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Stochastic Programming Modeling

IMA New Directions Short Course on Mathematical Optimization

Jeff Linderoth

Department of Industrial and Systems EngineeringUniversity of Wisconsin-Madison

August 8, 2016

Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 1 / 77

Week #2

The first week focused on theoryand algorithms for continuousoptimization problems whereproblem parameters are knownwith certainty.

This week we will focus on twodifferent topics:

1 Stochastic Programming:Used for Optimization underdata uncertainty

2 Integer Programming: Usedfor modeling discrete decisions


Today’s Outline

About This Week

About Us

About You

Stochastic Programming

What is it?/Why Should we Do it?

A Newsvendor

Recourse Models and Extensive Form

How to implement in a modeling language


This Week

Resources

Our exercises will be done with AMPL: A Mathematical ProgrammingLanguage

We added you all to a Dropbox: There you can get AMPL, templatesfor the exercises, and the lecture slides.


This Week

The Dream Team


This Week

Optimization “Dream Team”

Monday: Dave Morton, Northwestern, Sample AverageApproximation

Tuesday: Shabbir Ahmed, Georgia Tech, Multistage StochasticProgramming

Wednesday: Robert Hildebrand, IBM, Lenstra’s Algorithm

Thursday: Santanu Dey, Georgia Tech, Cutting Plane Theory

Friday: Dan Bienstock, Columbia, Mixed Integer NonlinearProgramming


This Week

Week Overview—Social Events!

Monday: Stub and Herb’s

Wednesday: Twins Game

Thursday: Surly Brewing Company


This Week

Recommended Texts


?: Very good. Requires strong math background

?: A more gentle introduction, but still covers the whole field quitewell.

?: FREE!. It’s in the Dropbox

Integer Programming

?: Classic reference.

?: A more gentle treatment

?: Very nice geometric intuition

?: My (new) favorite book


This Week

Course Level/Expectations

We will use AMPL (www.ampl.com) to solve problems and prototypealgorithms:

If nothing else, you can get to learn a new language for modeling andsolving mathematical optimization problems

We will do a few proofs, but we will not require significantmathematical sophistication beyond a reasonable understanding of LPduality

We assume some basic background in probability theory (no measuretheory required) – what is a random variable, expected value, law oflarge numbers, some basic statistics (CLT)

We will expect some basic linear algebra knowledge


www.ampl.com

This Week

About us...

B.S. (G.E.), UIUC, 1992.

M.S., OR, GA Tech, 1994.

Ph.D., GA Tech, 1998

1998-2000 : MCS, ANL

2000-2002 : Axioma, Inc.

2002-2007 : Lehigh University

Research Areas: Large ScaleOptimization, High PerformanceComputing.

Married. One child, Jacob. Now 13.He is awesome.

Hobbies: Golf, Integer Programming,Human Pyramids.


This Week

About Jim...

B.S. (I.E.), UW-Madison, 2001

M.S., OR, GA Tech, 2004.

Ph.D., GA Tech, 2007

2007-2008 : IBM

2008-2016 : UW-Madison

Research Areas: DiscreteOptimization, StochasticOptimization, Applications

Married. Three children: Rowan,Camerson, Remy. They are awesome

Hobbies: Boxing, IntegerProgramming, Human Pyramids.


This Week

Picture Time


This Week

About You – Quiz #1!

1 Name

2 Nationality

3 Education Background.

4 Research Interests/Thesis Topic?

5 (Optimization) Modeling Languages you know: (AMPL, GAMS,Mosel, CVX, . . .

6 Programming Languages you know: (C, Python, Matlab, Julia,FORTRAN, Java, . . .)

7 Anything specific you hope to accomplish/learn this week?

8 One interesting fact about yourself you think we should know.

9 Do you like human pyramids? :-)


Introduction to SP Background


$64 Question

What does “Programming” mean in “MathematicalProgramming”, “Linear Programming”, etc...?

A. Planning.

Mathematical Programming (Optimization) is about decision making,or planning.

