Sensitivity of the Objective Function Coefficients Linear Programming Sensitivity of the Objective Function Coefficients

Sensitivity of Any Coefficients Any coefficient in a linear programming model might change because: They may be only approximations or best estimates. The problem may be one in a dynamic environment where coefficients are subject to (frequent) changes. The decision maker may simply wish to ask, “what-if” a certain change is made – how will that affect the optimal solution. When only one coefficient changes at a time this is called “marginal” or “sensitivity” analysis of the coefficient.

Finding the Optimal Point - Review X2 1000 900 800 700 600 500 400 300 200 100 X1 Max 8X1 + 5X2 s.t. 2X1 + 1X2 ≤ 1000 3X1 + 4X2 ≤ 2400 (Redundant) 1X1 + 1X2 ≤ 700 1X1 - 1X2 ≤ 350 X1, X2 ≥ 0 OPTIMAL POINT (320,360)

Change the Objective Function Coefficient of X1 to 5 1000 900 800 700 600 500 400 300 200 100 X1 Max 5X1 + 5X2 s.t. 2X1 + 1X2 ≤ 1000 3X1 + 4X2 ≤ 2400 (Redundant) 1X1 + 1X2 ≤ 700 1X1 - 1X2 ≤ 350 X1, X2 ≥ 0 OPTIMAL POINT (320,360) STILL!

Change the Objective Function Coefficient of X1 to 9 1000 900 800 700 600 500 400 300 200 100 X1 Max 9X1 + 5X2 s.t. 2X1 + 1X2 ≤ 1000 3X1 + 4X2 ≤ 2400 (Redundant) 1X1 + 1X2 ≤ 700 1X1 - 1X2 ≤ 350 X1, X2 ≥ 0 OPTIMAL POINT (320,360) STILL!

But Change the Objective Function Coefficient of X1 to 1 1000 900 800 700 600 500 400 300 200 100 X1 Max 1X1 + 5X2 s.t. NEW OPTIMAL POINT (0,600) 2X1 + 1X2 ≤ 1000 3X1 + 4X2 ≤ 2400 (Redundant) 1X1 + 1X2 ≤ 700 1X1 - 1X2 ≤ 350 X1, X2 ≥ 0

Or Change the Objective Function Coefficient of X1 to 12 1000 900 800 700 600 500 400 300 200 100 X1 Max 12X1 + 5X2 s.t. 2X1 + 1X2 ≤ 1000 3X1 + 4X2 ≤ 2400 (Redundant) 1X1 + 1X2 ≤ 700 1X1 - 1X2 ≤ 350 X1, X2 ≥ 0 NEW OPTIMAL POINT (450, 100)

Range of Optimality Thus the objective function coefficient for X1 can decrease or increase by a certain amount and the optimal solution will not change! The optimal profit will change. For example if it changes from 8 to 6, the optimal profit changes from 8(320) + 5(360) = 4360 to 6(320) + 5(360) = 3720. The amount by which the coefficient can decrease or increase is what Excel called the ALLOWABLE DECREASE and the ALLOWABLE INCREASE of the objective function coefficient. The range of values of this coefficient from the (Original Coefficient – Allowable Decrease) to (Original Coefficient + Allowable Increase) is called the Range of Optimality for the coefficient.

How Much Can The Coefficient Change So That (320,360) is Still Optimal? X2 1000 900 800 700 600 500 400 300 200 100 X1 Max 8X1 + 5X2 s.t. 3.75X1 5X1 6X1 8X1 2X1 + 1X2 ≤ 1000 3X1 + 4X2 ≤ 2400 3X1 + 4X2 = 2400 (Redundant) 1X1 + 1X2 ≤ 700 1X1 - 1X2 ≤ 350 X1, X2 ≥ 0 OPTIMAL POINT (320,360) Until it becomes parallel to (has the same slope) as

How Much Can The Coefficient Change So That (320,360) is Still Optimal? X2 1000 900 800 700 600 500 400 300 200 100 X1 Max 8X1 + 5X2 s.t. 9X1 10X1 8X1 2X1 + 1X2 ≤ 1000 2X1 + 1X2 = 1000 3X1 + 4X2 ≤ 2400 (Redundant) 1X1 + 1X2 ≤ 700 1X1 - 1X2 ≤ 350 X1, X2 ≥ 0 OPTIMAL POINT (320,360) Or until it becomes parallel to (has the same slope) as

