Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. 1 Chapter 5 Sensitivity...

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

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Chapter 5Sensitivity Analysis: An Applied

Approach to accompany

Introduction to Mathematical Programming: Operations Research, Volume 14th edition, by Wayne L. Winston and Munirpallam Venkataramanan

Presentation: H. Sarper


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5.1 – A Graphical Approach to Sensitivity Analysis

Sensitivity analysis is concerned with how changes in an LP’s parameters affect the optimal solution.

max z = 3x1 + 2x2

2 x1 + x2 ≤ 100 (finishing constraint)

x1 + x2 ≤ 80 (carpentry constraint)

x1 ≤ 40 (demand constraint)

x1,x2 ≥ 0 (sign restriction)

Where:

x1 = number of soldiers produced each week

x2 = number of trains produced each week.

Reconsider the Giapetto problem from Chapter 3 shown to the right:


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The optimal solution for this LP was z = 180, x1 = 20, x2 = 60 (point B in the figure to the right) and it has x1, x2, and s3 (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? X1

X2

10 20 40 50 60 80

2040

6080

1 00

finishing constraintSlope = -2

carpentry constraintSlope = -1

demand constraint

Feasible RegionA

D

Isoprofit line z = 120 Slope = -3/2

C

B


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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.


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The values of the contribution to profit for soldiers for which the current optimal basis (x1,x2,s3) will remain optimal can be determined as follows:

Let c1 be the contribution ($3 per soldier) to the profit. For what values of c1 does the current basis remain optimal?

3x1 + 2x2 = constant

Rearranging:

At present c1 = 3 and each isoprofit line has the form:

Since -2 < slope < -1: 2c 1

2 1

x 23

2x 1

1

2constant

c 1

2x 1

1

2constant

Solving for c1 yields: 2 c 1 4Note: the profit will change in this range of c1


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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 b1 = number of available finishing hours.

The current optimal solution (point B) is where the carpentry and finishing constraints are binding. 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.


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X1

X2

20 40 50 60 80

20

40

60

80

100

finishing constraint, b1 = 100

carpentry constraint

demand constraint

Feasible Region

A

D

Isoprofit line z = 120

C

B



In the Giapetto problem to the right, we see that if b1 > 120, x1 will be greater than 40 and 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.


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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, 100 + finishing hours are available (assuming the current basis remains optimal). The LP’s 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 hours will increase the optimal z-value by $1. So, the shadow price for the first (finishing hours) constraint is $1.


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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.


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5.2 – 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.


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Consider the following maximization problem. Winco sells four types of products. The resources needed to produce one unit of each are:

Product 1 Product 2 Product 3 Product 4 Available

Raw material 2 3 4 7 4600

Hours of labor 3 4 5 6 5000

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 xi = number of units of product i produced by Winco.


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The Winco LP formulation:

max z = 4x1 + 6x2 +7x3 + 8x4

s.t. x1 + x2 + x3 + x4 = 950

x4 ≥ 400

2x1 + 3x2 + 4x3 + 7x4 ≤ 4600

3x1 + 4x2 + 5x3 + 6x4 ≤ 5000

x1,x2,x3,x4 ≥ 0


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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 <= 4600 5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000 END

LP OPTIMUM FOUND AT STEP 4

OBJECTIVE FUNCTION VALUE 1) 6650.000

VARIABLE VALUE REDUCED COST X1 0.000000 1.000000 X2 400.000000 0.000000 X3 150.000000 0.000000 X4 400.000000 0.000000

ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 3.000000 3) 0.000000 -2.000000 4) 0.000000 1.000000 5) 250.000000 0.000000

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.


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5.2 – The Computer and Sensitivity Analysis RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE CURRENT ALLOWABLE ALLOWABLE

COEF INCREASE DECREASE

X1 4.000000 1.000000 INFINITY

X2 6.000000 0.666667 0.500000

X3 7.000000 1.000000 0.500000

X4 8.000000 2.000000 INFINITY

RIGHTHAND SIDE RANGES

ROW CURRENT ALLOWABLE ALLOWABLE

RHS INCREASE DECREASE

2 950.000000 50.000000 100.000000

3 400.000000 37.500000 125.000000

4 4600.000000 250.000000 150.000000

5 5000.000000 INFINITY 250.000000

LINDO sensitivity analysis example(s).

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


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Shadow prices are shown in the Dual Prices section of LINDO output.






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).


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Interpretation of shadow prices for the Winco LP

ROW SLACK OR SURPLUS DUAL PRICES

2) 0.000000 3.000000 (overall demand)

3) 0.000000 -2.000000 (product 4 demand)

4) 0.000000 1.000000 (raw material availability)

5) 250.000000 0.000000 (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.


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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.


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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 ∞


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Degeneracy and Sensitivity AnalysisWhen 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 + X3 + 0.5 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.


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Since the LP has four constraints and in the optimal solution only two variables are positive, the optimal solution is a degenerate bfs.