Stochastic Programming is about decision making under uncertainty.

View it as “Mathematical Programming with random parameters”


Introduction to SP Background

Dealing With Randomness

In most applications of optimization, randomness is ignored

Otherwise, it is dealt with via:

Sensitivity analysis

For large-scale problems, sensitivity analysis is useless

“Careful” determination of instance parameters

No matter how careful you are, you can’t get rid of inherentrandomness.

Stochastic Programming is the way!1

1This is not necessarily true, but we will assume it to be so for the next two daysJeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 15 / 77

Introduction to SP Newsvendor

Hot Off the Presses

A paperboy (newsvendor) needs to decide how many papers to buy inorder to maximize his profit.

He doesn’t know at the beginning of the day how many papers he cansell (his demand).

Each newspaper costs c.He can sell each newspaper for a price of s.He can return each unsold newspaper at the end of the day for r.(Note that s > c > r).The demand (unknown when we purchase papers) is D

Newsvendor Profit

F (x,D) =

(s− c)x if x ≤ DsD + r(x−D)− cx if x > D



Pictures of Function

Marginal profit: (s− c) if can sell all: x ≤ DMarginal loss: (c− r) if have to salvage

xx = D

F (x,D)



What Should We Do?

Optimize, silly:maxx≥0

F (x,D).

http://en.wikipedia.org/wiki/

Chewbacca_defense

This problem does not makesense!

You can’t optimize somethingrandom!


http://en.wikipedia.org/wiki/Chewbacca_defense

http://en.wikipedia.org/wiki/Chewbacca_defense


The Function is “Random”

xx = D1

F (x,D1)

xx = D2

F (x,D2)

One x can’t simultaneously optimize both functions



(Silly) Idea #1

Suppose D is a random variable with cdf H(t)def= P(D ≤ t)

“Silly” Idea: Plan for Average Case

Let µdef= E[D] be the mean value of demand

In this case: (proof by picture)

maxx≥0

F (x, µ)⇒ x∗ = µ.

In this case, the optimal policy is to purchase µ

We will see that this can be far from optimal when your problemtakes more uncertainty into account



Idea #2 – RobustPlan for Worst Case

Suppose D ∈ [`, u], and we wish to do the best we can given thatthe worst outcome for our objective will occur:

maxx≥0

minD∈[`,u]

F (x,D)

Note that we can write:

F (x,D) = min(s− c)x,D(s− r) + (r − c)x

maxx≥0

minD∈[`,u]

F (x,D) = maxx≥0

minD∈[`,u]

min(s− c)x,D(s− r) + (r − c)x

= maxx≥0

min(s− c)x, `(s− r) + (r − c)x

= maxx≥0

F (x, `)⇒ x∗ = `

Robust optimization – say some nice things. But we will not cover indetail this week. (Give reference?)



Idea #3: Maximize Long-Run Profit

The “best” idea

Treat F (x,D) as a proper random variable, and maximize long-runprofit.

i.e. solve the optimization problem:

maxx≥0

E[F (x,D)].

In this case, the objective may make sense. The newsvendor will makea purchase every day



Optimizing for the Newsvendor

Given only knowledge of the random variable D, given as the cdfHD(t), how many newspapers should the newsvendor buy?

With some old-school calculus (Chain rule, Fundmental theorem ofcalculus), one can show that the optimal closed form solution to theNewsvendor problem is

x∗ = H−1(s− cs− r

)the (s− c)/(s− r) quantile of the distribution H

It Ain’t Always “That Easy”

The newsvendor is about the only stochastic program that admitssuch a simple “closed form” solution.

In general, we must solve instances numerically (and alsoapproximately)



Simulating (with Scenarios)

newsboy.xls

s = 2, c = 0.3, r = 0.05

Demand: Normally distributed. µ = 100, σ = 20

Mean Value Solution

Buy 100. (Duh!)TRUE long run profit ≈ 154

Stochastic Solution

Buy 123TRUE long run profit ≈ 162

The difference between the two solutions (162− 154) is called thevalue of the stochastic solution.



