Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. 1 Chapter 5 Sensitivity Analysis: An Applied Approach to accompany Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan Presentation: H. Sarper

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – A Graphical Approach to Sensitivity Analysis Sensitivity analysis is concerned with how changes in an LP’s parameters affect the optimal solution. max z = 3x 1 + 2x 2 2 x 1 + x 2 ≤ 100 (finishing constraint) x 1 + x 2 ≤ 80 (carpentry constraint) x 1 ≤ 40 (demand constraint) x 1,x 2 ≥ 0 (sign restriction) Where: x 1 = number of soldiers produced each week x 2 = number of trains produced each week. Reconsider the Giapetto problem from Chapter 3 shown to the right:

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – A Graphical Approach to Sensitivity Analysis The optimal solution for this LP was z = 180, x 1 = 20, x 2 = 60 (point B in the figure to the right) and it has x 1, x 2, and s 3 (the slack variable for the demand constraint. How would changes in the problem’s objective function coefficients or right-hand side values change this optimal solution?

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – A Graphical Approach to Sensitivity Analysis Graphical analysis of the effect of a change in an objective function value for the Giapetto LP shows: By inspection, we can see that making the slope of the isoprofit line more negative than the finishing constraint (slope = -2) will cause the optimal point to switch from point B to point C. Likewise, making the slope of the isoprofit line less negative than the carpentry constraint (slope = -1) will cause the optimal point to switch from point B to point A. Clearly, the slope of the isoprofit line must be between -2 and -1 for the current basis to remain optimal.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – A Graphical Approach to Sensitivity Analysis The values of the contribution to profit for soldiers for which the current optimal basis (x 1,x 2,s 3 ) will remain optimal can be determined as follows: Let c 1 be the contribution ($3 per soldier) to the profit. For what values of c 1 does the current basis remain optimal? 3x 1 + 2x 2 = constant Rearranging: At present c 1 = 3 and each isoprofit line has the form: Since -2 < slope < -1: Solving for c 1 yields: Note: the profit will change in this range of c 1

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – A Graphical Approach to Sensitivity Analysis Graphical Analysis of the Effect of a Change in RHS on the LP’s Optimal Solution (using the Giapetto problem). A graphical analysis can also be used to determine whether a change in the rhs of a constraint will make the current basis no longer optimal. For example, let b 1 = number of available finishing hours. The current optimal solution (point B) is where the carpentry and finishing constraints are binding. If the value of b 1 is changed, then as long as where the carpentry and finishing constraints are binding, the optimal solution will still occur where the carpentry and finishing constraints intersect.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – A Graphical Approach to Sensitivity Analysis In the Giapetto problem to the right, we see that if b 1 > 120, x 1 will be greater than 40 and will violate the demand constraint. Also, if b 1 < 80, x 1 will be less than 0 and the nonnegativity constraint for x 1 will be violated. Therefore: 80 ≤ b 1 ≤ 120 The current basis remains optimal for 80 ≤ b 1 ≤ 120, but the decision variable values and z-value will change.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – A Graphical Approach to Sensitivity Analysis Shadow Prices (using the Giapetto problem) It is often important to determine how a change in a constraint’s rhs changes the LP’s optimal z-value. We define: The shadow price for the i th constraint of an LP is the amount by which the optimal z-value is improved (increased in a max problem or decreased in a min problem) if the rhs of the i th constraint is increased by one. This definition applies only if the change in the rhs of constraint i leaves the current basis optimal. For the finishing constraint,  finishing hours are available (assuming the current basis remains optimal). The LP’s optimal solution is then x 1 = 20 +  and x 2 = 60 –  with z = 3x 1 + 2x 2 = 3(20 +  ) + 2(60 -  ) = . Thus, as long as the current basis remains optimal, a one-unit increase in the number of finishing hours will increase the optimal z-value by $1. So, the shadow price for the first (finishing hours) constraint is $1.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – A Graphical Approach to Sensitivity Analysis Importance of Sensitivity Analysis Sensitivity analysis is important for several reasons: Values of LP parameters might change. If a parameter changes, sensitivity analysis shows it is unnecessary to solve the problem again. For example in the Giapetto problem, if the profit contribution of a soldier changes to $3.50, sensitivity analysis shows the current solution remains optimal. Uncertainty about LP parameters. In the Giapetto problem for example, if the weekly demand for soldiers is at least 20, the optimal solution remains 20 soldiers and 60 trains. Thus, even if demand for soldiers is uncertain, the company can be fairly confident that it is still optimal to produce 20 soldiers and 60 trains.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis If an LP has more than two decision variables, the range of values for a rhs (or objective function coefficient) for which the basis remains optimal cannot be determined graphically. These ranges can be computed by hand but this is often tedious, so they are usually determined by a packaged computer program. LINDO will be used and the interpretation of its sensitivity analysis discussed.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis Consider the following maximization problem. Winco sells four types of products. The resources needed to produce one unit of each are: Product 1Product 2Product 3Product 4Available Raw material Hours of labor Sales price$4$6$7$8 To meet customer demand, exactly 950 total units must be produced. Customers demand that at least 400 units of product 4 be produced. Formulate an LP to maximize profit. Let x i = number of units of product i produced by Winco.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis The Winco LP formulation: max z = 4x 1 + 6x 2 +7x 3 + 8x 4 s.t. x 1 + x 2 + x 3 + x 4 = 950 x 4 ≥ 400 2x 1 + 3x 2 + 4x 3 + 7x 4 ≤ x 1 + 4x 2 + 5x 3 + 6x 4 ≤ 5000 x 1,x 2,x 3,x 4 ≥ 0

