01e Sensitivity Analysis and Duality

7/25/2019 01e Sensitivity Analysis and Duality

1/42

Sensitivity Analysis and Duality

1Sasadhar Bera, IIM Ranchi


2/42

Standard form of Linear Programming (LP)


Objective function = ZMin= Minimum (c1 x1+ c2x2+ . . . . +cn xn)

subject to

a11x1 + a12 x2+ . . . + a1n xn = b1

a21x1 + a22 x2 + . . . + a2n xn = b2

. . . . . . .

. . . . . . .

am1x1 + am2 x2 + . . . + amn xn= bmx1, x2, x3, .., xn0

Notation: c1, c2, . . .,cn are cost coefficients.

b1, b2, . . .,bm are available resources.

a11, a12, . . ., amn are technological coefficients.

In matrix notation:

ZMin= C1nXn1

subject to

AmnXn1= bm1

Xn1 0

Minimization Problem


3/42

Revisiting Product Mix Problem

3

A company wishes to schedule the production of a kitchen

appliance that requires two resources, labor and material.The company is considering 3 models (A, B, and C) and its

production engineering department has furnished the

data given below. Formulate the following problem and

solve.

Model

Resource

Resource

requirement

Availability

A B C

Labour (Hrs/Unit)

7

3

6

150 Hrs

Material (Kg/Unit)

4

4

5

200 Kg

Profit (Rs. /Unit) 4 2 3

Sasadhar Bera, IIM Ranchi


4/42

Revisiting Product Mix Problem (contd.)

4

Decision Variables

X1 = Number of A type model producedX2 = Number of B type model produced

X3 = Number of C type model produced

Objective function: Total profit maximization (ZMAX.)

ZMax.= 4X1+2X2+3X3

Subject to

7X1+3X

2+6X

3150

4X1+4X2+5X3200

X1, X2, X30

Objective function

Labour constraint

Material constraint

Boundary Constraint



5/42

Revisiting Product Mix Problem (contd.)

5

Primal Problem:

ZMax.= 4 X1+2X2+3X3

Subject to

7 X1+3 X2+6 X3 150

4 X1+4 X2+5X3 200

X1, X2, X30

X1, X2, X3 are decision variables


RHS of a constraint

oravailable resources

Profit coefficients

Technological coefficients


6/42

Standard Form of Product Mix Problem (contd.)

6

Objective function: Total profit maximization (ZMAX.)

ZMax.= 4X1+2X2+3X3

Subject to

7X1+3X2+6X3150

4X1+4X2+5X3200

X1, X2, X30

Standard form of above LP:

ZMax= 4x1+ 2x2+ 3x3

subject to

7x1 + 3x2 + 6x3 + s1 = 1504x1 + 4x2+ 5x3 + s2= 200

x1, x2, x3, s1, s2 0

s1,s2 are called slack variables



7/42

MS Excel Output


Target Cell (Max)

Cell Name Original Value Final Value

$D$7 Total Profit 12 100

Adjustable Cells

Cell Name Original Value Final Value

$E$5 Nos of Production A 1 0

$F$5 Nos of Production B 1 50

$G$5 Nos of Production C 2 0

Constraints

Cell Name Cell Value Formula Status Slack

$I$10 Labour Constraint 150 $I$10


8/42

MS Excel Output (contd.)


For first constraint:

7X1+3X2+6X3= 7*0 + 3*50 + 6*0 = 150 = RHS value of firstconstraint. Hence total labour resource is fully utilized. It is

called binding constraint.

For second constraint:4X1+4X2+5X3= 4*0 + 4*50 + 5*0 = 200 = RHS value of second

constraint. Hence total raw material is fully utilized. It is also a

binding constraint.

In case of nonbinding constraint LHS value is not equal to RHS

value.


9/42

MS Excel Output


Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$E$5 Nos of Production A 0 -0.667 4 0.667 Infinity

$F$5 Nos of Production B 50 0 2 Infinity 0.286

$G$5 Nos of Production C 0 -1 3 1 Infinity

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$I$10 Labour Constraint 150 0.667 150 0 150

$I$11 Material Constraint 200 0 200 infinity 0

Sensitivity Analysis Output


10/42

MS Excel Output Illustration


Reduced cost: The reduced cost indicates how much each

objective function coefficient has to improve (increase for

maximization problem and decrease for minimization

problem) before the corresponding decision variable could

assume apositive value in optimal solution.

