2.3 Simplex Method and Sensitivity Analysis
4 min read•july 30, 2024
Linear programming is a powerful tool in operations research for optimizing . The , a cornerstone algorithm, efficiently solves these problems by iteratively improving solutions. This topic explores its mechanics and the crucial role of sensitivity analysis in understanding solution robustness.
Mastering these concepts equips industrial engineers to tackle complex optimization challenges. By learning to interpret results and perform sensitivity analysis, you'll gain insights into resource valuation and decision-making under varying conditions, essential skills for effective operations management.
Linear Programming in Standard Form
Converting Constraints and Objective Function
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- Standard form requires all constraints expressed as equalities with non-negative right-hand sides
- Add slack variables to inequality constraints converting them into equalities
- Introduce surplus variables and artificial variables for greater-than-or-equal-to constraints
- Convert to maximization form by multiplying minimization problems by -1
- Replace variables without sign restrictions with the difference of two non-negative variables
- Ensure coefficient matrix of constraints has full row rank for unique solution
Example of Standard Form Conversion
- Original constraint: 2x + 3y ≤ 10
- Standard form: 2x + 3y + s = 10 (s is a non-negative )
- Original objective: Minimize z = 4x - 5y
- Standard form: Maximize z = -4x + 5y
- Variable without sign restriction: x (unrestricted)
- Standard form: x = x₁ - x₂ (x₁ and x₂ are non-negative)
Simplex Algorithm for Optimization
Iterative Process and Basic Concepts
- Simplex method moves from one basic to another, improving objective function value
- Obtain initial basic feasible solution by setting non-basic variables to zero and solving for basic variables
- Select entering variable based on most negative in objective function row
- Determine leaving variable using minimum ratio test to maintain feasibility
- Perform pivot operations to update simplex tableau and move to next basic feasible solution
- Reach optimality when all reduced costs in objective function row are non-negative for maximization problems
- Degeneracy occurs when multiple basic variables are zero, potentially causing cycling in algorithm
Simplex Tableau and Pivot Operations
- Construct initial simplex tableau with objective function coefficients, constraint coefficients, and right-hand side values
- Identify pivot element at intersection of entering variable column and leaving variable row
- Update tableau rows using row operations ()
- Continue pivoting until optimality conditions are met or infeasibility/unboundedness is detected
- Example pivot operation:
- Pivot element: 2
- Old row: [1 2 3 4]
- Pivot row: [0 2 1 6]
- New row: [1 0 2.5 -2]
Optimal Solutions and Shadow Prices
Interpreting the Final Simplex Tableau
- Read directly from final tableau, with basic variables' values in right-hand side column
- Identify reduced costs in final tableau indicating rate of change in objective function for non-basic variables
- Find shadow prices (dual variables) in bottom row of final tableau, corresponding to original constraints
- Locate objective function value in bottom-right cell of final tableau
- Identify binding constraints by non-zero shadow prices in final tableau
- Determine slack and surplus variables' values, indicating whether constraints are binding or have slack
Economic Interpretation of Shadow Prices
- Shadow prices represent marginal value of resources in constraints
- Positive indicates potential improvement in objective function by relaxing constraint
- Zero shadow price suggests with no immediate impact on objective value
- Example: Shadow price of 5 for each additional unit of raw material
Sensitivity Analysis of Parameters
Ranges of Optimality and Feasibility
- Examine how changes in coefficients affect optimal solution without re-solving entire problem
- Determine range of optimality for objective function coefficients before optimal solution changes
- Calculate range of feasibility for right-hand side values before current basis becomes infeasible
- Analyze dual prices (shadow prices) representing rate of change in objective function value per unit change in right-hand side of constraint
- Apply 100% rule for simultaneous changes in right-hand side values to determine if optimal basis remains unchanged
- Re-solve problem or use for sensitivity analysis of new variables or constraints
Examples of Sensitivity Analysis
- Objective coefficient range: Coefficient of x₁ can vary from 2 to 5 without changing optimal solution
- Right-hand side range: Resource 1 can increase by 10 units or decrease by 5 units without affecting optimal basis
- Simultaneous changes: Increase Resource 1 by 3 units and Resource 2 by 2 units (total percentage change: 60% < 100%, optimal basis unchanged)
