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15.053 February 26, 2002

Sensitivity Analysis

presented as FAQs Points illustrated on a running example of

glass manufacturing.

If time permits, we will also consider the

financial example from Lecture 2.

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Glass Example

x1 = # of cases of 6-oz juice glasses (in 100s)

x2 = # of cases of 10-oz cocktail glasses (in 100s)

x3 = # of cases of champagne glasses (in 100s)

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs)

10 x1 + 20 x2 + 10 x3 150 (wareh. cap. in ft2)

x1 8 (6-0z. glass dem.)

x1 0, x2 0, x3 0

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FAQ. Could you please remind me what a

shadow price is?

Let us assume that we are maximizing.

A shadow price is the increase in theoptimum objective value per unit increase

in a RHS coefficient, all other data

remaining equal.

The shadow price is valid in an interval.

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FAQ. Of course, I knew that. But can you

please provide an example.

Certainly. Let us recall the glass example given in the

book. Lets look at the objective function if we change

the production time from 60 and keep all other valuesthe same.

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More changes in the RHS

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FAQ. What is the intuition for the shadow

price staying constant, and then changing? Recall from the simplex method that the

simplex method produces a basic

feasible solution. The basis can often bedescribed easily in terms of a brief verbal

description.

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The verbal description for the

optimum basis for the glass problem:

1. Produce Juice Glasses

and cocktail glasses only

2. Fully utilize production

and warehouse capacity

z = 5 x1 + 4.5 x26 x1 + 5 x2 = 60

10 x1 + 20 x2 = 150

x1 = 6 3/7

x2 = 4 2/7

z = 51 3/7

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The verbal description for the

optimum basis for the glass problem:

1. Produce Juice Glasses

and cocktail glasses only

2. Fully utilize production

and warehouse capacity

z = 5 x1 + 4.5 x26 x1 + 5 x2 = 60 +

10 x1 + 20 x2 = 150

x1

= 6 3/7 + 2/7

x2 = 4 2/7 /7

z =

51 3/7 + 11/14

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FAQ. How can shadow prices be used

for managerial interpretations?

Let me illustrate with the previous

example.

How much should you be willing to pay for

an extra hour of production?

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FAQ. Does the shadow price always

have an economic interpretation?

The answer is no, unless one wants

to really stretch what is meant by an

economic interpretation.

Consider ratio constraints

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Apartment Development

x1 = number of 1-bedroom apartments built

x2 = number of 2-bedroom apartments built

x3 = number of 3-bedroom apartments build

x1/(x1 + x2 + x3) .5 x1 .5x1 + .5x2 + .5x3

.5x1 5.x2 - .5x3 0

The shadow price is the impact of increasing

the 0 to a 1.

This has no obvious managerial interpretation.

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FAQ. Right now, Im new to this. But

as I gain experience will interpretationsof the shadow prices always be obvious?

No.

But they should become straightforward forexamples given in 15.053.

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FAQ. In the book, they sometimes use

dual price and we use shadow price.Is there any difference?

No

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FAQ. Excel gives a report known as

the Sensitivity report. Does thisprovide shadow prices?

Yes, plus lots more.

In particular, it gives the range for which

the shadow price is valid.

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FAQ. I have heard that Excel

occasionally gives incorrect shadowprices. Is this true?

There is the possibility that the interval in

which the shadow price is valid is empty.

Excel can also give incorrect Shadowprices under certain circumstances that

will not occur in spreadsheets for 15.053.

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FAQ. You have told me that Excel sometimesmakes mistakes. Also, I can do sensitivity

analysis by solving an LP a large number oftimes, with varying data. So, what good is theSensitivity Report?

For large problems it is much more efficient, and forLP models used in practice, it will be accurate.

For large problems it can be used to identifyopportunities.

It can identify which coefficients are most sensitiveto changes in value (their accuracy is the mostimportant).

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FAQ. Would you please summarizewhat we have learned so far.

Of course. Here it is. The shadow price is the unit change in the

optimal objective value per unit change in theRHS.

Shadow prices usually but not always haveeconomic interpretations that are manageriallyuseful.