Physical interpretation of reduced cost: The reduced cost

for each variable (here each product) equals its per unit

marginal profit minus the per unit cost of the resources it

consumes.

Increasing or decreasing the objective function coefficient of

a decision variable equal to reduced cost has resulted an

alternative solution.

What is Reduced Cost?


11/42



The sensitivity analysis output table shows that the Final

Value of X2 already has positive value. Thus the reduced

cost is zero.

What is Reduced Cost? (contd.)

Adjustable Cells

Final Reduced Objective Allowable AllowableCell Name Value Cost Coefficient Increase Decrease





12/42



In case of X1 and X3, FinalValueare zero. Thus the reduced

cost is non-zero.

For X1, it means that unless the profit contribution (c1) of A

type of model is increased to (4+0.667 =) 4.667 or more, the

value of X1will not come as nonzero in optimal solution. If c1is exactly increased to 4.667, then there will have an


Similarly, for X3, it means that unless the profit contribution(c3) of C type of model is increased to (3+1 =) 4 or more,

the value of X3will not come as nonzero in optimal solution.

If c3 is exactly increased to 4, then there will have an


What is Reduced Cost? (contd.)


13/42



Range of Optimality

The range of optimality is calculated using Allowable

Increase and Allowable Decrease columns in AdjustmentCellsof sensitivity output table.

Range of optimality: The range of value for each coefficient

of an objective function over which the solution will remain

optimal (i.e. optimal values of the decision variableswould

not change).

100% rule: There may be simultaneous change of more than

one objective function coefficients. If the sum of theabsolute percent change (with respect to allowable change)

of all the coefficients does not exceed 100%, then the

original optimal solution was still be optimal. If it changes by

more than 100%, we cannot be sure.


14/42



Range of Optimality (contd.)

For above output table, optimum value for X1 (Model type A)

is 0, the objective coefficient is 4, allowable increase is

0.667, and allowable decrease is . Hence the range ofoptimality of c1is: c1(4+0.667)

Similarly, the range of optimality for c2(model type B) is:

(2-0.286) c2 +

Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease





15/42



Range of Optimality (contd.)

Similarly, the range of optimality for c3(model type C) is:

c3 (3+1)

Range of insignificance: The range in value over which an

objective function coefficient can change without causing

the corresponding decision variable to take a nonzero value.


16/42


16

Shadow Price: The shadow price of a constraint indicates

the change in the optimal value of the objective function

when the right hand side (RHS) of the same constraint

changes by one unit, assuming all other coefficients

remain constant. Shadow price may be positive or

negative.

Range of Feasibility: The range of feasibility for RHS of a

constraint is the range for which the shadow priceremains unchanged for that particular constraint.


Shadow Price and Range of Feasibility


17/42



Shadow Price (contd.)

The FinalValuecolumn represents the Final LHS of labour

and material constraint each separately.

The Shadow Price column provides the shadow price of

each constraint.

The Constraint R.H. Side column provides available

resources for labour and material.

Constraints






18/42


18

For first constraint, shadow price is +0.667, which indicates

that if we decreases number of labour (b1) from 150 to 149

then the objective function value (profit) decreases to 100

to 99.333. (i.e. 100 -1*(+0.667) = 99.333)

For second constraint, shadow price is 0, which indicates

that if number of material unit (b2) increases from 200 to

201 then there will be no change in the objective function

value (profit).




19/42



Some software (LINGO) provides dual price which is used todescribe the shadow price. Shadow price and Dual price are

same in sign for maximization problem. In case of

minimization problem, Shadow price and Dual price are in

opposite sign.

It is to be noted that shadow price of a non-binding

constraintis zero.



20/42



The range of feasibility is calculated using Allowable Increaseand Allowable Decrease columns of above sensitivity output

table.

For labour constraint, range of feasibility:

Lower bound = 150 - 150 = 0, Upper bound = 150 + 0 = 1500 b1 150

For material constraint, range of feasibility:

Lower bound = 2000 = 200, Upper bound= Infinity

200 b2 +

Range of Feasibility

Constraints






21/42

Sensitivity Analysis


Sensitivity analysis is the study of how the optimal solution

will be impacted with the changes in the different

coefficients of a linear program. Using the sensitivityanalysis we can answer how optimal solution affect under

the following conditions:

1. Change in the objective function coefficient (ci

)

2. Change in resources (RHS of a constraint) (bi)

3. Change in technological coefficients (aij)

4. Addition of a new decision variable5. Addition of a new constraint

Sensitivity analysis is also referred to as postoptimality

analysis.


