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.

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

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 the optimum objective value per unit increase in a RHS coefficient, all other data remaining equal.

• The shadow price is valid in an interval.

FAQ. Of course, I knew that. But can you please provide an example.

• Certainly. Let us recall the glass example given in the book. Let’s look at the objective function if we change the production time from 60 and keep all other values the same.

More changes in the RHS

FAQ. What is the intuition for the shadow price staying constant, and then changing? Recall from the simplex method that the simple

x method produces a “basic feasible solution.” The basis can often be described easily in terms of a brief verbal description.

The verbal description for the optimum basis for the glass problem:

1. Produce Juice Glassesand cocktail glasses only

2. Fully utilize productionand warehouse capacity

z = 5 x1 + 4.5 x2

6 x1 + 5 x2 = 60 10 x1 + 20 x2 = 150

x1 = 6 3/7

x2 = 4 2/7

z = 51 3/7

The verbal description for the optimum basis for the glass problem:

1. Produce Juice Glassesand cocktail glasses only

2. Fully utilize productionand warehouse capacity

z = 5 x1 + 4.5 x2

6 x1 + 5 x2 = 60 +Δ 10 x1 + 20 x2 = 150

x1 = 6 3/7 + 2Δ/7x2 = 4 2/7 – Δ/7z =51 3/7 + 11/14 Δ

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?

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

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.

FAQ. Right now, I’m new to this. But as I gain experience will interpretations of the shadow prices always be obvious?

No.

But they should become straightforward for examples given in 15.053.

FAQ. In the book, they sometimes use “dual price” and we use shadow price. Is there any difference?

No

FAQ. Excel gives a report known as the Sensitivity report. Does this provide shadow prices?

Yes, plus lots more.

In particular, it gives the range for which the shadow price is valid.

FAQ. I have heard that Excel occasionally gives incorrect shadow prices. Is this true?

There is the possibility that the interval in which the shadow price is valid is empty.

Excel can also give incorrect Shadow prices under certain circumstances that will not occur in spreadsheets for 15.053.

FAQ. You have told me that Excel sometimes makes mistakes. Also, I can do sensitivity analysis by solving an LP a large number of times, with varying data. So, what good is the Sensitivity Report?

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

For large problems it can be used to identify opportunities.

It can identify which coefficients are most sensitive to changes in value (their accuracy is the most important).

FAQ. Would you please summarize what 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 the RHS.

– Shadow prices usually but not always have economic interpretations that are managerially useful.

– 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

Overview of what is to come

Using insight from managerial situations to obtain properties of shadow prices

reduced costs and pricing out

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 ≥ 0The shadow price is the “increase” in the optimal value perunit 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 inoptimal objective value, then the shadow price is negative.

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 capacityconstraint cannot be negative

Reason: any feasible solution for this problem remainsfeasible after the production capacity increases. So, theincrease in production capacity cannot cause the optimumobjective value to go down.

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” constraintcannot be positive.Reason: Let x* be the solution if we replace the constraint“x1 ≥ 0” with the constraint “x1 ≥ 1”. Then x* is feasiblefor the original problem, and thus the original problem hasat least as high an objective value.

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.

Signs of Shadow Prices forminimization problems

The shadow price for a minimization problem is the “increase” in the objective function per unit increase in the RHS.

“ ≤ constraint” . The shadow price is … ?

“ ≥ constraint” . The shadow price is … ?

“ = constraint”. The shadow price could be zero or positive or negative.

Please answer with your partner.

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, onedoes not get a better solution by permitting x1 > 8.

FAQ. The shadow price is valid if only oneright hand side changes. What if multipleright hand side coefficients change?

The shadow prices are valid if multipleRHS coefficients change, but the rangesare no longer valid.

FAQ. Do the non-negativity constraintsalso have shadow prices?

Yes. They are very special and are calledreduced costs?

Look at the reduced costs for– Juice glasses reduced cost = 0– Cocktail glasses reduced cost = 0– Champagne glasses red. cost = -4/7

FAQ. Does Excel provide informationon the reduced costs?

Yes. They are also part of the sensitivityreport.

FAQ. What is the managerialinterpretation of a reduced cost?

There are two interpretations. Here is one of them.

We are currently not producing champagneglasses. 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.

FAQ. Why are they called the reducedcosts? Nothing appears to be “reduced”

That is a very astute question. Thereduced costs can be obtained by treatingthe shadow prices are real costs. Thisoperation is called “pricing out.”

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 ≤ 8x1 ≥ 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

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 ≤ 8x1 ≥ 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

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 ≤ 8x1 ≥ 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

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 ≤ 8x1 ≥ 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

FAQ. Can we use pricing out to figureout whether a new type of glass shouldbe 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 ≤ 150

1 x1 ≤ 8x1 ≥ 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

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

……y1

……y2

………

……ym

Reduced cost of xj = ?

Please complete withyour partner.

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).

Would you please summarize what wehave learned this lecture?

I’d be happy to.

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 thereduced cost.

Shadow prices usually but not always haveeconomic 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

The Financial Problem from Lecture 2

Sarah has $1.1 million to invest in five differentprojects for her firm.

Goal: maximize the amount of money that isavailable 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.

Return on investments

(undiscounted dollars)

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

The verbal description of theoptimum 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 asmuch 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

A graph for the financial Problem

• Any additional money in2002 is invested in A.

• Any additional money in2003 is invested in CD2003.

• Any additional money in2004 is invested in E.

Shadow Price Interpretation

Constraint: cash flow into2004 is all invested.

Shadow price: -1.2

Interpretation: an extra$1 in 2004 would beworth $1.20 in 2005.

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

Shadow Price Interpretation

Constraint: cash flow into2003 is all invested.

Shadow price: -1.26

Interpretation: an extra$1 in 2003 would beworth $1.26 in 2005.

Shadow Price Interpretation

Constraint: all $1.1 million isinvested in 2002.

Shadow price: -1.464

Interpretation: an extra$1 in 2002 would beworth $1.46 in 2005.

.4 x 1.05 x 1.2 + .8 x 1.2 = 1.464