Duality in Linear Programming
Primal-Dual Relationships
- Every linear programming problem (primal) has associated with reciprocal relationship
- Dual problem provides economic interpretation of 's constraints and variables
- Weak duality theorem states any feasible dual solution provides bound on optimal objective value of primal
- Strong duality theorem asserts if either primal or dual has optimal solution, both do, with equal optimal values
- Complementary slackness conditions relate primal and dual solutions, indicating binding constraints
- Dual simplex method solves dual problem implicitly by working with primal tableau, useful for sensitivity analysis and re-optimization
- Duality principles fundamental in developing solution methods and understanding economic implications of linear programming problems
Constructing and Interpreting Dual Problems
- Primal constraints become dual variables
- Primal variables become dual constraints
- Objective function coefficients and right-hand side values swap positions
- Inequality directions reverse (≤ becomes ≥, and vice versa)
- Example:
- Primal: Maximize 3x₁ + 2x₂ subject to x₁ + x₂ ≤ 4, 2x₁ + x₂ ≤ 5, x₁, x₂ ≥ 0
- Dual: Minimize 4y₁ + 5y₂ subject to y₁ + 2y₂ ≥ 3, y₁ + y₂ ≥ 2, y₁, y₂ ≥ 0
Key Terms to Review (19)
Artificial variable: An artificial variable is a mathematical construct introduced in linear programming problems to facilitate finding an initial basic feasible solution, especially when constraints do not naturally allow for a feasible solution. These variables are added to the original system of equations to create an auxiliary problem that can be solved using methods like the simplex method. They help in ensuring that all constraints are satisfied while searching for optimal solutions.
Binding constraint: A binding constraint refers to a condition or limitation in a linear programming model that restricts the feasible region and directly affects the optimal solution. When a constraint is binding, it means that the solution to the optimization problem is determined by that constraint, as it touches the feasible region at a specific point. The importance of binding constraints lies in their role in sensitivity analysis, where changes to these constraints can lead to significant shifts in the optimal solution.
Decision variable: A decision variable is a variable used in mathematical programming that represents a choice or decision to be made within a model. These variables are crucial for defining the constraints and objectives of optimization problems, as they are what you actually control in order to achieve the best outcome. In the context of optimization methods, such as the simplex method, decision variables are manipulated to find optimal solutions while considering various constraints.
Dual problem: The dual problem refers to a formulation derived from a linear programming problem, which provides insights into the original problem's constraints and objective function. Each primal linear programming problem has a corresponding dual problem that encapsulates the relationship between variables and constraints, highlighting how changes in resources or costs impact the optimal solution. Understanding the dual problem is crucial for sensitivity analysis, as it helps in assessing the effect of changes in parameters on the original solution.
Dual Simplex Method: The dual simplex method is a variation of the traditional simplex algorithm used for solving linear programming problems, focusing on maintaining dual feasibility while allowing primal infeasibility. This approach is particularly useful when dealing with problems where constraints are modified, such as in transportation and assignment scenarios, enabling the solution to progress even when primal solutions may not be feasible. By updating the solution iteratively, the dual simplex method effectively navigates through the feasible region of the dual problem, providing insights into resource allocation and optimization strategies.
Feasible Solution: A feasible solution refers to a set of decision variables that satisfies all the constraints imposed by a mathematical model. It is critical in optimization problems as it identifies potential solutions that adhere to restrictions such as resource limitations, capacity bounds, or demand requirements. Understanding feasible solutions helps in evaluating the performance of various operational strategies and finding optimal outcomes.
Non-binding constraint: A non-binding constraint is a limitation in an optimization problem that does not affect the feasible region or optimal solution because it is not fully utilized. In other words, at the optimal solution, the resources specified by the constraint are either not needed or are available in excess. Understanding non-binding constraints is crucial for evaluating the effectiveness of resource allocation and helps in sensitivity analysis.
