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.

Change the Objective Function Coefficient of X 1 C 1By changing the objective function coefficient C 1 of X 1, result is displayed as follows:

Range of Optimality optimal solution will not change!Thus the objective function coefficient for X 1 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) = ALLOWABLE DECREASEALLOWABLE INCREASEThe 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. Range of OptimalityThe 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.

Range of Optimality On Excel Here is the printout out of the sensitivity analysis dealing with the objective function coefficients for the original Galaxy Industries problem. Range of Optimality for C 1 8 – 4.25   10 Range of Optimality for C 2 5 – 1   Range of Optimality 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.

Reduced Cost 1 Max. 1X 1 + 5X 2 ST 2X 1 + 1X 2 ≤ 1000 (Plastic) 3X 1 + 4X 2 ≤ 2400 (Time) 1X 1 + 1X 2 ≤ 700 (Limit) 1X 1 - 1X 2 ≤ 350 (Product mix) X 1, X 2 ≥ 0 X 1 = 0, X 2 = 600, = 3,000 The optimal solution is X 1 = 0, X 2 = 600, objective function = 3,000 No product 1 were produced because its unit profit was not large enough.

Questions Question 1 How much would the cost have to be reduced (profit be increased) so that it would be profitable to make X 1 ’s? Question 2 If X 1 were at least 1, how would the profit be affected?

Answer to Question 1 When slope of objective function line equals slope of time constraint line, optimal solutions exist with X 1 >0: Objective function: C 1 Objective function: C 1 X 1 + 5X 2 Time Constraint: 3X 1 + 4X 2 ≤ 2400 C1C1 Thus, C1/5 = 3/4 or C1 = 3.75 Per unit profit of product 1 has to be increased to $3.75 to make production of 1 profitable; profit has to be increased Increased Profit = 3.75 – 1 = 2.75Increased Profit = 3.75 – 1 = 2.75 Reduced Cost-2.75Reduced Cost = 1 – 3.75 = -2.75

Answer to Question 2 If X 1 ≥ 1, how is the profit affected? 1 Max. 1X 1 + 5X 2 ST 2X 1 + 1X 2 ≤ 1000 (Plastic) 3X 1 + 4X 2 ≤ 2400 (Time) 1X 1 + 1X 2 ≤ 700 (Limit) 1X 1 - 1X 2 ≤ 350 (Product mix) X 1 ≥ 1 (Requirement of X 1 ) 1, , Solution is X 1 =1, X 2 = , New Optimal Profit = Reduced Cost Reduced Cost = – 3000 = -2.75

Reduced Cost on Excel Here is the printout out of the sensitivity analysis dealing with the objective function coefficients for the problem. Reduced Costs

Complementary Slackness MAX 1X 1 + 5X 2 Sensitivity report for the problem For X 1, Final Value = 0, but Reduced Cost ≠ 0. For X 2, Final Value ≠ 0, but Reduced Cost = 0.

Complementary Slackness MAX 8X 1 + 5X 2 For Space Rays, Final Value > 0, but Reduced Cost = 0. For Zappers, Final Value > 0, but Reduced Cost = 0. Sensitivity report for the problem

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.

Review Reasons for Sensitivity Analyses –Approximations –Dynamic Changes –What-If Range of Optimality for Objective Function Coefficients –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