Do You Feel Lucky, Punk?

Should we always optimize the randomvariable F (x,D) in expectation?

We may be “risk-averse”

minx≥0

ρ[F (x,D)]

If ρ(a) = E[a]: Standard stochasticprogram

If ρ(a) = E[a] + λV(a) for λ ∈ R, wehave a “mean-variance” stochasticprogram

Risk measures are discussed in thesecond lecture



Another Possible Newsvendor Problem

Suppose the newsvendor is lazy. He just wants to usually makeenough money to go to Stub and Herb’s, but he doesn’t want to hurthis back carrying too may papers

Chance Constraints

minx≥0x | PF (x,D) ≥ b ≥ 1− α

Minimize the number of papers to purchase to ensure that theprobability that you make at least b in profit is at least 1− αNote that F (x,D) is a random variable

Jim will discuss this a bit as well



Take Away Message

The “Flaw” of Averages

The flaw of averages occurs whenuncertainties are replaced by “singleaverage numbers” planning.

Joke: Did you hear the one aboutthe statistician who drowned fordinga river with an average depth ofthree feet.



Take-away Message: Point Estimates

If you are planning using point estimates, then you are planningsub-optimally

It doesn’t matter how carefully you choose the point estimate— it isimpossible to hedge against future uncertainty by considering onerealization of the uncertainty in your planning process


Stages

Stages and Decisions

The newsvendor problem is a classical “recourse problem”:

1 We make a decision now (first-period decision)

2 Nature makes a random decision (“stuff” happens)

3 We make a second period decision that attempts to repair the havocwrought by nature in (2). (recourse)

Key Idea

The evolution of information is of paramount importance


Stages

Newsvendor Again

Newsvendor Profit

F (x,D) = min(s− c)x, (s+ r)D + (r − c)x

D a random variable with cdf HD(t)

We showed that

x∗ = H−1(s− cs− r

).

Suppose thatΩ = d1, d2, . . . d|S|

So there are a finite set of scenarios S, each with associatedprobability pj . (

∑j∈S pj = 1)


Stages

Newsvendor SP

Parameters

ds: Demand for newspapers in scenario s

ps: Probability of scenario s

Writing an optimization model for the newsvendor

Variables

x: Number to purchase

ys: Number to sell in scenario s

zs: Number to salvage in scenario s


Stages

Newsvendor Stochastic LP

max−cx+∑s∈S

ps(qys + rzs)

s.t.

ys ≤ ds ∀s ∈ Sx− ys − zs = 0 ∀s ∈ S

x ≥ 0

ys, zs ≥ 0 ∀s ∈ S


Stages

Put Another Way

We could write the objective for the newsvendor problem in the form:

F (x,D) = −cx+ EQ(x,D),

where

Q(x,D) = maxy≥0,z≥0

qy + rz | y ≤ D, y + z = x.

Q(x,D) is the optimal recourse function: Given that we have chosenx and observed demand D, what should I do to maximize profit?


Stages

It’s Not Always So Easy

For the newsvendor the recourse function: Q(x,D) has a simpleclosed form:

Q(x,D) = minsx, sD + r(x−D)

In general the recourse function may not be simple

In fact, for two-stage stochastic linear programs, the recourse functionwill be the optimal value of a linear program


Stages

Scenario Modeling

The most common representation of uncertainty (in stochasticprogramming) is via a list of scenarios, which are specificrepresentations of how the future will unfold.

Think of these as random variables ξ1, ξ2, . . . ξS , with ξj ∈ Ξ

What we CAN’T do

Planners often generate a solution for each scenariogenerated—“What-if” analysis.

Each solution yields a prescription of what should be done if thescenario occurs, but there is no theoretical guidance about thecompromise between those prescriptions

Can we “combine” these prescriptions in a natural way?Stochastic Programming does this!