22/42

Why Sensitivity Analysis is Important?


1) Values of LP parameters might change

If a parameter changes, sensitivity analysis shows

whether is it necessary to re-solve the problem

again?

2) Uncertainty about LP parameters

Even if demand is uncertain, manager of a

company can be fairly confident that it can still

produce optimal amounts of products.


23/42



Range of optimality of each coefficients of an objective

function provides us the sensitivity of objective function

coefficient.

Values in the AllowableIncreaseand AllowableDecrease

columns in adjustment Cells of sensitivity analysis reportindicate the amounts by which an objective function

coefficient can be changed without changing the optimal

solution, assuming all other coefficients remain constant.

It is to be noted that the objective function value (Z) would

change due to change in profit (or cost ) coefficient within

the range of optimality.

Change in only one coefficient (ci) of a objective function


24/42



You would like to know what would happen to your optimal

solution when multiple profit (or cost) coefficients are

different than what you expected.

In the above situation, 100% rule is applicable. This rule says

that if the sum of the absolute percent change (with respect

to allowable increase or decease) of all the coefficients does

not exceed 100%, then the original optimal solution will still

be optimal. If it changes by more than 100%, we cannot be

sure.

Change in more than one coefficient (ci) simultaneously


25/42



The percentage change for a coefficient value can be

calculated as:

Absolute[(New Value Old Value) / Allowable increase or

decrease]

For example: when coefficient value 300 changes to 600

and the allowable increase is 900 you get a proportional

change of (600-300)/900 which equals approximately

33.33%.

Calculating a Percentage Change


26/42



The RHS value of a constraint represents the resources

available to the firm. The resources could be labour hours,

machine hours, money, and material etc.

Sensitivity analysis of these resources help to answer howadditional resources could be used to realize higher profit.

If the RHS of a constraint is changed, the feasible region

will change (unless the constraint is redundant) and oftenoptimal solution changes.

Change in resources (RHS of a constraint) (bi)


27/42



Shadow prices (or dual price) only indicates the change

that occur in the objective function value that results from

one unit change in RHS value of a constraint.

Shadow prices for nonbinding constraints are always

zero.

Changing a RHS value for a binding constraint also

changes the feasible region and the optimal solution.

To find the optimal solution after changing a binding

RHS value, we must re-solve the problem.

Change in resources (RHS of a constraint) (bi) (contd.)


28/42



Technological coefficients reflect changes in coefficients in

the LHS of a constraint because of labour, raw material andtechnology etc. Changes of technological coefficients can

significantly changes the shape of the feasible region and

hence in optimal profit or cost value (Z) .

The changes of technological coefficient can be two types:

1) Change of aij coefficient in nonbasic columns

2) Change of aij coefficient in basic columns

Refer book Operations Research by H. M. Taha for above two

types of sensitivity analysis.

Change in Technological Coefficient (aij)


29/42



After addition of a column vector with a new decision

variable (Xn+1) we have to calculate the (zjcj) in 0throw for

(n+1)th variable. If (zj cj) 0 for minimization problem

then the current solution is optimal. On the other hand, if

(zj cj) 0 then Xn+1 is introduced into the basis and the

simplex method continues to find the new optimalsolution.

For understanding of (zjcj) value and 0throw refer to Operations

Research by H. M. Taha.

Adding a new decision variable or activity

l


30/42



If the optimal solutions satisfy the new constraint then

current optimal solution is still be best solution.

If the optimal solutions does not satisfy the new constraint

then dual simplex method is used to find the new optimalsolution to an LP with added constraint.

For dual simplex method refer Operations Research by H. M.

Taha.

Adding a new constraint

i i i l i i i f l


31/42

Sensitivity Analysis using Microsoft Solver


Microsoft solvers sensitivity analysis report performs

two types of sensitivity analysis:

i. On the coefficient of the objective function

ii. On the right hand side of a constraint


32/42

Duality


Every linear programming (LP) problem can have two

forms:

1) Primal

2) Dual

The original formulation of a linear programming problem

is called Primal or Primal Problem.

Another linear program associated with Primal is called its

dual which is involving a different set of variables, butsharing the same data.

