1、Chapter 5 Sensitivity Analysis: An Applied Approach,to accompany Operations Research: Applications and Algorithms 4th edition by Wayne L. Winston,Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.,5.1 A Graphical Introduction to Sensitivity Analysis,Sensitivity analysis is concerne
2、d with how changes in an LPs parameters affect the optimal solution. Reconsider the Giapetto problem from Chapter 3. Where: x1 = number of soldiers produced each week x2 = number of trains produced each week.,max z = 3x1 + 2x2 2 x1 + x2 100 (finishing constraint)x1 + x2 80 (carpentry constraint)x1 4
3、0 (demand constraint)x1,x2 0 (sign restriction),The optimal solution for this LP was z = 180, x1=20, x2= 60 (point B) and it has x1, x2, and s3 (the slack variable for the demand constraint) as basic variables. How would changes in the problems objective function coefficients or right-hand side valu
4、es change this optimal solution?,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 switc
5、h 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 opti
6、mal.,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 b1 = number of available finishing hours. The current optimal solution (point B) is where the carpentry and finishing constraints are bi
7、nding. If the value of b1 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.,In the Giapetto problem to the right, we see that if b1 120, x1 will be greater than 40 and
8、will violate the demand constraint. Also, if b1 80, x1 will be less than 0 and the nonnegativity constraint for x1 will be violated. Therefore: 80 b1 120 The current basis remains optimal for 80 b1 120, but the decision variable values and z-value will change.,It is often important to determine how
9、a change in a constraints rhs changes the LPs optimal z-value. The shadow price for the ith constraint of an LP is the amount by which the optimal z-value is improved if the rhs of the ith constraint is increased by one. This definition applies only if the change in the rhs of constraint i leaves th
10、e current basis optimal. For the finishing constraint, 100 + finishing hours are available.The LPs optimal solution is then x1 = 20 + and x2 = 60 with z = 3x1 + 2x2 = 3(20 + ) + 2(60 - ) = 180 + . Thus, as long as the current basis remains optimal, a one-unit increase in the number of finishing hour
11、s will increase the optimal z-value by $1. So, the shadow price for the first (finishing hours) constraint is $1.,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 aga
12、in. 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 solu
13、tion 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.,5.2 The Computer and Sensitivity Analysis,If an LP has more than two decision variables, the range of values for
14、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
15、analysis discussed.,Example 1: Winco Products 1,Winco sells four types of products. The resources needed to produce one unit of each are known.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 maxi
16、mize profit.,Example 1: Solution,Let xi = number of units of product i produced by Winco. The Winco LP formulation:,max z = 4x1 + 6x2 +7x3 + 8x4 s.t. x1 + x2 + x3 + x4 = 950x4 4002x1 + 3x2 + 4x3 + 7x4 46003x1 + 4x2 + 5x3 + 6x4 5000x1,x2,x3,x4 0,Ex. 1 Solution continued,The LINDO output. Reduced cost
17、 is the amount the objective function coefficient for variable i would have to be increased for there to be an alternative optimal solution.,MAX 4 X1 + 6 X2 + 7 X3 + 8 X4SUBJECT TO2) X1 + X2 + X3 + X4 = 9503) X4 = 4004) 2 X1 + 3 X2 + 4 X3 + 7 X4 = 46005) 3 X1 + 4 X2 + 5 X3 + 6 X4 = 5000ENDLP OPTIMUM
18、 FOUND AT STEP 4OBJECTIVE FUNCTION VALUE1) 6650.000VARIABLE VALUE REDUCED COSTX1 0.000000 1.000000X2 400.000000 0.000000X3 150.000000 0.000000X4 400.000000 0.000000ROW SLACK OR SURPLUS DUAL PRICES2) 0.000000 3.0000003) 0.000000 -2.0000004) 0.000000 1.0000005) 250.000000 0.000000NO. ITERATIONS= 4,Ex.
19、 1 Solution continued,LINDO sensitivity analysis output Allowable range (w/o changing basis) for the x2 coefficient (c2) is: 5.50 c2 6.667 Allowable range (w/o changing basis) for the rhs (b1) of the first constraint is: 850 b1 1000,RANGES IN WHICH THE BASIS IS UNCHANGED:OBJ COEFFICIENT RANGESVARIAB
20、LE CURRENT ALLOWABLE ALLOWABLECOEF INCREASE DECREASEX1 4.000000 1.000000 INFINITYX2 6.000000 0.666667 0.500000X3 7.000000 1.000000 0.500000X4 8.000000 2.000000 INFINITYRIGHTHAND SIDE RANGESROW CURRENT ALLOWABLE ALLOWABLERHS INCREASE DECREASE2 950.000000 50.000000 100.0000003 400.000000 37.500000 125
21、.0000004 4600.000000 250.000000 150.0000005 5000.000000 INFINITY 250.000000,Ex. 1 Solution continued,Shadow prices are shown in the Dual Prices section of LINDO output. Shadow prices are the amount the optimal z-value improves if the rhs of a constraint is increased by one unit (assuming no change i
22、n basis).,MAX 4 X1 + 6 X2 + 7 X3 + 8 X4SUBJECT TO2) X1 + X2 + X3 + X4 = 9503) X4 = 4004) 2 X1 + 3 X2 + 4 X3 + 7 X4 = 46005) 3 X1 + 4 X2 + 5 X3 + 6 X4 = 5000ENDLP OPTIMUM FOUND AT STEP 4OBJECTIVE FUNCTION VALUE1) 6650.000VARIABLE VALUE REDUCED COSTX1 0.000000 1.000000X2 400.000000 0.000000X3 150.0000
23、00 0.000000X4 400.000000 0.000000ROW SLACK OR SURPLUS DUAL PRICES2) 0.000000 3.0000003) 0.000000 -2.0000004) 0.000000 1.0000005) 250.000000 0.000000NO. ITERATIONS= 4,Shadow price signs,Constraints with symbols will always have nonpositive shadow prices. Constraints with will always have nonnegative
24、shadow prices. Equality constraints may have a positive, a negative, or a zero shadow price.,For any inequality constraint, the product of the values of the constraints slack/excess variable and the constraints shadow price must equal zero. This implies that any constraint whose slack or excess vari
25、able 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:,When the optimal solution is degenerate (a bfs is degenerate
26、if at least one basic variable in the optimal solution equals 0), caution must be used when interpreting the LINDO output. 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.,MAX 6 X1 + 4 X2 + 3 X3 + 2 X
27、4SUBJECT TO2) 2 X1 + 3 X2 + X3 + 2 X4 = 4003) X1 + X2 + 2 X3 + X4 = 1504) 2 X1 + X2 + X3 + 0.5 X4 = 2005) 3 X1 + X2 + X4 = 250,5.3 Managerial Use of Shadow Prices,The managerial significance of shadow prices is that they can often be used to determine the maximum amount a manger should be willing to
28、 pay for an additional unit of a resource.,Example 5: Winco Products 2,Reconsider the Winco to the right. What is the most Winco should be willing to pay for additional units of raw material or labor?,5.4 What happens to the Optimal z-Value if the Current Basis Is No Longer Optimal?,Shadow prices we
29、re 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 f
30、eature can be used to determine how the shadow price of a constraint and optimal z-value change.,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 constraints shadow price. For co
31、nstraint, 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,A graph of the optimal objective function value as a function of a variables
32、objective function coefficient can be created. When the slope of the line is portrayed graphically, the graph is a piecewise linear function. The slope of each line segment is equal to the value of x1 in the optimal solution.,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.,