Objective Function: An objective function is a mathematical expression that defines the goal of an optimization problem, usually to maximize or minimize a particular quantity. It serves as the core of optimization techniques, where various strategies and applications aim to find the best solution under given constraints. This function is crucial in decision-making processes and is represented in various forms, such as linear programming, to ensure effective resource allocation.
Optimal Solution: An optimal solution refers to the best possible outcome that satisfies all constraints in a given problem, maximizing or minimizing an objective function. In various applications, this concept is crucial for decision-making and resource allocation, as it provides the most efficient way to achieve desired goals under specific limitations. Understanding optimal solutions allows for effective planning, scheduling, and distribution of resources across different scenarios.
Primal problem: The primal problem is a foundational concept in linear programming, representing the original optimization problem that seeks to maximize or minimize a linear objective function subject to a set of linear constraints. This problem is essential because it forms the basis for solving other related problems, such as the dual problem, and is crucial for understanding the Simplex Method and sensitivity analysis. The structure of the primal problem helps determine feasible solutions and optimality conditions that are key to effective decision-making in various applications.
Production Scheduling: Production scheduling is the process of planning and organizing the production activities of a manufacturing facility to optimize resource usage and meet demand. This involves determining what needs to be produced, when it should be produced, and allocating resources effectively, which is essential for achieving efficiency and maximizing output. It plays a critical role in coordinating operations and ensuring timely delivery of products while minimizing waste and costs.
Reduced cost: Reduced cost is a concept in linear programming that indicates how much the objective function coefficient of a non-basic variable must improve before that variable can enter the solution basis. It highlights the opportunity cost of including a non-basic variable in the optimal solution, which is crucial for determining feasibility and optimality during optimization processes.
Resource allocation: Resource allocation is the process of assigning available resources, such as time, money, personnel, and equipment, to various projects or tasks in a manner that maximizes efficiency and effectiveness. This concept is critical as it influences decision-making and operational strategies across different domains, ultimately impacting productivity and success.
Sensitivity report: A sensitivity report is a document that provides insights into how changes in the parameters of a linear programming model can affect the optimal solution. It highlights the range of values for objective function coefficients and right-hand side constraints within which the current solution remains optimal, making it an essential tool in sensitivity analysis.
Shadow price: Shadow price is the implicit value of a constraint in an optimization problem, representing the rate at which the objective function would increase if the constraint were relaxed by one unit. This concept helps in assessing how much an additional unit of a resource is worth in the context of maximizing or minimizing an objective, providing insights into resource allocation and efficiency. Understanding shadow prices aids decision-making by quantifying the trade-offs associated with constraints in various operational scenarios.
Simplex method: The simplex method is an algorithm for solving linear programming problems, which involves optimizing a linear objective function subject to linear equality and inequality constraints. It efficiently navigates the vertices of the feasible region defined by the constraints to find the optimal solution. This method is crucial for maximizing or minimizing resources in various applications such as transportation, manufacturing, and finance.
Slack variable: A slack variable is a non-negative variable added to a linear programming constraint to convert an inequality into an equality. It represents the difference between the left-hand side and right-hand side of a less-than-or-equal-to constraint, effectively allowing for the allocation of unused resources in optimization problems. By introducing slack variables, it becomes easier to analyze feasible solutions and their impact on the objective function during the solution process.
Surplus variable: A surplus variable is a variable used in linear programming to represent excess resources in a constraint that is greater than or equal to a certain amount. This concept is critical in optimization problems where resources such as time, money, or materials exceed the required limits, allowing for flexibility in the solution. Surplus variables are added to less-than-or-equal-to constraints to convert them into equalities, making them essential for the Simplex Method.
What-if analysis: What-if analysis is a technique used to evaluate the potential outcomes of different scenarios by altering the input variables in a model. This method helps in understanding how changes in certain parameters can impact the results, making it essential for decision-making and strategic planning. It connects to the evaluation of various options and helps assess risk by identifying possible consequences before they occur.