P i l d D l


33/42

Primal and Dual


Primal: ZMax= c1 x1+ c2 x2+ . . .+cn xn

subject to

a11x1 + a12 x2+ . . . + a1n xn b1a21x1 + a22 x2 + . . . + a2n xn b2

. . . . . . .

am1x1 + am2 x2 + . . . + amn xnbm

x1, x2, .., xn0

Dual: YMin= w1 b1+ w2 b2+ . . .+ wm bm

subject to

a11w1 + a21 w2+ . . . + am1 wm c1

a12

w1

+ a22

w2

+ . . . + am2

wm

c2

. . . . . . .

a1nw1 + a2n w2 + . . . + amn wncn

w1, w2, .., wm0

wi indicates price paid for per unit of ithresource

Profit per unit

Resources

Dual variable

E i I i f D l


34/42

Economic Interpretation of Dual


Primal objective function: Maximization of profit subject to

availability of limited resources.

Dual objective function: Minimization of the total implicit

value of resources consumed by the different activities.

xj: Quantity of jthtype product, j = 1, 2, . .,n.

wi: Price paid for per unit of ithresource, i = 1, 2, . .,m.

The dual variables are interpreted as thecontribution to profit

per unit of resource. For this reason, dual variables are often

referred to as resource shadow prices.

Dual variables are used to determine the marginal values of

resources i. e. how much profit for one unit of each resource is

equivalent to.


35/42

Economic Interpretation of Dual (contd.)


Meaning of primal constraint:

(ai1 x1 + ai2 x2 + . . . + ain xn) bi represents totalconsumed resources should be at most available resource,

where i = 1, 2, . .,m.

Meaning of dual constraint:

(a1j w1 + a2j w2 + . . . + amj wm) cj represents the

minimum cj unit profit should be paid for the resources

needed to produce the jth type of product.


36/42

Formulation of Dual


If Primal is Minimization problem then dual is Maximization

Problem and vice versa.

There is exactly one dual variable for each primal constraint

and exactly one dual constraint for each primal variable.

Primal-dual relationship table (shown in next slide) is useful

to put dual variable restriction and inequality sign (,or = )

in each constraint.


37/42

Primal and Dual Relationship


Minimization

Problem

Maximization

Problem

Variables

0

Constraints0

Unrestricted =

Constraints

0

Variables 0

=

Unrestricted


38/42

Formulation of Dual (contd.)


Ex1:

Primal: Maximize 6x1 + 8x2

subject to 3x1+ x2 4

5x1+ 2x27

x1, x20

Dual: Minimize 4w1+ 7w2

subject to 3w1 + 5w2 6

w1 + 2w2 8w1, w20


39/42

Formulation of Dual(contd.)


EX2:

Primal: Minimize 6x1+ 8x2

subject to 3x1 + x2 - x3 = 4

5x1 + 2x2 - x4 = 7

x1, x2, x3, x4 0

Dual: Maximize 4w1+ 7w2

subject to 3w1 + 5w26

w1 + 2w28- w1 0

- w2 0

w1, w2 unrestricted

Formulation of Dual


40/42

Formulation of Dual (contd.)


EX3:

Primal: Minimize -12 x1+ 13 x2+ 15 x3

subject to-2x1 + x2 + 3x3 + x4 15

2x1 + x3 +3x4 14

+2x2 + x3 + x4 = 16

x10 x

2, x

30, x

4unrestricted

Dual: Maximize 15 w1+ 14 w2+ 16 w3

subject to

-2w1 + 2w2 -12

w1 + 2w3 13

3w1 + w2 + w3 15

w1 + 3w2 + w3 = 0

w10 w20, w3unrestricted

Ad t f D l Li P i


41/42

Advantages of Dual Linear Programming


The optimal value of the objective function of the primal

problem equals the optimal value of the objective function

of the dual problem.

Solving the dual might be computationally more efficient

when the primal has large number of constraints and few

variables.

EX: Let us consider a Primal Linear Programming problem

has 8 variables and 800 constraints. Maximum number of

iterations required to solve Primal is high as the number of

constraints is large. In such situation, Dual LP converts thesame optimization problem with 800 variables and 8

constraints. Hence, computational time is less for dual LP as

it has fewer constraints.


42/42

Advantages of Dual LP (contd.)

Lets consider a situation where an analyst has overlooked

a constraint in the model during problem formulationstage. In the solution development stage, analyst wants to

incorporate this new constraint. In such instances, it is

sometimes difficult to find a starting basic solution that is

feasible to linear programming. Using dual LP, it is oftenpossible to find out optimal solution.

Documents

01e Sensitivity Analysis and Duality