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis MAX 4 X1 + 6 X2 + 7 X3 + 8 X4 SUBJECT TO 2) X1 + X2 + X3 + X4 = 950 3) X4 >= 400 4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= ) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000 END LP OPTIMUM FOUND AT STEP 4 OBJECTIVE FUNCTION VALUE 1) VARIABLE VALUE REDUCED COST X X X X ROW SLACK OR SURPLUS DUAL PRICES 2) ) ) ) NO. ITERATIONS= 4 LINDO output and sensitivity analysis example(s). Reduced cost is the amount the objective function coefficient for variable i would have to be increased for there to be an alternative optimal solution.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X INFINITY X X X INFINITY RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE INFINITY LINDO sensitivity analysis example(s). Allowable range (w/o changing basis) for the x 2 coefficient (c 2 ) is: 5.50  c 2  Allowable range (w/o changing basis) for the rhs (b 1 ) of the first constraint is: 850  b 1  1000

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis Shadow prices are shown in the Dual Prices section of LINDO output. MAX 4 X1 + 6 X2 + 7 X3 + 8 X4 SUBJECT TO 2) X1 + X2 + X3 + X4 = 950 3) X4 >= 400 4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= ) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000 END LP OPTIMUM FOUND AT STEP 4 OBJECTIVE FUNCTION VALUE 1) VARIABLE VALUE REDUCED COST X X X X ROW SLACK OR SURPLUS DUAL PRICES 2) ) ) ) NO. ITERATIONS= 4 Shadow prices are the amount the optimal z- value improves if the rhs of a constraint is increased by one unit (assuming no change in basis).

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis Interpretation of shadow prices for the Winco LP ROW SLACK OR SURPLUS DUAL PRICES 2) (overall demand) 3) (product 4 demand) 4) (raw material availability) 5) (labor availability) Assuming the allowable range of the rhs is not violated, shadow (Dual) prices show: $3 for constraint 1 implies that each one-unit increase in total demand will increase net sales by $3. The -$2 for constraint 2 implies that each unit increase in the requirement for product 4 will decrease revenue by $2. The $1 shadow price for constraint 3 implies an additional unit of raw material (at no cost) increases total revenue by $1. Finally, constraint 4 implies any additional labor (at no cost) will not improve total revenue.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis Shadow price signs 1.Constraints with  symbols will always have nonpositive shadow prices. 2.Constraints with  will always have nonnegative shadow prices. 3.Equality constraints may have a positive, a negative, or a zero shadow price.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis Sensitivity Analysis and Slack/Excess Variables For any inequality constraint, the product of the values of the constraint’s slack/excess variable and the constraint’s shadow price must equal zero. This implies that any constraint whose slack or excess variable > 0 will have a zero shadow price. Similarly, any constraint with a nonzero shadow price must be binding (have slack or excess equaling zero). For constraints with nonzero slack or excess, relationships are detailed in the table below: Type of Constraint Allowable Increase for rhs Allowable Decrease for rhs ≤ ∞= value of slack ≥ = value of excess∞