The Range of Optimality for C1 Let us denote the objective function coefficient for X1 as C1. The slope of a line written as: aX1 + bX2 = d is: Thus, - 2/1 ≤ -C1/5 ≤ - 3/4 -a/b Slope of first constraint Slope of objective function Slope of second constraint Multiplying by -5 gives: 10 ≥ C1 ≥ 3.75 or 3.75 ≤ C1 ≤ 10

Reduced Cost REDUCED COST OPTIMAL POINT (0,600) X2 1000 900 800 700 600 500 400 300 200 100 X1 Recall that if the objective function were MAX 1X1 + 5X2 we had: Max 1X1 + 5X2 s.t. 2X1 + 1X2 ≤ 1000 OPTIMAL POINT (0,600) 3X1 + 4X2 ≤ 2400 1X1 + 1X2 ≤ 700 (Redundant) 1X1 - 1X2 ≤ 350 X1, X2 ≥ 0 No X1’s were produced because its unit profit was not large enough. 1. How much would the cost have to be reduced (profit increased) so that it would be profitable to make X1’s? 2. If X1 were at least 1, how would the profit be affected? Answer to both: REDUCED COST

How much would the cost have to be reduced so that it would be profitable to make X1’s? 1000 900 800 700 600 500 400 300 200 100 X1 3.75X1 2.5X1 Max 1X1 + 5X2 s.t. 1X1 2X1 + 1X2 ≤ 1000 OPTIMAL POINT (0,600) 3X1 + 4X2 ≤ 2400 1X1 + 1X2 ≤ 700 (Redundant) 1X1 - 1X2 ≤ 350 X1, X2 ≥ 0 When objective function line is parallel with 3X1 + 4X2 = 2400, optimal solutions exist with X1>0. Thus, C1/5 = 3/4 or C1 = 3.75. Reduced Cost = 1 – 3.75 = -2.75 Original Coefficient

If X1 ≥ 1*, how is the profit affected? 1000 900 800 700 600 500 400 300 200 100 X1 Max 1X1 + 5X2 s.t. 2X1 + 1X2 ≤ 1000 OPTIMAL POINT (0,600) 3X1 + 4X2 ≤ 2400 1X1 + 1X2 ≤ 700 (Redundant) 1X1 - 1X2 ≤ 350 Optimal Profit = 3000 X1 ≥ 1 (Added) NEW OPTIMAL POINT (1,599.25) X1, X2 ≥ 0 New Optimal Profit = 2997.25 Reduced Cost = 2997.25 – 3000 = -2.75 * Actually reduced cost is the instantaneous per unit increase in the objective function value as X1 increases from 0. The above approach is valid as long as no other constraints “kick in” before X1 = 1.

Range of Optimality for C1 Range of Optimality for C2 Comparison With Excel Here is the printout out of the sensitivity analysis dealing with the objective function coefficients for the original Galaxy Industries problem. Reduced Costs are both 0 because already X1 > 0 and X2 > 0 Range of Optimality for C1 8 – 4.25  8 + 2 3.75  10 Range of Optimality for C2 5 – 1  5 + 5.666667 4  10.666667 Range of Optimality is the range of values that an objective function coefficient can assume without changing the optimal solution as long as no other changes are made.

Complementary Slackness Sensitivity report for the problem of MAX 1X1 + 5X2 For X1, Final Value = 0, but Reduced Cost ≠ 0. For X2, Final Value ≠ 0, but Reduced Cost = 0.

Complementary Slackness An important concept in linear programming is that of complementary slackness. It states: It can happen, that both are 0. Complementary Slackness For Objective Function Coefficients For each variable, either its value or its reduced cost will be 0.

Complementary Slackness Max. 8X1+5X2 For Space Rays, Final Value > 0, but Reduced Cost = 0. For Zappers, Final Value > 0, but Reduced Cost = 0.

Review Reasons for Sensitivity Analyses Approximations Dynamic Changes What-If Range of Optimality for Objective Function Coefficients By Graph Excel Reduced Cost – Two Meanings/Calculations How much an objective coefficient must change before the variable can be positive. Change to profit for a 1-unit increase in a variable whose optimal value is 0. Complementary Slackness