Shadow prices are valid in an interval, which is

provided by the Excel Sensitivity Report. Excel provides correct shadow prices for our

LPs but can be incorrect in other situations

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Overview of what is to come

Using insight from managerial situations

to obtain properties of shadow prices

reduced costs and pricing out

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Illustration with the glass example:

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs)

10 x1 + 20 x2 + 10 x3 150 (wareh. cap. in ft2)

x1 8 (6-0z. glass dem.)

x1 0, x2 0, x3 0

The shadow price is the increase in the optimal value per

unit increase in the RHS.

If an increase in RHS coefficient leads to an increase inoptimal objective value, then the shadow price is positive.

If an increase in RHS coefficient leads to a decrease in

optimal objective value, then the shadow price is negative.

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Illustration with the glass example:

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs)

10 x1 + 20 x2 + 10 x3 150 (wareh. cap. in ft2)

x1 8 (6-0z. glass dem.)

x1 0, x2 0, x3 0

Claim: the shadow price of the production capacity

constraint cannot be negative

Reason: any feasible solution for this problem remains

feasible after the production capacity increases. So, the

increase in production capacity cannot cause the optimum

objective value to go down.

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Illustration with the glass example:

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs)

10 x1 + 20 x2 + 10 x3 150 (wareh. cap. in ft2)

x1 8 (6-0z. glass dem.)

x1 0, x2 0, x3 0

Claim: the shadow price of the x1 0 constraint

cannot be positive.

Reason: Let x* be the solution if we replace the constraint

x1 0 with the constraint x1 1. Then x* is feasible

for the original problem, and thus the original problem has

at least as high an objective value.

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Signs of Shadow Prices for

maximization problems

constraint . The shadow price is non-negative.

constraint . The shadow price is non-positive.

= constraint. The shadow price could be zero

or positive or negative.

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Signs of Shadow Prices forminimization problems

The shadow price for a minimization problem is theincrease in the objective function per unit increasein the RHS.

constraint . The shadow price is ?

constraint . The shadow price is ?

= constraint. The shadow price could be zeroor positive or negative.

Please answer with your partner.

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The shadow price of a non-binding constraint is 0.

This is known as Complementary Slackness.

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs)

10 x1 + 20 x2 + 10 x3 150 (wareh. cap. in ft2)

x1 8 (6-0z. glass dem.)

x1 0, x2 0, x3 0

In the optimum solution, x1 = 6 3/7.

Claim: The shadow price for the constraint x1 8 is zero.

Intuitive Reason: If your optimum solution has x1 < 8, one

does not get a better solution by permitting x1 > 8.

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FAQ. The shadow price is valid if only one

right hand side changes. What if multipleright hand side coefficients change?

The shadow prices are valid if multipleRHS coefficients change, but the ranges

are no longer valid.

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FAQ. Do the non-negativity constraints

also have shadow prices?

Yes. They are very special and are called

reduced costs?

Look at the reduced costs for Juice glasses reduced cost = 0

Cocktail glasses reduced cost = 0 Champagne glasses red. cost = -4/7

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FAQ. Does Excel provide information

on the reduced costs?

Yes. They are also part of the sensitivity

report.

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FAQ. What is the managerialinterpretation of a reduced cost?

There are two interpretations. Here is one of them.

We are currently not producing champagne

glasses. How much would the profit of champagneglasses need to go up for us to producechampagne glasses in an optimum solution?

The reduced cost for champagne classes is 4/7. Ifwe increase the revenue for these glasses by 4/7(from 6 to 6 4/7), then there will be an alternativeoptimum in which champagne glasses areproduced.

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FAQ. Why are they called the reduced

costs? Nothing appears to be reduced

That is a very astute question. The

reduced costs can be obtained by treating

the shadow prices are real costs. Thisoperation is called pricing out.

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Pricing Out

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60

10 x1 + 20 x2 + 10 x3 150

1 x1 8

x1 0, x2 0, x3 0

Pricing out treats shadow prices as

though they are real prices. The

result is the reduced costs.

