Sensitivity analysis - Part 2

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This video is a lecture on Sensitivity Analysis in Linear Programming, presented by an instructor at a whiteboard. The instructor begins by stating the objective function Max Z = 3x + 2y and the constraints x + y ≤ 4, 2x + y ≤ 6, with x, y ≥ 0. He then presents the final simplex tableau, which shows the optimal solution with basic variables x and s2, and a maximum Z value of 10. The core of the lecture focuses on analyzing how changes to the objective function coefficients (c1 and c2) affect the optimality of the current solution. The instructor demonstrates that the current solution remains optimal as long as the reduced costs (cj - zj) are non-positive. He derives the condition for c1 (3+Δ) by ensuring the reduced cost for variable y (c2 - z2) remains ≤ 0, which leads to the inequality -1 - Δ ≤ 0, resulting in Δ ≥ -1. Similarly, for c2 (2+Δ), he uses the reduced cost for variable x (c1 - z1) to derive the condition 1 - Δ ≤ 0, resulting in Δ ≤ 1. The final analysis shows that the current optimal solution is valid for a range of c1 from 2 to 4 and c2 from 1 to 3. The instructor also briefly touches upon the analysis of changes to the right-hand side (RHS) of the constraints, showing the formula for the new solution values (10+2Δ and 10+Δ) and the condition for feasibility (Δ ≥ -3).

Chapters

1. 0:00 – 2:00 00:00-02:00

   The video opens with a whiteboard displaying a Linear Programming Problem and its final simplex tableau. The instructor, standing to the left, begins the lecture on Sensitivity Analysis. He clearly states the problem: Max Z = 3x + 2y, subject to the constraints x + y ≤ 4, 2x + y ≤ 6, and x, y ≥ 0. He then points to the final simplex tableau, which shows the optimal solution with basic variables x and s2, and a maximum Z value of 10. The instructor explains that the current solution is optimal and the goal is to determine the range of values for the objective function coefficients (c1 and c2) for which this solution remains optimal.

2. 2:00 – 5:00 02:00-05:00

   The instructor focuses on the sensitivity of the objective function coefficient c1. He introduces a change Δ to c1, making it 3+Δ. He explains that for the current solution to remain optimal, the reduced cost of the non-basic variable y (c2 - z2) must be ≤ 0. He calculates the new reduced cost as (2 - (3+Δ) * 0 - 2 * 1) = -1 - Δ. He then sets up the inequality -1 - Δ ≤ 0, which simplifies to Δ ≥ -1. This means c1 can decrease by at most 1 (from 3 to 2) and still remain optimal. He then moves to c2, introducing a change Δ to make it 2+Δ, and calculates the reduced cost for the non-basic variable x (c1 - z1) as (3 - (2+Δ) * 1 - 3 * 0) = 1 - Δ.

3. 5:00 – 10:00 05:00-10:00

   The instructor completes the analysis for c2. He sets the reduced cost for variable x (1 - Δ) ≤ 0, which gives the condition Δ ≤ 1. This means c2 can increase by at most 1 (from 2 to 3) and the current solution remains optimal. He then summarizes the range for c1 as 3 + Δ where Δ ≥ -1, so c1 ∈ [2, ∞), and for c2 as 2 + Δ where Δ ≤ 1, so c2 ∈ (-∞, 3]. He then discusses the range for c1, stating that the current solution is optimal for c1 ≥ 2. He also mentions that the range for c2 is c2 ≤ 3. The instructor then begins to discuss the sensitivity analysis for the right-hand side (RHS) of the constraints, writing the formula for the new solution values as 10 + 2Δ and 10 + Δ, and the condition for feasibility as Δ ≥ -3.

4. 10:00 – 11:36 10:00-11:36

   The instructor concludes the lecture by summarizing the findings. He states that the current optimal solution remains optimal for c1 in the range [2, 4] and c2 in the range [1, 3]. He then discusses the feasibility of the solution when the RHS of the constraints changes. He shows that the new solution values are 10 + 2Δ and 10 + Δ, and for the solution to remain feasible, both values must be ≥ 0. This leads to the condition Δ ≥ -3. He also mentions that the range for the RHS of the first constraint is [1, 7] and for the second is [3, 9]. The video ends with the instructor summarizing the key points of sensitivity analysis.

The video provides a comprehensive, step-by-step demonstration of sensitivity analysis in linear programming. It begins with a clear statement of the problem and its optimal solution. The core of the lesson is the method for determining the range of objective function coefficients (c1 and c2) for which the current optimal solution remains optimal. This is achieved by ensuring the reduced costs of all non-basic variables remain non-positive. The instructor systematically derives the conditions for c1 and c2, showing that the solution is optimal for c1 ≥ 2 and c2 ≤ 3. The lecture also briefly covers the analysis of changes to the right-hand side of the constraints, demonstrating how to calculate the new solution values and the range of feasibility. The entire process is clearly illustrated on the whiteboard, making it an excellent resource for understanding the practical application of sensitivity analysis.