Farmer Ted Background

Farmer Ted

In this example, the farmer has recourse that is, he can do somethingat step (3). Not just sell his newspapers.

Farmer Ted can grow Wheat, Corn, or Beans on his 500 acres.

Farmer Ted requires 200 tons of wheat and 240 tons of corn to feedhis cattle

These can be grown on his land or bought from a wholesaler.



More Constraints

Any excess production can be sold for $170/ton (wheat) and$150/ton (corn)

Any shortfall must be bought from the wholesaler at a cost of$238/ton (wheat) and $210/ton (corn).

Farmer Ted can also grow beans

Beans sell at $36/ton for the first 6000 tonsDue to economic quotas on bean production, beans in excess of 6000tons can only be sold at $10/ton



The Data

500 acres available for planting

Wheat Corn Beans

Yield (T/acre) 2.5 3 20Planting Cost ($/acre) 150 230 260

Selling Price 170 150 36 (≤ 6000T)10 (>6000T)

Purchase Price 238 210 N/AMinimum Requirement 200 240 N/A



Formulate the LP – Decision Variables

xW,C,B Acres of Wheat, Corn, Beans Planted

wW,C,B Tons of Wheat, Corn, Beans sold (at favorable price).

eB Tons of beans sold at lower price

yW,C Tons of Wheat, Corn purchased.

Note that Farmer Ted has recourse. After he observes the weatherevent, he can decide how much of each crop to sell or purchase!



Formulation

max−150xW − 230xC − 260xB − 238yW + 170wW

−210yC + 150wC + 36wB + 10eB

subject to

xW + xC + xB ≤ 500

2.5xW + yW − wW = 200

3xC + yC − wC = 240

20xB − wB − eB = 0

wB ≤ 6000

xW , xC , xB, yW , yC , eB, wW , wC , wB ≥ 0



Solution with (expected) yields

Wheat Corn Beans

Plant (acres) 120 80 300Production 300 240 6000

Sales 100 0 6000Purchase 0 0 0

Profit: $118,600



It’s the Weather, Stupid!

Farmer Ted knows well enough to know that his yields aren’t alwaysprecisely Y = (2.5, 3, 20). He decides to run two more scenarios

Good weather: 1.2Y

Bad weather: 0.8Y .


Farmer Ted Making the SuperModel

Creating a Stochastic Model

Here is a general procedure for making a (scenario-based) 2-stagestochastic optimization problem

For a “nominal” state of nature (scenario), formulate an appropriateLP model

Decide which decisions are made before uncertainty is revealed, andwhich are decided after

All second stage variables get “scenario” index

Constraints with scenario indices must hold for all scenarios

Second stage variables in the objective function should be weightedby the probability of the scenario occurring



What does this mean in our case?

First stage variables are the x (or planting variables)

Second stage variables are the y, w, e (purchase and sale variables)

We have one copy of the y, w, e for each scenario!

Attach a scenario subscript s = 1, 2, 3 to each of the purchase andsale variables.

1: Good, 2: Average, 3: Bad

wC2 : Tons of corn sold at favorable price in scenario 2

eB3 : Tons of beans sold at unfavorable price in scenario 3.



Expected Profit

The second stage cost for each submodel appears in the overallobjective function weighted by the probability that nature will choosethat scenario

−150xW − 230xC − 260xB

+1/3(−238yW1 + 170wW1 − 210yC1 + 150wC1 + 36wB1 + 10eB1)

+1/3(−238yW2 + 170wW2 − 210yC2 + 150wC2 + 36wB2 + 10eB2)

+1/3(−238yW3 + 170wW3 − 210yC3 + 150wC3 + 36wB3 + 10eB3)



Constraints

xW + xC + xB ≤ 500

3xW + yW1 − wW1 = 200

2.5xW + yW2 − wW2 = 200

2xW + yW3 − wW3 = 200



Constraints (cont.)

