Sensitivity Analysis: An Applied Approach

1
Chapter 5

• Sensitivity Analysis An Applied Approach

2
5.1 A Graphical Introduction to Sensitivity
Analysis

• Sensitivity analysis is concerned 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 40 (demand
constraint) x1,x2 0 (sign
restriction)
3

• The optimal solution for this LP was z 180,
  x120, 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 values change this optimal
  solution?

4

Graphical Analysis of the Effect of a Change in
an Objective Function Coefficient

• Lets determine the values of the contribution to
  profit for soldiers for which the current basis
  will remain optimal.
• Let c1 contribution to profit by each soldier
• For what values of c1 does the current basis
  remain optimal?
• c1 (current) 3 (coef. of x1 in z)
• Isoprofit line 3x12x2constant, or
• x2(-3/2)x1(constant/2)
• Slope of the line (current) -3/2
• Slope of each isoprofit line -c1/2
• Changing the objective function coefficient of a
  variable changes the slope of the
  isoprofit/isocost line.

5

• 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 (steeper)
  (-c1/2lt-2, or c1gt4) than the finishing constraint
  (slope -2) will cause the optimal point to
  switch from point B to point C (40,20).
• Likewise, making the slope of the isoprofit line
  less negative (flatter) (-c1/2gt-1, or c1lt2) than
  the carpentry constraint (slope -1) will cause
  the optimal point to switch from point B to point
  A (0, 80).
• Clearly, the slope of the isoprofit line must be
  between -2 and -1 (ie., 2 c1 4) for the
  current basis to remain optimal.
• And Giapetto should still manufature 20 soldiers
  and 60 trains.
• Profit may change e.g., c14, z 200 instead of
  180.

6

• The current basis remains optimal as long as the
  current optimal solution is the last point in the
  feasible region to make contact with
  isoprofit/isocost lines as we move in the
  direction of
• increasing (for a max problem) and
• decreasing (for a min problem).
• If the current basis remains optimal
• the values of the decision variables remain
  unchanged,
• but the optimal z-value may change.

7

Graphical Analysis of the Effect of a Change in a
Right-Hand-Side on the LPs Optimal Solution

• 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.
• Consider the Giapetto problem again
• Let b1 number of available finishing hours.
• b1(current)100

max z 3x1 2x2 2 x1 x2 100 (finishing
constraint) x1 x2 80 (carpentry
constraint) x1 40 (demand
constraint) x1,x2 0 (sign
restriction)
8

• The current optimal solution (point B) is where
  the carpentry and finishing constraints are
  binding (intersect).
• A change in a bi value shifts the corresponding
  constraint parallel to its current position.
• If the value of b1 is changed, then as long as
  where the carpentry and finishing constraints are
  binding remains feasible, the optimal solution
  will still occur where the carpentry and
  finishing constraints intersect.

9

• In the Giapetto problem to the right, we see that
• if b1 gt 120,
• x1 gt 40 (violates the demand constraint).
• if b1 lt 80 (A),
• x1 lt 0 (nonnegativity constraint for x1 is
  violated).
• Therefore 80 b1 120
• The current basis remains optimal for 80 b1
  120, but the decision variable values and z-value
  may change.
• For 80 b1 100 (optimal sol any point on line
  AB)
• For 100 b1 120 (optimal sol any point on line
  BC)

10
If the current basis remains optimal, determine
how a change in the rhs of a constraint changes
the values of the decision variables

• Let b1 number of available finishing hours.
• b1100 (current value)
• Lets change b1 100 ? ? current basis remains
  optimal for
• -20? ? ?20 (80 b1 120).
• ? New values of decision variables are found
• by solving
• 2x1 x2 100 ? (finishing constraint)
• x1 x2 80 (carpentry
  constraint)
• x1 20 ? and x2 60 - ?
• That is, an increase in no. of available
  finishing hours (b1 100 ?) results in an
  increase in no. of soldiers produced (x1) and a
  decrease in no. of trains produced (x2).