shadow price

11/14

1/35

.0

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Pricing Out of x1

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60

10 x1 + 20 x2 + 10 x3 150

1 x1 8

x1 0, x2 0, x3 0

shadow price

11/14

1/35

.0

Reduced cost of x1 =5

- 6 x 11/14

- 10 x 1/35

- 1 x 0

= 5 33/7 2/7 = 0

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Pricing Out of x2

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60

10 x1 + 20 x2 + 10 x3 150

1 x1 8

x1 0, x2 0, x3 0

shadow price

11/14

1/35

.0

Reduced cost of x2 =4.5

- 5 x 11/14

- 20 x 1/35- 0 x 0

= 4.5 55/14 4/7 = 0

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Pricing Out of x3

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60

10 x1 + 20 x2 + 10 x3 150

1 x1 8

x1 0, x2 0, x3 0

shadow price

11/14

1/35

.0

Reduced cost of x3 =6

- 8 x 11/14

- 10 x 1/35- 0 x 0

= 6 44/7 2/7 = -4/7

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FAQ. Can we use pricing out to figure

out whether a new type of glass should

be produced?

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60

10 x1 + 20 x2 + 10 x3

1501 x1 8

x1 0, x2 0, x3 0

shadow price

11/14

1/35

.0

Reduced cost of x4 =

7

- 8 x 11/14- 20 x 1/35

- 0 x 0

= 7 44/7 4/7 = 1/7

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Pricing Out of xj

max 5 x1 + 4.5 x2 + cj xj ($100s)

s.t 6 x1 + 5 x2 + a1j xj 60

10 x1 + 20 x2 + a2j xj 150

... + amjxj = bm

x1 0, x2 0, x3 0

shadow price

y1y2

ym

Reduced cost of xj = ?

Please complete with

your partner.

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Brief summary on reduced costs

The reduced cost of a non-basic variable xj is the

increase in the objective value of requiring thatxj >= 1.

The reduced cost of a basic variable is 0. The reduced cost can be computed by treating

shadow prices as real prices. This operation isknown as pricing out.

Pricing out can determine if a new variable wouldbe of value (and would enter the basis).

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Would you please summarize what we

have learned this lecture?

Id be happy to.

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Summary

The shadow price is the unit change in the optimalobjective value per unit change in the RHS. The shadow price for a 0 constraint is called the

reduced cost. Shadow prices usually but not always have

economic interpretations that are manageriallyuseful.

Non-binding constraints have a shadow price of 0. The sign of a shadow price can often be determined

by using the economic interpretation Shadow prices are valid in an interval, which is

provided by the Excel Sensitivity Report. Reduced costs can be determined by pricing out

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The Financial Problem from Lecture 2

Sarah has $1.1 million to invest in five different

projects for her firm.

Goal: maximize the amount of money that is

available at the beginning of 2005. (Returns on investments are on the next slide).

At most $500,000 in any investment

Can invest in CDs, at 5% per year.

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Return on investments

(undiscounted dollars)

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The LP formulation

Max .8 xB + 1.5 xD + 1.2 xE + 1.05 xCD04

s.t. -xA xC xD xCD02 = -1.1

.4 xA

xB

+ 1.2 xD

+ 1.05 xCD02

xCD03

= 0

.8 xA + .4 xB - xE + 1.05 xCD03 xCD04 = 0

.8 xA + .4 xB - xE + 1.05 xCD03 xCD04 = 0

0 xj .5 for j = A, B, C, D, E, CD02

CD03, and CD04

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The verbal description of the

optimum basis

1. Invest as much as possible in C and D in2002. Invest the remainder in A.

2. Take the returns in 2003 and invest as

much as possible in B. Invest theremainder in CDs

3. Take all returns in 2004 and invest themin E.

Note: if an extra dollar became available inyears 2002 or 2003 or 2004, we wouldinvest it in A or 2003CDs or E

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A graph for the financial Problem

Any additional money in2002 is invested in A.

Any additional money in

2003 is invested in CD2003.

Any additional money in

2004 is invested in E.

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Shadow Price Interpretation

Constraint: cash flow into

2004 is all invested.

Shadow price: -1.2

Interpretation: an extra

$1 in 2004 would be

worth $1.20 in 2005.

.8 xA + .4 xB - xE + 1.05 xCD03 xCD04 = 0

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Shadow Price Interpretation

Constraint: cash flow into

2003 is all invested.

Shadow price: -1.26

Interpretation: an extra

$1 in 2003 would be

worth $1.26 in 2005.

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Shadow Price Interpretation

Constraint: all $1.1 million isinvested in 2002.

Shadow price: -1.464

Interpretation: an extra

$1 in 2002 would be

worth $1.46 in 2005.