3.6xC + yC1 − wC1 = 240

3xC + yC2 − wC2 = 240

2.4xC + yC3 − wC3 = 240

24xB − wB1 − eB1 = 0

20xB − wB2 − eB2 = 0

16xB − wB3 − eB3 = 0

wB1, wB2, wB3 ≤ 6000

All vars ≥ 0



Optimal Solution

Wheat Corn Beans

s Plant (acres) 170 80 250

1 Production 510 288 60001 Sales 310 48 60001 Purchase 0 0 0




Farmer Ted Statistics:VSS

The Value of the Stochastic Solution (VSS)

Suppose we just replaced the “random” quantities (the yields) by their meanvalues and solved that problem.

Would we get the same expected value for the Farmer’s profit?

How can we check?

Solve the “mean-value” problem to get a first stage solution x.Fix the first stage solution at that value x, and solve all the scenariosto see Farmer Ted’s profit in each.Take the weighted (by probability) average of the optimal objectivevalue for each scenario

Alternatively (and probably faster), we can fix the x variables and solve thestochastic programming problem we created.



Computing FT’s VSS

Mean yields Y = (2.5, 3, 20)

(We already solved this problem).

xW = 120, xC = 80, xB = 300



Fixed Policy – Average Yield Scenario

maximize

−150xW − 230xC − 260xB − 238yW + 170wW − 210yC + 150yC + 36wB + 10eB

subject to

xW = 120

xC = 80

xB = 300

xW + xC + xB ≤ 500

2.5xW + yW − wW = 200

3xC + yC − wC = 240

20xB − wB − eB = 0

wB ≤ 6000

xW , xC , xB , yW , yC , eB , wW , wC , wB ≥ 0



Fixed Policy – Average Yield Scenario Solution

Wheat Corn Beans

Plant (acres) 120 80 300Production 300 240 6000

Sales 100 0 6000Purchase 0 0 0

Profit: $118,600



Fixed Policy – Bad Yield Scenariomaximize


subject to

xW = 120

xC = 80

xB = 300

xW + xC + xB ≤ 500

2xW + yW − wW = 200

2.4xC + yC − wC = 240

16xB − wB − eB = 0

wB ≤ 6000

Objective Value: $55,120



Fixed Policy – Good Yield Scenariomaximize


subject to

xW = 120

xC = 80

xB = 300

xW + xC + xB ≤ 500

3xW + yW − wW = 200

3.6xC + yC − wC = 240

24xB − wB − eB = 0

wB ≤ 6000

Objective Value: $148,000



What’s it Worth to Model Randomness?

If Farmer Ted implemented the policy based on using only “average”yields, he would plant xW = 120, xC = 80, xB = 300

He would expect in the long run to make an average profit of...

1/3(118600) + 1/3(55120) + 1/3(148000) = 107240

If Farmer Ted implemented the policy based on the solution to thestochastic programming problem, he would plantxW = 170, xC = 80, xB = 250.

From this he would expect to make 108390



VSS

The difference of the values 180390-107240 is the Value of theStochastic Solution : $1150.

It would pay off $1150 per growing season for Farmer Ted to use the“stochastic” solution rather than the “mean value” solution.

$1150 is precisely the “value” of implementing a planting policy basedon the “stochastic solution”, rather than the mean-value solution.



(General) Stochastic Programming

A Stochastic Programminx∈X

f(x)def= Eω[F (x, ξ(ω))]

2 Stage Stochastic LP w/Recourse

F (x, ω)def= cTx+Q(x, ω)

cTx: Pay me now

Q(x, ω): Pay me later

The Recourse Problem

Q(x, ω)def= min q(ω)T y

W (ω)y = h(ω)− T (ω)x

y ≥ 0


Extensive Form

Two Stage Stochastic Linear Program

Assume Ω = ω1, ω2, . . . ωS ⊆ Rr, P(ω = ωs) = ps,∀s = 1, 2, . . . , S

Tsdef= T (ωs), hs

def= h(ωs), qs

def= q(ωs),Ws = W (ωs)

