Study Notes on Sensitivity Analysis in Management Science

Introduction to Sensitivity Analysis

• Sensitivity analysis examines how the optimal solution of a linear programming problem is affected by changes in the model's parameters within specified limits, specifically:

  • The coefficients of the objective function.

  • The right-hand side (RHS) values.

• It is also referred to as postoptimality analysis, as it considers the effects after an optimal solution has been found.

• Applications include:

  • Identifying critical coefficients that may influence the decision.

  • Understanding the value of additional resources and the point at which diminishing returns begin.

Chapter Contents

• 3-1 Introduction to Sensitivity Analysis

• 3-2 Graphical Sensitivity Analysis

• 3-3 Sensitivity Analysis: Computer Solution

• 3-4 Limitations of Classical Sensitivity Analysis

• 3-5 The Electronic Communications Problem

Chapter Objectives (1 of 2)

• LO 3.1: Perform sensitivity analysis using graphical methods for linear programs.

• LO 3.2: Use software tools such as Excel Solver to conduct sensitivity analysis on linear programs and interpret the outcomes.

• LO 3.3: Understand dual values (shadow prices) of constraints in linear programs.

• LO 3.4: Define and interpret ranges of optimality for objective function coefficients.

Chapter Objectives (2 of 2)

• LO 3.5: Define and interpret ranges of feasibility for constraints in linear programs.

• LO 3.6: Discuss assumptions of traditional sensitivity analysis and how to update the model when these assumptions are not adhered to.

• LO 3.7: Interpret reduced cost values for variables in linear programming.

• LO 3.8: Distinguish between sunk costs and relevant costs, integrating these concepts into linear programming.

Detailed Explanation of Key Concepts

3-1 Introduction to Sensitivity Analysis

• The significance of sensitivity analysis arises due to the variability of real-world problems, enabling decision-makers to adapt in changing conditions.

• Key aspects of sensitivity analysis include:

  • Providing information to address coefficient changes without needing complete resolution of the original linear problem.

  • Determining which coefficients are essential by examining their ranges of optimality. For example, if a coefficient's optimality range is narrow, it may need reassessment.

  • Assessing the economic worth of resources and understanding the limits of additional resources before it triggers diminishing returns.

3-2 Changes in the Objective Function Coefficients

Graphical Solution of Par, Inc.

• Example with Par, Inc.: Changes in the coefficients influence the optimal solutions:

  • Extreme Point ③ remains optimal as long as the slope lies between the slopes of constraints defined as:

  • Line A: $y = - rac{7}{10}x + 630$

  • Line B: $y = - rac{3}{2}x + 1062$

• Therefore, extreme point ③ is optimal if:
  $- rac{3}{2} ext{ (slope of Line B)} ext{ } ext{ to } ext{ } - rac{7}{10} ext{ (slope of Line A)}$

Ranges of Optimality for Extreme Point ③

• The profit for the standard bag can vary between:

  • Lower bound: $6.3$

  • Upper bound: $13.5$

• This range indicates that as long as the contribution does not fall outside of these limits, production levels recommended will remain optimal.

Ranges of Optimality for Deluxe Bag (CD)

• Similar analysis for the deluxe bag shows:

  • Profit contribution for deluxe bags can vary from $6.67$ to $14.29$ while keeping other coefficients fixed.

Rotation of the Objective Function Line

• If the objective slope becomes vertical due to adjustments, it poses no upper or lower limits, e.g.:
  $- rac{9}{10} ext{ (if CS = 9} ext{ is considered) } ext{ leading to } 13.5 ext{ to } ext{ ∞}$

Simultaneous Changes

• Illustrative example: If the profit contribution per standard bag reaches $13$ and deluxe bag decreases to $8$, we can determine the new slopes and clarity on when the solution alters, reiterating that simultaneous variances must be resolved separately.

Changes in Right-Hand Sides

• Example: An increase of 10 hours in production time leads to:

  • New RHS for cutting constraint upgraded to $640$.

  • Resulting in new optimal points and profit rise.

Dual Value

• Indicates the value of the optimal solution per unit increase in the right-hand side increases.

• E.g.: When RHS increases in cutting and dyeing, the dual value indicates a profit increase per hour on both sides.

3-3 Sensitivity Analysis: Computer Solution

• Computer models aid in resolving complex linear programs where graphical methods fall short.

• Tools like Excel Solver can derive sensitivity interpretations including reduced costs and dual values.

  • For example, changing nonnegativity constrains yields dual values.

3-4 Limitations of Classical Sensitivity Analysis

• Assumes only one coefficient changes, exposing it to limitations during simultaneous adjustments, and does not addressed necessary changes in other calculations. Non-intuitive dual values can lead to misunderstanding.

3-5 The Electronic Communications Problem

• Exploration into new product strategies for distribution - requiring robust optimization given defined constraints.

• The problem involves four distribution channels with profit analysis relative to advertising and sales effort while maintaining a production level and contract obligations.