11

• Shadow Prices
• How 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 (increased in a max and decreased in a
  min problem) if the rhs of the ith constraint is
  increased (or decreased) by 1.
• This definition applies only if the change in the
  rhs of constraint i leaves the current basis
  optimal.
• Finding the shadow price for the first (finishing
  hours) constraint (Giapetto problem)
• If 100 ? finishing hours are available (and
  current basis remains optimal)
• Then, the LPs optimal solution is found by
  solving 1st and 2nd constraints as x1 20
  ? and x2 60 - ? with
• z 3x1 2x2 3(20 ?) 2(60 - ?)
  180 ?.
• 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.

12
Finding the shadow price for the second
(carpentry hours) constraint

• If b2 80 ? carpentry hours are available (and
  current basis remains optimal)
• Then, the LPs optimal solution is found by
  solving 1st and 2nd constraints x1 20 -
  ? and x2 60 2? with
• z 3x1 2x2 3(20 - ?) 2(60 2?) 180
  ?.
• A one-unit increase in the number of carpentry
  hours will increase the optimal z-value by 1.
• So the shadow price for the second (carpentry
  hours) constraint is 1.

13
Finding the shadow price for the third (demand)
constraint

• If b3 40 ? soldiers are produced per week
  (and current basis remains optimal)
• Optimal values of the decision variables remain
  unchanged.
• The optimal value of z-also remains unchanged.
• ? Shadow price 0
• Recall the optimal basic variables were x1, x2,
  and s3
• Whenever the slack or excess variable for a
  constraint is positive in an LPs optimal
  solution,
• ? shadow price (for the constraint) 0

14

• Let ?bi increment on rhs of the ith constraint
• (?bi lt0 decrement on rhs of the ith
  constraint)
• ? Each unit increase on rhs of the ith
  constraint will
• increase optimal z-value (for a max
  problem) and decrease
• optimal z-value (for a min problem) by
  the
• shadow price.
• Thus, the new optimal z-value is found by
• For max problem (New optimal z-value) (old
  optimal z-value)
• 
  (constraint is shadow price) ?bi
• For min problem (New optimal z-value) (old
  optimal z-value)
• -
  (constraint is shadow price) ?bi
• Example if 110 finishing hours are available,
  then
• ?b1110-10010 and z-value18010(1)190

15

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 often unnecessary to solve the
  problem again.
• For example in the Giapetto problem, if the
  profit contribution of a soldier changes from 3
  to 3.50, sensitivity analysis shows that it is
  unnecessary to solve the Giapetto problem again,
  because the current solution remains optimal.
• Uncertainty about LP parameters.
• Even if demand is uncertain, the company can be
  fairly confident that it can still produce
  optimal amounts of products.
• In the Giapetto problem for example, if the
  weekly demand for soldiers is at least 20 (x1
  optimal), 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.

16
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.
• To obtain a sensitivity report in LINDO
• Select YESwhen asked (after solving LP) whether
  you want a RANGE ANALYSIS

17
Example 1 Winco Products 1

• Winco sells four types of products. The
  resources needed to produce one unit of each are
  known.
• Currently, 4600 units of raw material and 5000
  labor hours are available.
• 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.

18
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 950 x4 400
2x1 3x2 4x3 7x4 4600 3x1
4x2 5x3 6x4 5000
x1,x2,x3,x4 0
SOLVE by LINDO!
19
Ex. 1 Solution (contd)

• The LINDO output.
• For any nonbasic variable xk, the REDUCED COST is
  the amount by which the objective function
  coefficient of xk must be improved before the LP
  will have alternative optimal solutions at least
  one in which
• xk is a basic variable, and
• at least one in which
• xk is not a basic variable.