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis Degeneracy and Sensitivity Analysis When the optimal solution is degenerate (a bfs is degenerate if at least one basic variable in the optimal solution equals 0), caution must be used when interpreting the LINDO output. MAX 6 X1 + 4 X2 + 3 X3 + 2 X4 SUBJECT TO 2) 2 X1 + 3 X2 + X3 + 2 X4 <= 400 3) X1 + X2 + 2 X3 + X4 <= 150 4) 2 X1 + X2 + X X4 <= 200 5) 3 X1 + X2 + X4 <= 250 For an LP with m constraints, if the optimal LINDO output indicates less than m variables are positive, then the optimal solution is degenerate bfs. Consider the LINDO LP formulation shown to the right and the LINDO output on the next slide.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis Since the LP has four constraints and in the optimal solution only two variables are positive, the optimal solution is a degenerate bfs. LP OPTIMUM FOUND AT STEP 3 OBJECTIVE FUNCTION VALUE 1) VARIABLE VALUE REDUCED COST X X X X ROW SLACK OR SURPLUS DUAL PRICES 2) ) ) )

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X X X X INFINITY RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE INFINITY

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis THE TABLEAU ROW (BASIS) X1 X2 X3 X4 SLK 2 1 ART X X SLK X ROW SLK 3 SLK 4 SLK LINDO TABLEAU command indicates the optimal basis is RV = { x 1,x 2,x 3,s 4 }.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – The Computer and Sensitivity Analysis Oddities that may occur when the optimal solution found by LINDO is degenerate are: 1.In the RANGE IN WHICH THE BASIS IS UNCHANGED at least one constraint will have a 0 AI or AD. This means that for at least one constraint the DUAL PRICE can tell us about the new z-value for either an increase or decrease in the rhs, but not both. 2.For a nonbasic variable to become positive, a nonbasic variable’s objective function coefficient may have to be improved by more than its RECDUCED COST. 3.Increasing a variable’s objective function coefficient by more than its AI or decreasing it by more than its AD may leave the optimal solution the same.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – Managerial Use of Shadow Prices MAX 4 X1 + 6 X2 + 7 X3 + 8 X4 SUBJECT TO 2) X1 + X2 + X3 + X4 = 950 3) X4 >= 400 4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= ) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000 END LP OPTIMUM FOUND AT STEP 4 OBJECTIVE FUNCTION VALUE 1) VARIABLE VALUE REDUCED COST X X X X ROW SLACK OR SURPLUS DUAL PRICES 2) ) ) ) NO. ITERATIONS= 4 The managerial significance of shadow prices is that they can often be used to determine the maximum amount a manger should be willing to pay for an additional unit of a resource. Reconsider the Winco to the right. What is the most Winco should be willing to pay for additional units of raw material or labor? raw material labor

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – Managerial Use of Shadow Prices MAX 4 X1 + 6 X2 + 7 X3 + 8 X4 SUBJECT TO 2) X1 + X2 + X3 + X4 = 950 3) X4 >= 400 4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= ) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000 END LP OPTIMUM FOUND AT STEP 4 OBJECTIVE FUNCTION VALUE 1) VARIABLE VALUE REDUCED COST X X X X ROW SLACK OR SURPLUS DUAL PRICES 2) ) ) ) NO. ITERATIONS= 4 The shadow price for raw material constraint (row 4) shows an extra unit of raw material would increase revenue $1. Winco could pay up to $1 for an extra unit of raw material and be as well off as it is now. Labor constraint’s (row 5) shadow price is 0 meaning that an extra hour of labor will not increase revenue. So, Winco should not be willing to pay anything for an extra hour of labor.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – What happens to the Optimal z-Value if the Current Basis Is No Longer Optimal? In Section 5.2 shadow prices were used to determine the new optimal z- value if the rhs of a constraint was changed but remained within the range where the current basis remains optimal. Changing the rhs of a constraint to values where the current basis is no longer optimal can be addressed by the LINDO PARAMETRICS feature. This feature can be used to determine how the shadow price of a constraint and optimal z- value change. The use of the LINDO PARAMETICS feature is illustrated by varying the amount of raw material in the Winco example. Suppose we want to determine how the optimal z-value and shadow price change as the amount of raw material varies between 0 and 10,000 units. With 0 raw material, we then obtain from the RANGE and SENSITIVTY ANALYSIS results that show Row 4 has an ALLOWABLE INCREASE of This indicates at least 3900 units of raw material are required to make the problem feasible.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – What happens to the Optimal z-Value if the Current Basis Is No Longer Optimal? OBJECTIVE FUNCTION VALUE 1) VARIABLE VALUE REDUCED COST X X X X ROW SLACK OR SURPLUS DUAL PRICES 2) ) ) ) RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X INFINITY X INFINITY X INFINITY X INFINITY RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE INFINITY THE TABLEAU ROW (BASIS) X1 X2 X3 X4 SLK 3 SLK 4 1 ART X X X SLK ART ART SLK Raw Material rhs = 3900 optimal solution