OBJECTIVE FUNCTION VALUE

1) 700.0000


ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 0.500000 3) 0.000000 1.250000 4) 0.000000 0.000000 5) 0.000000 1.250000


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RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGESVARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE

X1 6.000000 3.000000 3.000000 X2 4.000000 5.000000 1.000000 X3 3.000000 3.000000 2.142857 X4 2.000000 1.500000 INFINITY

RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 400.000000 0.000000 200.000000 3 150.000000 0.000000 0.000000 4 200.000000 INFINITY 0.000000 5 250.000000 0.000000 120.000000


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THE TABLEAU ROW (BASIS) X1 X2 X3 X4 SLK 2 1 ART 0.000 0.000 0.000 1.500 0.500 2 X2 0.000 1.000 0.000 0.500 0.500 3 X3 0.000 0.000 1.000 0.167 -0.167 4 SLK 4 0.000 0.000 0.000 -0.500 0.000 5 X1 1.000 0.000 0.000 0.167 -0.167 ROW SLK 3 SLK 4 SLK 5 1 1.250 0.000 1.250 700.000 2 -0.250 0.000 -0.250 100.000 3 0.583 0.000 -0.083 0.000 4 -0.500 1.000 -0.500 0.000 5 0.083 0.000 0.417 50.000

LINDO TABLEAU command indicates the optimal basis is RV = {x1,x2,x3,s4}.


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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.


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5.3 – Managerial Use of Shadow Prices




VARIABLE VALUE REDUCED COST

X1 0.000000 1.000000 X2 400.000000 0.000000 X3 150.000000 0.000000 X4 400.000000 0.000000


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


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5.3 – Managerial Use of Shadow Prices




VARIABLE VALUE REDUCED COST

X1 0.000000 1.000000 X2 400.000000 0.000000 X3 150.000000 0.000000 X4 400.000000 0.000000


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.


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5.4 – 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 -3900. This indicates at least 3900 units of raw material are required to make the problem feasible.


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OBJECTIVE FUNCTION VALUE

1) 5400.000



RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X1 4.000000 1.000000 INFINITY X2 6.000000 INFINITY 0.500000 X3 7.000000 1.000000 INFINITY X4 8.000000 6.000000 INFINITY

RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 950.000000 0.000000 183.333328 3 400.000000 0.000000 137.500000 4 3900.000000 550.000000 0.000000 5 5000.000000 INFINITY 950.000000

THE TABLEAU

ROW (BASIS) X1 X2 X3 X4 SLK 3 SLK 4 1 ART 0.000 0.000 1.000 0.000 6.000 2.000 2 X1 1.000 0.000 -1.000 0.000 -4.000 -1.000 3 X4 0.000 0.000 0.000 1.000 -1.000 0.000 4 X2 0.000 1.000 2.000 0.000 5.000 1.000 5 SLK 5 0.000 0.000 0.000 0.000 -2.000 -1.000 ART ART 0.000 0.000 1.000 0.000 6.000 2.000

SLK 50.000 5400.0000.000 550.0000.000 400.0000.000 0.0001.000 950.0000.000 0.000

Raw Material rhs = 3900 optimal solution


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RIGHTHANDSIDE PARAMETRICS REPORT FOR ROW: 4

VAR VAR PIVOT RHS DUAL PRICE OBJOUT IN ROW VAL BEFORE PIVOT VAL

3900.00 2.00000 5400.00 X1 X3 2 4450.00 2.00000 6500.00 SLK 5 SLK 3 5 4850.00 1.00000 6900.00 X3 SLK 4 2 5250.00 -0.333067E-15 6900.00 10000.0 -0.555112E-16 6900.00

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 rm be the amount of available raw material. If rm < 3900, we know the LP is infeasible. From the figure above, from 3899 < rm < 4450, the shadow price (DUAL) is $2, switches to $1 from 4449 < rm < 4849, and finally to $0 at 4850.


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LINDO Parametric Feature Graphical Output (z-value vs. Raw Material rhs from 3900 to 10000)


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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


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Effect of change in Objective Function Coefficient on Optimal z-value

max z = 3x1 + 2x2

2 x1 + x2 ≤ 100 (finishing constraint)

x1 + x2 ≤ 80 (carpentry constraint)

x1 ≤ 40 (demand constraint)

x1,x2 ≥ 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 c1 = objective coefficient of x1. Currently, c1 = 3 and we want to determine how the optimal z-value depend upon c1..


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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 c1x1 + 2x2 = k, we know the slope of the isoprofit line is just -c1/2. This implies:

1. Point A is optimal if -c1/2 ≥ -1 or 0 ≤ c1 ≤ 2 ( -1 is the carpentry constraint slope).

2. Point B is optimal if -2 ≤ -c1/2 ≤ -1 or 2 ≤ c1 ≤ 4 (between the slopes of the carpentry and finishing constraint slopes).

3. Point C is optimal if -c1/2 ≤ -2 or c1 ≥ 4 ( -2 is the finishing constraint slope).

This piecewise function is shown on the next page.


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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.

0 2 4 6 8 10

100

200

300

400

500

z-value

Optimal z-Value vs c1

C1

Opt

imal

z-V

alue

440

0

z c 1

100 c 1

z c 1 160 0 c 1 2if

120 20c 1 2 c 1 4if

40 40 c 1 c 1 4if

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. 1 Chapter 5 Sensitivity...

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