min c>x+∑S

s=1 psQs(x)

s.t. Ax ≥ bx ∈ Rn1

+

where for s = 1, . . . , S

Qs(x)def= Q(x, ωs) = min q>s y

s.t. Wsy = hs − Tsxy ∈ Rn2

+


Extensive Form

Extensive Form

When we have a finite number of scenarios, or if we approximate theproblem with a finite number of scenarios2, we can write anequivalent extensive form linear program:

cTx + p1qT1 y1 + p2q

T2 y2 + · · · + psq

Ts ys

s.t.Ax = bT1x + W1y1 = h1T2x + W2y2 = h2

... +. . .

...TSx + WSys = hsx ∈ X y1 ∈ Y y2 ∈ Y ys ∈ Y

2Stay Tuned for Dave Morton’s LectureJeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 59 / 77

Extensive Form

The Upshot

This is just a larger linear program

It is a larger linear program that also has special structure

Jim explains how to exploit this structure tomorrow

cTx + p1qT1 y1 + p2q

T2 y2 + · · · + psq

Ts ys

s.t.Ax = bT1x + W1y1 = h1T2x + W2y2 = h2

... +. . .

...TSx + WSys = hsx ∈ X y1 ∈ Y y2 ∈ Y ys ∈ Y


Extensive Form

Building the Supermodel

Weird Science

A general technique for creating two-stage resource problems.

1 Write a nominal (one scenario) model

2 Decide which variables are first stage,and second stage

3 Give s scenario index to all secondstage variables and random parameters

4 “Give context” to all scenarios


Facility Location Example

Facility Location and Distribution

Facilities: I

Customers: J

Fixed cost fi, capacity ui for facility i ∈ IDemand dj : for j ∈ JPer unit Delivery cost: cij ∀i ∈ J, j ∈ J

min∑i∈I

fixi +∑i∈I

∑j∈J

cijyij

∑i∈I

yij ≥ dj ∀j ∈ J∑j∈J

yij − uixi ≤ 0 ∀i ∈ I

xi ∈ 0, 1, yij ≥ 0 ∀i ∈ I, ∀j ∈ J



AMPL for FL

AMPL Code

1 var xI binary;

2 var yI,J >= 0;

3

4 minimize Cost:

5 sumi in I f[i]*x[i] + sumi in I, j in J c[i,j]*y[i,j] ;

6

7 subject to MeetDemandj in J:

8 sumi in I y[i,j] >= d[j] ;

9

10 subject to FacCapacityi in I:

11 sumj in J y[i,j] - u[i]*x[i] <= 0 ;



Evolution of Information

1 Build facilities now

2 Demand becomes known. One of the scenarios S = d1, d2, . . . d|S|happens

3 Meet demand from open facilities

First stage variables: xi

Second stage variables: yijs



The SuperModel

min∑i∈I

fixi +∑s∈S

ps∑i∈I

∑j∈J

cijyijs

∑i∈I

yijs ≥ djs ∀j ∈ J ∀s ∈ S∑j∈J

yijs − uixi ≤ 0 ∀i ∈ I, ∀s ∈ S

xi ∈ 0, 1, yijs ≥ 0 ∀i ∈ I, ∀j ∈ J, ∀s ∈ S



Modeling Discussion

Do we always want to meet demand?

Regardless of the outcome ds?

What happens on the off chance that our product is so popular thatwe can’t possibly meet demand, even if we opened all of the facilities?

Does the world end?