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – What happens to the Optimal z-Value if the Current Basis Is No Longer Optimal? RIGHTHANDSIDE PARAMETRICS REPORT FOR ROW: 4 VAR VAR PIVOT RHS DUAL PRICE OBJ OUT IN ROW VAL BEFORE PIVOT VAL X1 X SLK 5 SLK X3 SLK E E Changing Row 4’s rhs to 3900, resolving the LP, and selecting the REPORTS PARAMTERICS feature. In this feature we choose Row 4, setting the Value to 10000, and select text output. We then obtain the output below: Let r m be the amount of available raw material. If r m < 3900, we know the LP is infeasible. From the figure above, from 3899 < r m < 4450, the shadow price (DUAL) is $2, switches to $1 from 4449 < r m < 4849, and finally to $0 at 4850.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – What happens to the Optimal z-Value if the Current Basis Is No Longer Optimal? LINDO Parametric Feature Graphical Output (z-value vs. Raw Material rhs from 3900 to 10000)

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – What happens to the Optimal z-Value if the Current Basis Is No Longer Optimal? For any LP, the graph of the optimal objective function value as a function a rhs will be a piecewise linear function. The slope of each straight line segment is just the constraint’s shadow price. 1.For < constraints in a maximization LP, the slope of each segment must be nonnegative and the slopes of successive line segments will be nonincreasing. 2.For a > constraint, in a maximization problem, the graph of the optimal function will again be piecewise linear function. The slope of each line segment will be nonpositive and the slopes of successive segments will be nonincreasing

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – What happens to the Optimal z-Value if the Current Basis Is No Longer Optimal? Effect of change in Objective Function Coefficient on Optimal z-value max z = 3x 1 + 2x 2 2 x 1 + x 2 ≤ 100 (finishing constraint) x 1 + x 2 ≤ 80 (carpentry constraint) x 1 ≤ 40 (demand constraint) x 1,x 2 ≥ 0 (sign restriction) A graph of the optimal objective function value as a function of a variable’s objective function coefficient can be created. Consider again the Giapetto LP shown to the right. Let c 1 = objective coefficient of x 1. Currently, c 1 = 3 and we want to determine how the optimal z-value depend upon c 1..

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – What happens to the Optimal z-Value if the Current Basis Is No Longer Optimal? Recall from the Giapetto problem, if the isoprofit line is flatter than the carpentry constraint, Point A(0,80) is optimal. Point B(20,60) is optimal if the isoprofit line is steeper than the carpentry constraint but flatter than the finishing constraint. Finally, Point C(40,20) is optimal if the slope of the isoprofit line is steeper than the slope of the finishing constraint. Since a typical isoprofit line is c 1 x 1 + 2x 2 = k, we know the slope of the isoprofit line is just -c 1 /2. This implies: 1.Point A is optimal if -c 1 /2 ≥ -1 or 0 ≤ c 1 ≤ 2 ( -1 is the carpentry constraint slope). 2.Point B is optimal if -2 ≤ -c 1 /2 ≤ -1 or 2 ≤ c 1 ≤ 4 (between the slopes of the carpentry and finishing constraint slopes). 3.Point C is optimal if -c 1 /2 ≤ -2 or c 1 ≥ 4 ( -2 is the finishing constraint slope). This piecewise function is shown on the next page.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc – What happens to the Optimal z-Value if the Current Basis Is No Longer Optimal? In a maximization LP, the slope of the graph of the optimal z- value as a function of an objective function coefficient will be nondecreasing. In a minimization LP, the slope of the graph of the optimal z- value as a function of an objective function coefficient will be nonincreasing.