MAX 4 X1 6 X2 7 X3 8 X4 SUBJECT TO
2) X1 X2 X3 X4 950 3)
X4 gt 400 4) 2 X1 3 X2 4 X3
7 X4 lt 4600 5) 3 X1 4 X2 5 X3
6 X4 lt 5000 END LP OPTIMUM FOUND AT STEP
4 OBJECTIVE FUNCTION VALUE 1)
6650.000 VARIABLE VALUE
REDUCED COST X1 NBV
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
(Shadow p.) 2) 0.000000
3.000000 3) 0.000000
-2.000000 4) 0.000000
1.000000 5) s4 250.000000
0.000000 NO. ITERATIONS 4
250 hrs of labor unused
BVs x2, x3, x4, s4
20
Ex. 1 Solution (contd)

• LINDO SENSITIVITY ANALYSIS output
• ALLOWABLE RANGES for the obj. function
  coefficients of x1 (c1) and x2 (c2) are (with
  current basis remaining optimal)
• -? ? c1 ? 5 (41)
• 5.50 ? c2 ? 6.667
• Ex. Suppose Winco raises the price of product 2
  by 60 cent per unit. What is the new optimal
  solution?
• AI for c20.666667gt60 (current basis remains
  optimal x10, x2400, x3150, x4400)
• z-value (new)z-value (old) x2(.6)
  66502406890

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
21

• ALLOWABLE RANGES for the rhs of the first (b1)
  and second constraints (b2) are (with current
  basis remaining optimal)
• 850 ? b1 ? 1000
• 275 ? b2 ? 437.5

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
22
Ex. 1 Solution (contd)

• SHADOW PRICES are shown in the DUAL PRICES
  section of LINDO output.
• Row 2 Each additional unit of demand increases
  revenues by 3.
• Ex. Suppose 990 units must be produced (instead
  of 950). Determine the new optimal z-value?
• ?b1 990 950 40 (AI50)
• z-optimal (new)
• 665040(3) 6740

MAX 4 X1 6 X2 7 X3 8 X4 SUBJECT TO
2) X1 X2 X3 X4 950 3)
X4 gt 400 4) 2 X1 3 X2 4 X3
7 X4 lt 4600 5) 3 X1 4 X2 5 X3
6 X4 lt 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
23

Shadow Price Signs (For both max and min LPs)

• 1. Constraints with ³ symbols will always have
  nonpositive shadow prices.
• z-optimal decreases or stays the same
• 2. Constraints with will always have
  nonnegative shadow prices.
• z-optimal increases or stays the same
• 3. Equality constraints may have a positive, a
  negative, or a zero shadow price.

MAX 4 X1 6 X2 7 X3 8 X4 SUBJECT TO
2) X1 X2 X3 X4 950 3)
X4 gt 400 4) 2 X1 3 X2 4 X3
7 X4 lt 4600 5) 3 X1 4 X2 5 X3
6 X4 lt 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
24

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.
• 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 X4 SUBJECT TO
2) 2 X1 3 X2 X3 2 X4 lt 400
3) X1 X2 2 X3 X4 lt 150 4)
2 X1 X2 X3 0.5 X4 lt 200 5)
3 X1 X2 X4 lt 250
25

• 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) 700.0000 VARIABLE
VALUE REDUCED COST X1
50.000000 0.000000 X2
100.000000 0.000000 X3
0.000000 0.000000 X4
0.000000 1.500000 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
26
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 pay
  for an additional unit of a resource.

27
Example 5 Winco Products 2

• Reconsider the Winco problem.
• What is the most Winco should be willing to pay
  for additional units of raw material or labor?

28
Example 5 Solution

• 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 constraints (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.

MAX 4 X1 6 X2 7 X3 8 X4 SUBJECT TO
2) X1 X2 X3 X4 950 3)
X4 gt 400 4) 2 X1 3 X2 4 X3
7 X4 lt 4600 5) 3 X1 4 X2 5 X3
6 X4 lt 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