Two Ideas

1 We could penalize not meeting demand of customers.

2 We only want to meet demand “most of the time”. (Chanceconstraint)



SP Definitions

A 2-stage stochastic optimization problem has complete recourse iffor every scenario, there always exists a feasible second solution:

Qs(x) < +∞ ∀x ∈ Rn,∀s = 1, . . . , S

A 2-stage stochastic optimization problem has relatively completerecourse if for every scenario, and for every feasible first stagesolution, there always exists a feasible second solution:

Qs(x) < +∞ ∀x ∈ X, ∀s = 1, . . . , S



Penalize Shortfall: A Recourse Formulation

min∑i∈I

fixi +∑s∈S

ps

∑i∈I

∑j∈J

cijyijs + λejs

∑i∈I

yijs + ejs ≥ djs ∀j ∈ J ∀s ∈ S∑j∈J

yijs − uixi ≤ 0 ∀i ∈ I, ∀s ∈ S

xi ∈ 0, 1, yij ≥ 0 ∀i ∈ I, ∀j ∈ J


AMPL Hints

Stop. AMPL Time.


AMPL Hints

AMPL Hints

1 All chapters of AMPL book are available for download: http:

//ampl.com/resources/the-ampl-book/chapter-downloads/

2 You can change the solver with the command option solver

cplex; (or replace cplex with baron, conopt, gurobi, knitro,

loqo, minos, snopt, xpress.)

3 Use var to declare variables; You may also put >= 0 on the same lineif the variables are constrained to be non-negative.

4 One your AMPL model is complete, you can type model

<filename>; at the ampl: prompt. This will tell you if you havesyntax errors.

5 If you have syntax errors. Fix them. Save the file, and type reset;

Then go to 4.

6 If no errors, type solve;


http://ampl.com/resources/the-ampl-book/chapter-downloads/


AMPL Hints

AMPL Entities

Data

Sets: lists of products, materials, etc.Parameters: numerical inputs such as costs, etc.

Model

Variables: The values to be decided upon.Objective Function.Constraints.

Data and Model typically stored in different files!


AMPL Hints

Template of Typical AMPL File

Define Sets

Define Parameters

Define Variables

Also can define variable bound constraints in this section

Define Objective

Define Constraints


AMPL Hints

Important AMPL Keywords/Syntax

model file.mod;

data file.mod;

reset;

quit;

set

param

var

maximize (minimize)

subject to


AMPL Hints

Important AMPL Notes

The # character starts a comment

All statements must end in a semi-colon;

Names must be unique!

A variable and a constraint cannot have the same name

AMPL is case sensitive. Keywords must be in lower case.

Even if the AMPL error message is cryptic, look at the location whereit shows an error – this will often help you deduce what is wrong.

Learning Data Input

Look at examples

Look at Chapter 9 of AMPL Book: http:

//ampl.com/resources/the-ampl-book/chapter-downloads/




AMPL Hints

Some AMPL Tips

option show stats 1; shows the problem size


AMPL Hints

Conclusions

Replacing uncertain parameters with point estimates may lead tosub-optimal planning: the flaw of averages

Two-stage recourse problems: Decision → Event → Decision

The Value of the Stochastic Solution

Creating the extensive form/supermodel


AMPL Hints

VSS: Value of the Stochastic Solution

Let zs be the optimal solution value to

zsdef= min

x∈XE[F (x, ξ(ω))]

Let xmv be an optimal solution to the “mean-value” problem:

xmv ∈ arg minx∈X

F (x,E[ξ(ω)])

Let zmv be the long run cost if you plan based on the policy obtainedfrom the ’average’ scenario:

zmvdef= EF (xmv, ξ(ω))

Value of Stochastic Solution

vssdef= zmv − zs

Simple HW: Prove vss ≥ 0


AMPL Hints

VSS: Value of the Stochastic Solution

Let zs be the optimal solution value to

zsdef= min

x∈XE[F (x, ξ(ω))]

Let xmv be an optimal solution to the “mean-value” problem:

xmv ∈ arg minx∈X

F (x,E[ξ(ω)])

Let zmv be the long run cost if you plan based on the policy obtainedfrom the ’average’ scenario:

zmvdef= EF (xmv, ξ(ω))

Value of Stochastic Solution

vssdef= zmv − zs

Simple HW: Prove vss ≥ 0Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 77 / 77

stochastic programming modeling · stochastic programming modeling ima new directions short course...

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