFS is x1 = 0, x2 = 0, s1 = 18, s2 = 42, and z = 0. This corresponds to not producing any of the products and having all resources available. This is analogous to a support and resistance level, a starting point for analysis.

The Simplex Tableau

The Simplex algorithm is often implemented using a *simplex tableau*. This is a tabular representation of the LP problem in standard form. It simplifies the calculations and makes the process more organized.

Here's the initial tableau for our example:

| Basic Variable | x1 | x2 | s1 | s2 | RHS | |---|---|---|---|---|---| | s1 | 2 | 1 | 1 | 0 | 18 | | s2 | 2 | 3 | 0 | 1 | 42 | | z | -3 | -2 | 0 | 0 | 0 |

The tableau represents the equations:

  • s1 = 18 - 2x1 - x2
  • s2 = 42 - 2x1 - 3x2
  • z = -3x1 - 2x2

The objective is to increase 'z' (for maximization problems) by iteratively improving the solution.

The Simplex Iteration

The core of the Simplex algorithm is the *iteration* process. Each iteration involves the following steps:

1. **Identify the entering variable:** Select the non-basic variable with the most negative coefficient in the objective row (z row). This variable has the greatest potential to increase the objective function value. In our example, x1 is the entering variable. This is similar to identifying the strongest momentum indicator. 2. **Identify the leaving variable:** Calculate the *ratio* of the RHS values to the corresponding positive coefficients in the entering variable's column. The variable with the smallest non-negative ratio is the leaving variable. This determines which resource will become fully utilized. For s1, the ratio is 18/2 = 9. For s2, the ratio is 42/2 = 21. Therefore, s1 is the leaving variable. This is akin to analyzing a moving average convergence divergence (MACD) to determine potential exits. 3. **Pivot:** Perform row operations to make the entering variable's coefficient in the leaving variable's row equal to 1 and all other coefficients in the entering variable's column equal to 0. This involves dividing the pivot row by the pivot element (the element at the intersection of the entering variable's column and the leaving variable's row) and using row operations to eliminate the entering variable from other rows.

After pivoting on x1 and s1, the new tableau becomes:

| Basic Variable | x1 | x2 | s1 | s2 | RHS | |---|---|---|---|---|---| | x1 | 1 | 0.5 | 0.5 | 0 | 9 | | s2 | 0 | 1.5 | -1 | 1 | 24 | | z | 0 | 0.5 | 1.5 | 0 | 27 |

4. **Optimality Check**: Check if all coefficients in the z-row are non-negative. If they are, the current solution is optimal. If not, repeat steps 1-3.

In this case, all coefficients in the z-row are non-negative (0, 0.5, 1.5, 0). Therefore, the optimal solution is:

x1 = 9, x2 = 0, s1 = 0, s2 = 24, z = 27

This means that to maximize profit, the company should produce 9 units of product x1 and 0 units of product x2, resulting in a maximum profit of 27. This is comparable to identifying a breakout pattern and capitalizing on the subsequent price movement.

Dealing with Multiple Optimal Solutions

Sometimes, an LP problem may have multiple optimal solutions. This occurs when the objective function is parallel to an edge of the feasible region. The Simplex algorithm may cycle between these solutions. Techniques like *perturbation* or *sensitivity analysis* can be used to identify and handle such cases. This is similar to how traders use average true range (ATR) to understand price volatility and potential multiple outcomes.

The Two-Phase Simplex Method

If the initial basic feasible solution is not readily available (e.g., when dealing with minimization problems or constraints with equality), the *two-phase simplex method* is used.

  • **Phase I:** A new objective function is introduced to drive at least one basic variable to zero. The Simplex algorithm is then applied to this modified problem. If an optimal solution is found in Phase I with a zero value for the new objective function, it indicates that a feasible solution exists for the original problem.
  • **Phase II:** The original objective function is restored, and the Simplex algorithm is applied starting from the feasible solution found in Phase I. This is like using Elliott Wave Theory – first identifying the larger pattern before focusing on the details.

Unbounded Solutions

If the Simplex algorithm encounters a situation where the entering variable's column has no positive coefficients, the problem is *unbounded*. This means that the objective function can be increased (or decreased) indefinitely without violating the constraints. In practical terms, this indicates that the problem is not properly formulated or that there are no resource limitations. This is similar to identifying a double top or double bottom – a sign of potential reversal.

Degeneracy

  • Degeneracy* occurs when a basic variable has a value of zero. This can lead to *cycling*, where the Simplex algorithm repeatedly visits the same set of basic feasible solutions without making progress. Various techniques, such as *Bland's rule* (choosing the entering variable with the smallest subscript), can be used to prevent cycling. This is comparable to managing risk-reward ratio – mitigating potential losses.

Limitations of the Simplex Algorithm

Despite its widespread use, the Simplex algorithm has some limitations:

  • **Exponential worst-case time complexity:** In rare cases, the algorithm can take an exponentially long time to solve the problem.
  • **Cycling:** As mentioned earlier, degeneracy can lead to cycling.
  • **Sensitivity to data changes:** Small changes in the problem data can significantly impact the solution. This is similar to the impact of economic indicators on market movements.

Alternatives to the Simplex algorithm, such as *interior-point methods*, have been developed to address these limitations. These methods are often more efficient for large-scale LP problems. Understanding correlation analysis can help assess the impact of these changes.

Applications Beyond Optimization

The Simplex algorithm's principles extend beyond simple optimization. It's foundational to:

  • **Network Flow Problems:** Optimizing the flow of goods or information through a network.
  • **Transportation Problems:** Minimizing the cost of transporting goods from multiple sources to multiple destinations.
  • **Assignment Problems:** Assigning tasks to individuals in the most efficient way.
  • **Game Theory:** Finding Nash equilibria in games.
  • **Portfolio Optimization:** Selecting a portfolio of assets to maximize return for a given level of risk. This is closely related to technical analysis and understanding market trends.

Further Learning

Understanding these related concepts will provide a more comprehensive grasp of the Simplex algorithm and its applications. Learning about chart patterns and trading psychology can also enhance your analytical skills.

Start Trading Now

Sign up at IQ Option (Minimum deposit $10) Open an account at Pocket Option (Minimum deposit $5)

Join Our Community

Subscribe to our Telegram channel @strategybin to receive: ✓ Daily trading signals ✓ Exclusive strategy analysis ✓ Market trend alerts ✓ Educational materials for beginners

Баннер